{"id":62829,"date":"2021-08-23T14:48:00","date_gmt":"2021-08-23T09:18:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=62829"},"modified":"2021-08-26T10:18:45","modified_gmt":"2021-08-26T04:48:45","slug":"rd-sharma-solutions-class-11-maths-chapter-1-sets","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/","title":{"rendered":"RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets (Updated For 2021-22)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-119190\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-1.jpg\" alt=\"RD Sharma Solutions Class 11 Maths Chapter 1 \" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-1.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-1-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><span style=\"font-weight: 400;\"><strong>RD Sharma Solutions Class 11 Maths Chapter 1:<\/strong> \u00a0<\/span><span style=\"font-weight: 400;\">If you want to ace your Class 11 Maths exams then <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions\/\">RD Sharma Solutions<\/a> Class 11 Maths Chapter 1 is the key to unlock your success in class 11 Maths. It offers a detailed and stepwise solution to every problem no matter how hard or easy the problem is. You can download the Free PDF of <a href=\"https:\/\/www.kopykitab.com\/blog\/cbse-class-11-maths-rd-sharma-solutions\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 11 Maths<\/a> Chapter 1.\u00a0<\/span><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_47_1 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"ez-toc-toggle-icon-1\"><label for=\"item-69ea92dd90e07\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;direction:ltr;\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input  type=\"checkbox\" id=\"item-69ea92dd90e07\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-visibility-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#download-rd-sharma-solutions-class-11-maths-chapter-1-%e2%80%93-sets-pdf\" title=\"Download RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets PDF\">Download RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets PDF<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercises-rd-sharma-solutions-class-11-maths-chapter-1\" title=\"Exercises: RD Sharma Solutions Class 11 Maths Chapter 1\u00a0\">Exercises: RD Sharma Solutions Class 11 Maths Chapter 1\u00a0<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-12-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.2 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.2 Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-13-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.3 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.3 Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-14-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.4 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.4 Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-15-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.5 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.5 Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-16-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.6 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.6 Solution<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#rd-sharma-class-11-chapter-1-exercise-17-ex-18-solution\" title=\"RD Sharma Class 11 Chapter 1 Exercise 1.7 &amp; Ex 1.8 Solution\">RD Sharma Class 11 Chapter 1 Exercise 1.7 &amp; Ex 1.8 Solution<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#access-rd-sharma-solutions-class-11-maths-chapter-1\" title=\"Access RD Sharma Solutions Class 11 Maths Chapter 1\u00a0\">Access RD Sharma Solutions Class 11 Maths Chapter 1\u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-11-page-no-12\" title=\"EXERCISE 1.1 PAGE NO: 1.2\">EXERCISE 1.1 PAGE NO: 1.2<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-what-is-the-difference-between-a-collection-and-a-set-give-reasons-to-support-your-answer\" title=\"1. What is the difference between a collection and a set? Give reasons to support your answer.\">1. What is the difference between a collection and a set? Give reasons to support your answer.<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-which-of-the-following-collections-are-sets-justify-your-answer-i-a-collection-of-all-natural-numbers-less-than-50-ii-the-collection-of-good-hockey-players-in-india-iii-the-collection-of-all-the-girls-in-your-class-iv-the-collection-of-all-talented-writers-of-india-v-the-collection-of-difficult-topics-in-mathematics-vi-the-collection-of-novels-written-by-munshi-prem-chand-vii-the-collection-of-all-questions-of-this-chapter-viii-the-collection-of-all-months-of-a-year-beginning-with-the-letter-j-ix-a-collection-of-most-dangerous-animals-of-the-world-x-the-collection-of-prime-integers\" title=\"2.\u00a0Which of the following collections are sets? Justify your answer: (i) A collection\u00a0of all natural numbers less than 50. (ii) The collection\u00a0of good hockey players in India. (iii)\u00a0The collection of all the girls in your class. (iv)\u00a0The collection\u00a0of all talented writers of India. (v) The collection\u00a0of difficult topics in Mathematics. (vi)\u00a0The collection of novels written by Munshi Prem Chand. (vii)\u00a0The collection\u00a0of all questions of this chapter. (viii)\u00a0The collection of all months of a year beginning with the letter J. (ix) A collection\u00a0of most dangerous animals of the world. (x) The collection\u00a0of prime integers.\">2.\u00a0Which of the following collections are sets? Justify your answer: (i) A collection\u00a0of all natural numbers less than 50. (ii) The collection\u00a0of good hockey players in India. (iii)\u00a0The collection of all the girls in your class. (iv)\u00a0The collection\u00a0of all talented writers of India. (v) The collection\u00a0of difficult topics in Mathematics. (vi)\u00a0The collection of novels written by Munshi Prem Chand. (vii)\u00a0The collection\u00a0of all questions of this chapter. (viii)\u00a0The collection of all months of a year beginning with the letter J. (ix) A collection\u00a0of most dangerous animals of the world. (x) The collection\u00a0of prime integers.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-if-a012345678910-then-insert-the-appropriate-symbol-or-in-each-of-the-following-blank-spaces-i-4-%e2%80%a6%e2%80%a6a\" title=\"3. If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol\u00a0or\u00a0in each of the following blank spaces: (i) 4 \u2026\u2026A\">3. If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol\u00a0or\u00a0in each of the following blank spaces: (i) 4 \u2026\u2026A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-4%e2%80%a6%e2%80%a6a-iii-12%e2%80%a6a\" title=\"(ii) -4\u2026\u2026A (iii) 12\u2026..A\">(ii) -4\u2026\u2026A (iii) 12\u2026..A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-9%e2%80%a6a-v-0%e2%80%a6%e2%80%a6a\" title=\"(iv) 9\u2026..A (v) 0\u2026\u2026A\">(iv) 9\u2026..A (v) 0\u2026\u2026A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#vi-2%e2%80%a6%e2%80%a6a\" title=\"(vi) -2\u2026\u2026A\">(vi) -2\u2026\u2026A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-12-page-no-16\" title=\"EXERCISE 1.2 PAGE NO: 1.6\">EXERCISE 1.2 PAGE NO: 1.6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-describe-the-following-sets-in-roster-form\" title=\"1. Describe the following sets in Roster form:\">1. Describe the following sets in Roster form:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#i-x-x-is-a-letter-before-e-in-the-english-alphabet\" title=\"(i)\u00a0{x : x is a letter before e in the English alphabet}\">(i)\u00a0{x : x is a letter before e in the English alphabet}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-x-%e2%88%88-n-x2-%3c-25\" title=\"(ii) {x\u00a0\u2208 N: x2\u00a0&lt; 25}\">(ii) {x\u00a0\u2208 N: x2\u00a0&lt; 25}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-x-%e2%88%88-n-x-is-a-prime-number-10-%3c-x-%3c-20\" title=\"(iii) {x \u2208\u00a0N: x is a prime number, 10 &lt; x &lt; 20}\">(iii) {x \u2208\u00a0N: x is a prime number, 10 &lt; x &lt; 20}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-x-%e2%88%88-n-x-2n-n-%e2%88%88-n\" title=\"(iv) {x \u2208\u00a0N: x = 2n, n \u2208\u00a0N}\">(iv) {x \u2208\u00a0N: x = 2n, n \u2208\u00a0N}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#v-x-%e2%88%88-r-x-%3e-x\" title=\"(v) {x \u2208\u00a0R: x &gt; x}\">(v) {x \u2208\u00a0R: x &gt; x}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#vi-x-x-is-a-prime-number-which-is-a-divisor-of-60\" title=\"(vi) {x : x is a prime number which is a divisor of 60}\">(vi) {x : x is a prime number which is a divisor of 60}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#vii-x-x-is-a-two-digit-number-such-that-the-sum-of-its-digits-is-8\" title=\"(vii) {x : x is a two digit number such that the sum of its digits is 8}\">(vii) {x : x is a two digit number such that the sum of its digits is 8}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#viii-the-set-of-all-letters-in-the-word-%e2%80%98trigonometry%e2%80%99\" title=\"(viii) The set of all letters in the word \u2018Trigonometry\u2019\">(viii) The set of all letters in the word \u2018Trigonometry\u2019<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ix-the-set-of-all-letters-in-the-word-%e2%80%98better%e2%80%99\" title=\"(ix) The set of all letters in the word \u2018Better.&#8217;\">(ix) The set of all letters in the word \u2018Better.&#8217;<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-describe-the-following-sets-in-set-builder-form-i-a-1-2-3-4-5-6-ii-b-1-12-13-14-15-%e2%80%a6-iii-c-0-3-6-9-12%e2%80%a6-iv-d-10-11-12-13-14-15-v-e-0-vi-1-4-9-16%e2%80%a6100-vii-2-4-6-8%e2%80%a6-viii-5-25-125-625\" title=\"2.\u00a0Describe the following sets in set-builder form: (i) A = {1, 2, 3, 4, 5, 6} (ii) B = {1, 1\/2, 1\/3, 1\/4, 1\/5, \u2026..} (iii) C = {0, 3, 6, 9, 12,\u2026.} (iv) D = {10, 11, 12, 13, 14, 15} (v) E = {0} (vi) {1, 4, 9, 16,\u2026,100} (vii) {2, 4, 6, 8,\u2026.} (viii) {5, 25, 125, 625}\">2.\u00a0Describe the following sets in set-builder form: (i) A = {1, 2, 3, 4, 5, 6} (ii) B = {1, 1\/2, 1\/3, 1\/4, 1\/5, \u2026..} (iii) C = {0, 3, 6, 9, 12,\u2026.} (iv) D = {10, 11, 12, 13, 14, 15} (v) E = {0} (vi) {1, 4, 9, 16,\u2026,100} (vii) {2, 4, 6, 8,\u2026.} (viii) {5, 25, 125, 625}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-list-all-the-elements-of-the-following-sets\" title=\"3.\u00a0List all the elements of the following sets:\">3.\u00a0List all the elements of the following sets:<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#i-ax-x2%e2%89%a4-10-x-%e2%88%88-z\" title=\"(i) A={x : x2\u2264\u00a010, x\u00a0\u2208\u00a0Z}\">(i) A={x : x2\u2264\u00a010, x\u00a0\u2208\u00a0Z}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-b-x-x-12n-1-1-%e2%89%a4-n-%e2%89%a4-5\" title=\"(ii) B = {x : x = 1\/(2n-1), 1 \u2264 n\u00a0\u2264 5}\">(ii) B = {x : x = 1\/(2n-1), 1 \u2264 n\u00a0\u2264 5}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-c-x-x-is-an-integer-12-%3c-x-%3c-92\" title=\"(iii) C = {x : x is an integer, -1\/2 &lt; x &lt; 9\/2}\">(iii) C = {x : x is an integer, -1\/2 &lt; x &lt; 9\/2}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-dx-x-is-a-vowel-in-the-word-%e2%80%9cequation%e2%80%9d\" title=\"(iv) D={x : x is a vowel in the word \u201cEQUATION\u201d}\">(iv) D={x : x is a vowel in the word \u201cEQUATION\u201d}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#v-e-x-x-is-a-month-of-a-year-not-having-31-days\" title=\"(v) E = {x : x is a month of a year not having 31 days}\">(v) E = {x : x is a month of a year not having 31 days}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-35\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#vi-fx-x-is-a-letter-of-the-word-%e2%80%9cmississippi%e2%80%9d\" title=\"(vi) F={x : x is a letter of the word \u201cMISSISSIPPI\u201d}\">(vi) F={x : x is a letter of the word \u201cMISSISSIPPI\u201d}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-36\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-match-each-of-the-sets-on-the-left-in-the-roster-form-with-the-same-set-on-the-right-described-in-the-set-builder-form-i-aple-i-x-x55-x-%e2%88%88-z-ii-5-5-ii-x-x-is-a-prime-natural-number-and-a-divisor-of-10-iii-0-iii-x-x-is-a-letter-of-the-word-%e2%80%9crajasthan%e2%80%9d-iv-1-2-5-10-iv-x-x-is-a-natural-and-divisor-of-10-v-a-h-j-r-s-t-n-v-x-x2-%e2%80%93-25-0-vi-25-vi-x-x-is-a-letter-of-word-%e2%80%9capple%e2%80%9d\" title=\"4.\u00a0Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form: (i) {A,P,L,E} (i) {x : x+5=5, x\u00a0\u2208\u00a0z} (ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10} (iii) {0} (iii) {x : x is a letter of the word \u201cRAJASTHAN\u201d} (iv) {1, 2, 5, 10} (iv) {x : x is a natural and divisor of 10} (v) {A, H, J, R, S, T, N} (v) {x : x2\u00a0\u2013 25 =0} (vi) {2,5} (vi) {x : x is a letter of word \u201cAPPLE\u201d}\">4.\u00a0Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form: (i) {A,P,L,E} (i) {x : x+5=5, x\u00a0\u2208\u00a0z} (ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10} (iii) {0} (iii) {x : x is a letter of the word \u201cRAJASTHAN\u201d} (iv) {1, 2, 5, 10} (iv) {x : x is a natural and divisor of 10} (v) {A, H, J, R, S, T, N} (v) {x : x2\u00a0\u2013 25 =0} (vi) {2,5} (vi) {x : x is a letter of word \u201cAPPLE\u201d}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-37\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#6-write-the-set-of-all-positive-integers-whose-cube-is-odd\" title=\"6. Write the set of all positive integers whose cube is odd.\">6. Write the set of all positive integers whose cube is odd.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-38\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#7-write-the-set-12-25-310-417-526-637-750-in-the-set-builder-form\" title=\"7. Write the set {1\/2, 2\/5, 3\/10, 4\/17, 5\/26, 6\/37, 7\/50}\u00a0in the set-builder form.\">7. Write the set {1\/2, 2\/5, 3\/10, 4\/17, 5\/26, 6\/37, 7\/50}\u00a0in the set-builder form.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-39\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-which-of-the-following-are-examples-of-empty-set-i-set-of-all-even-natural-numbers-divisible-by-5-ii-set-of-all-even-prime-numbers-iii-x-x2%e2%80%9320-and-x-is-rational-iv-x-x-is-a-natural-number-x-%3c-8-and-simultaneously-x-%3e-12-v-x-x-is-a-point-common-to-any-two-parallel-lines\" title=\"1. Which of the following are examples of empty set? (i) Set of all even natural numbers divisible by 5. (ii) Set of all even prime numbers. (iii) {x: x2\u20132=0 and x is rational}. (iv) {x: x is a natural number, x &lt; 8 and simultaneously x &gt; 12}. (v) {x: x is a point common to any two parallel lines}.\">1. Which of the following are examples of empty set? (i) Set of all even natural numbers divisible by 5. (ii) Set of all even prime numbers. (iii) {x: x2\u20132=0 and x is rational}. (iv) {x: x is a natural number, x &lt; 8 and simultaneously x &gt; 12}. (v) {x: x is a point common to any two parallel lines}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-40\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-which-of-the-following-sets-are-finite-and-which-are-infinite-i-set-of-concentric-circles-in-a-plane-ii-set-of-letters-of-the-english-alphabets-iii-x-%e2%88%88-n-x-%3e-5-iv-x-%e2%88%88-n-x-%3c-200-v-x-%e2%88%88-z-x-%3c-5-vi-x-%e2%88%88-r-0-%3c-x-%3c-1\" title=\"2. Which of the following sets are finite and which are infinite? (i) Set of concentric circles in a plane. (ii) Set of letters of the English Alphabets. (iii) {x\u00a0\u2208\u00a0N: x &gt; 5} (iv) {x\u00a0\u2208\u00a0N: x &lt; 200} (v) {x\u00a0\u2208\u00a0Z: x &lt; 5} (vi) {x\u00a0\u2208\u00a0R: 0 &lt; x &lt; 1}.\">2. Which of the following sets are finite and which are infinite? (i) Set of concentric circles in a plane. (ii) Set of letters of the English Alphabets. (iii) {x\u00a0\u2208\u00a0N: x &gt; 5} (iv) {x\u00a0\u2208\u00a0N: x &lt; 200} (v) {x\u00a0\u2208\u00a0Z: x &lt; 5} (vi) {x\u00a0\u2208\u00a0R: 0 &lt; x &lt; 1}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-41\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-which-of-the-following-sets-are-equal-i-a-1-2-3-ii-b-x-%e2%88%88-r-x2%e2%80%932x10-iii-c-1-2-2-3-iv-d-x-%e2%88%88-r-x3-%e2%80%93-6x211x-%e2%80%93-6-0\" title=\"3. Which of the following sets are equal? (i) A = {1, 2, 3} (ii) B = {x\u00a0\u2208\u00a0R:x2\u20132x+1=0} (iii) C = (1, 2, 2, 3} (iv) D = {x\u00a0\u2208 R : x3\u00a0\u2013 6x2+11x \u2013 6 = 0}.\">3. Which of the following sets are equal? (i) A = {1, 2, 3} (ii) B = {x\u00a0\u2208\u00a0R:x2\u20132x+1=0} (iii) C = (1, 2, 2, 3} (iv) D = {x\u00a0\u2208 R : x3\u00a0\u2013 6x2+11x \u2013 6 = 0}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-42\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-are-the-following-sets-equal-ax-x-is-a-letter-in-the-word-reap\" title=\"4. Are the following sets equal? A={x: x is a letter in the word reap},\">4. Are the following sets equal? A={x: x is a letter in the word reap},<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-43\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#bx-x-is-a-letter-in-the-word-paper-cx-x-is-a-letter-in-the-word-rope\" title=\"B={x: x is a letter in the word paper}, C={x: x is a letter in the word rope}.\">B={x: x is a letter in the word paper}, C={x: x is a letter in the word rope}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-44\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#5-from-the-sets-given-below-pair-the-equivalent-sets-a-1-2-3-b-t-p-q-r-s-c-%ce%b1-%ce%b2-%ce%b3-d-a-e-i-o-u\" title=\"5. From the sets given below, pair the equivalent sets: A= {1, 2, 3}, B = {t, p, q, r, s}, C = {\u03b1,\u00a0\u03b2,\u00a0\u03b3}, D = {a, e, i, o, u}.\">5. From the sets given below, pair the equivalent sets: A= {1, 2, 3}, B = {t, p, q, r, s}, C = {\u03b1,\u00a0\u03b2,\u00a0\u03b3}, D = {a, e, i, o, u}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#6-are-the-following-pairs-of-sets-equal-give-reasons-i-a-2-3-b-x-x-is-a-solution-of-x2-5x-6-0\" title=\"6. Are the following pairs of sets equal? Give reasons. (i) A = {2, 3}, B = {x: x is a solution of x2\u00a0+ 5x + 6= 0}\">6. Are the following pairs of sets equal? Give reasons. (i) A = {2, 3}, B = {x: x is a solution of x2\u00a0+ 5x + 6= 0}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-46\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-ax-x-is-a-letter-of-the-word-%e2%80%9cwolf%e2%80%9d-bx-x-is-letter-of-word-%e2%80%9cfollow%e2%80%9d\" title=\"(ii) A={x: x is a letter of the word \u201cWOLF\u201d} B={x: x is letter of word \u201cFOLLOW\u201d}\">(ii) A={x: x is a letter of the word \u201cWOLF\u201d} B={x: x is letter of word \u201cFOLLOW\u201d}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#7-from-the-sets-given-below-select-equal-sets-and-equivalent-sets-a-0-a-b-1-2-3-4-c-4-8-12-d-3-1-2-4-e-1-0-f-8-4-12-g-1-5-7-11-h-a-b\" title=\"7. From the sets given below, select equal sets and equivalent sets. A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12}, D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12}, G = {1, 5, 7, 11}, H = {a, b}\">7. From the sets given below, select equal sets and equivalent sets. A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12}, D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12}, G = {1, 5, 7, 11}, H = {a, b}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-48\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#8-which-of-the-following-sets-are-equal-a-x-x-%e2%88%88-n-x-%3c-3-b-1-2-c-3-1-d-x-x-%e2%88%88-n-x-is-odd-x-%3c-5-e-1-2-1-1\" title=\"8. Which of the following sets are equal? A = {x: x\u00a0\u2208\u00a0N, x &lt; 3} B = {1, 2}, C= {3, 1} D = {x: x\u00a0\u2208\u00a0N, x is odd, x &lt; 5} E = {1, 2, 1, 1}\">8. Which of the following sets are equal? A = {x: x\u00a0\u2208\u00a0N, x &lt; 3} B = {1, 2}, C= {3, 1} D = {x: x\u00a0\u2208\u00a0N, x is odd, x &lt; 5} E = {1, 2, 1, 1}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-49\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#f-1-1-3\" title=\"F = {1, 1, 3}\">F = {1, 1, 3}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-50\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#9-show-that-the-set-of-letters-needed-to-spell-%e2%80%9ccataract%e2%80%9d-and-the-set-of-letters-needed-to-spell-%e2%80%9ctract%e2%80%9d-are-equal\" title=\"9. Show that the set of letters needed to spell \u201cCATARACT\u201d and the set of letters needed to spell \u201cTRACT\u201d are equal.\">9. Show that the set of letters needed to spell \u201cCATARACT\u201d and the set of letters needed to spell \u201cTRACT\u201d are equal.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-51\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-which-of-the-following-statements-are-true-give-a-reason-to-support-your-answer-i-for-any-two-sets-a-and-b-either-a-b-or-b-a-ii-every-subset-of-an-infinite-set-is-infinite-iii-every-subset-of-a-finite-set-is-finite-iv-every-set-has-a-proper-subset-v-a-b-a-b-a-b%e2%80%a6-is-an-infinite-set-vi-a-b-c-and-1-2-3-are-equivalent-sets-vii-a-set-can-have-infinitely-many-subsets\" title=\"1. Which of the following statements are true? Give a reason to support your answer. (i) For any two sets A and B either\u00a0A\u00a0B or B\u00a0A. (ii) Every subset of an infinite set is infinite. (iii) Every subset of a finite set is finite. (iv) Every set has a proper subset. (v) {a, b, a, b, a, b,\u2026.} is an infinite set. (vi) {a, b, c} and {1, 2, 3} are equivalent sets. (vii) A set can have infinitely many subsets.\">1. Which of the following statements are true? Give a reason to support your answer. (i) For any two sets A and B either\u00a0A\u00a0B or B\u00a0A. (ii) Every subset of an infinite set is infinite. (iii) Every subset of a finite set is finite. (iv) Every set has a proper subset. (v) {a, b, a, b, a, b,\u2026.} is an infinite set. (vi) {a, b, c} and {1, 2, 3} are equivalent sets. (vii) A set can have infinitely many subsets.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-52\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-state-whether-the-following-statements-are-true-or-false-i-1-%e2%88%88-123-ii-a-%e2%8a%82-bca-iii-a-%e2%88%88-abc-iv-a-b-a-a-b-b-a-v-the-set-x-x-8-8-is-the-null-set\" title=\"2.\u00a0State whether the following statements are true or false: (i) 1 \u2208\u00a0{ 1,2,3} (ii) a\u00a0\u2282\u00a0{b,c,a} (iii) {a} \u2208\u00a0{a,b,c} (iv) {a, b} = {a, a, b, b, a} (v) The set {x: x + 8 = 8} is the null set.\">2.\u00a0State whether the following statements are true or false: (i) 1 \u2208\u00a0{ 1,2,3} (ii) a\u00a0\u2282\u00a0{b,c,a} (iii) {a} \u2208\u00a0{a,b,c} (iv) {a, b} = {a, a, b, b, a} (v) The set {x: x + 8 = 8} is the null set.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-53\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-decide-among-the-following-sets-which-are-subsets-of-which-a-x-x-satisfies-x2-%e2%80%93-8x-120-b-246-c-2468%e2%80%a6-d-6\" title=\"3. Decide among the following sets, which are subsets of which: A = {x: x satisfies x2\u00a0\u2013 8x + 12=0}, B = {2,4,6}, C = {2,4,6,8,\u2026.}, D = {6}\">3. Decide among the following sets, which are subsets of which: A = {x: x satisfies x2\u00a0\u2013 8x + 12=0}, B = {2,4,6}, C = {2,4,6,8,\u2026.}, D = {6}<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-54\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-write-which-of-the-following-statements-are-true-justify-your-answer-i-the-set-of-all-integers-is-contained-in-the-set-of-all-rational-numbers-ii-the-set-of-all-crows-is-contained-in-the-set-of-all-birds-iii-the-set-of-all-rectangles-is-contained-in-the-set-of-all-squares-iv-the-set-of-all-rectangle-is-contained-in-the-set-of-all-squares-v-the-sets-p-a-and-b-a-are-equal-vi-the-sets-ax-x-is-a-letter-of-word-%e2%80%9clittle%e2%80%9d-and-b-x-x-is-a-letter-of-the-word-%e2%80%9ctitle%e2%80%9d-are-equal\" title=\"4.\u00a0Write which of the following statements are true? Justify your answer. (i) The set of all integers is contained in the set of all rational numbers. (ii) The set of all crows is contained in the set of all birds. (iii) The set of all rectangles is contained in the set of all squares. (iv) The set of all rectangle is contained in the set of all squares. (v) The sets P = {a} and B = {{a}} are equal. (vi) The sets A={x: x is a letter of word \u201cLITTLE\u201d} AND, b = {x: x is a letter of the word \u201cTITLE\u201d} are equal.\">4.\u00a0Write which of the following statements are true? Justify your answer. (i) The set of all integers is contained in the set of all rational numbers. (ii) The set of all crows is contained in the set of all birds. (iii) The set of all rectangles is contained in the set of all squares. (iv) The set of all rectangle is contained in the set of all squares. (v) The sets P = {a} and B = {{a}} are equal. (vi) The sets A={x: x is a letter of word \u201cLITTLE\u201d} AND, b = {x: x is a letter of the word \u201cTITLE\u201d} are equal.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-55\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#5-which-of-the-following-statements-are-correct-write-a-correct-form-of-each-of-the-incorrect-statements-i-a-%e2%8a%82-a-b-c-ii-a-a-b-c-iii-a-a-b-iv-a-%e2%8a%82-a-b-v-b-c-%e2%8a%82-ab-c-vi-a-b-%e2%8a%82-ab-c-vii-%cf%95-a-b-viii-%cf%95-%e2%8a%82-a-b-c-ix-x-x-3-3-%cf%95\" title=\"5.\u00a0Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) a\u00a0\u2282\u00a0{a, b, c} (ii) {a}\u00a0{a, b, c} (iii) a\u00a0{{a}, b} (iv) {a}\u00a0\u2282\u00a0{{a}, b} (v) {b, c}\u00a0\u2282\u00a0{a,{b, c}} (vi) {a, b}\u00a0\u2282\u00a0{a,{b, c}} (vii)\u00a0\u03d5\u00a0{a, b} (viii)\u00a0\u03d5\u00a0\u2282\u00a0{a, b, c} (ix) {x: x + 3 = 3}=\u00a0\u03d5\">5.\u00a0Which of the following statements are correct? Write a correct form of each of the incorrect statements. (i) a\u00a0\u2282\u00a0{a, b, c} (ii) {a}\u00a0{a, b, c} (iii) a\u00a0{{a}, b} (iv) {a}\u00a0\u2282\u00a0{{a}, b} (v) {b, c}\u00a0\u2282\u00a0{a,{b, c}} (vi) {a, b}\u00a0\u2282\u00a0{a,{b, c}} (vii)\u00a0\u03d5\u00a0{a, b} (viii)\u00a0\u03d5\u00a0\u2282\u00a0{a, b, c} (ix) {x: x + 3 = 3}=\u00a0\u03d5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-56\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#6-let-a-a-bc-d-e-which-of-the-following-statements-are-false-and-why-i-c-d-%e2%8a%82-a-ii-c-d-a-iii-c-d-%e2%8a%82-a-iv-a-a-v-a-%e2%8a%82-a-vi-a-b-e-%e2%8a%82-a-vii-a-b-e-a-viii-a-b-c-%e2%8a%82-a-ix-%cf%95-a-x-%cf%95-%e2%8a%82-a\" title=\"6.\u00a0Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) {c, d}\u00a0\u2282\u00a0A (ii) {c, d}\u00a0A (iii) {{c, d}}\u00a0\u2282\u00a0A (iv) a\u00a0A (v) a\u00a0\u2282\u00a0A. (vi) {a, b, e}\u00a0\u2282\u00a0A (vii) {a, b, e}\u00a0A (viii) {a, b, c}\u00a0\u2282\u00a0A (ix)\u00a0\u03d5\u00a0A (x) {\u03d5}\u00a0\u2282\u00a0A\">6.\u00a0Let A = {a, b,{c, d}, e}. Which of the following statements are false and why? (i) {c, d}\u00a0\u2282\u00a0A (ii) {c, d}\u00a0A (iii) {{c, d}}\u00a0\u2282\u00a0A (iv) a\u00a0A (v) a\u00a0\u2282\u00a0A. (vi) {a, b, e}\u00a0\u2282\u00a0A (vii) {a, b, e}\u00a0A (viii) {a, b, c}\u00a0\u2282\u00a0A (ix)\u00a0\u03d5\u00a0A (x) {\u03d5}\u00a0\u2282\u00a0A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-57\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#7-let-a-1-2-3-4-5-6-7-8-determine-which-of-the-following-is-true-or-false-i-1-%e2%88%88-a-ii-1-2-3-%e2%8a%82-a-iii-6-7-8-%e2%88%88-a-iv-4-5-%e2%8a%82-a-v-%cf%95-%e2%88%88-a-vi-%cf%95-%e2%8a%82-a\" title=\"7.\u00a0Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) 1 \u2208\u00a0A (ii) {1, 2, 3}\u00a0\u2282\u00a0A (iii) {6, 7, 8} \u2208\u00a0A (iv) {4,\u00a05}\u00a0\u2282\u00a0A (v) \u03d5\u00a0\u2208\u00a0A (vi) \u03d5 \u2282\u00a0A\">7.\u00a0Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false: (i) 1 \u2208\u00a0A (ii) {1, 2, 3}\u00a0\u2282\u00a0A (iii) {6, 7, 8} \u2208\u00a0A (iv) {4,\u00a05}\u00a0\u2282\u00a0A (v) \u03d5\u00a0\u2208\u00a0A (vi) \u03d5 \u2282\u00a0A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-58\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#8-let-a-%cf%95-%cf%95-1-1-%cf%95-2-which-of-the-following-are-true-i-%cf%95-%e2%88%88-a-ii-%cf%95-%e2%88%88-a-iii-1-%e2%88%88-a-iv-2-%cf%95-%e2%8a%82-a-v-2-%e2%8a%82-a-vi-2-1-%e2%8a%84a-vii-2-1-%e2%8a%84-a-viii-%cf%95-%cf%95-1-%cf%95-%e2%8a%82-a-ix-%cf%95-%e2%8a%82-a\" title=\"8.\u00a0Let A = {\u03d5, {\u03d5}, 1, {1, \u03d5}, 2}. Which of the following are true? (i)\u00a0\u03d5\u00a0\u2208\u00a0A (ii) {\u03d5}\u00a0\u2208\u00a0A (iii) {1}\u00a0\u2208\u00a0A (iv) {2,\u00a0\u03d5}\u00a0\u2282\u00a0A (v) 2\u00a0\u2282\u00a0A (vi) {2, {1}}\u00a0\u2284A (vii) {{2}, {1}}\u00a0\u2284 A (viii) {\u03d5, {\u03d5}, {1, \u03d5}}\u00a0\u2282\u00a0A (ix) {{\u03d5}}\u00a0\u2282\u00a0A\">8.\u00a0Let A = {\u03d5, {\u03d5}, 1, {1, \u03d5}, 2}. Which of the following are true? (i)\u00a0\u03d5\u00a0\u2208\u00a0A (ii) {\u03d5}\u00a0\u2208\u00a0A (iii) {1}\u00a0\u2208\u00a0A (iv) {2,\u00a0\u03d5}\u00a0\u2282\u00a0A (v) 2\u00a0\u2282\u00a0A (vi) {2, {1}}\u00a0\u2284A (vii) {{2}, {1}}\u00a0\u2284 A (viii) {\u03d5, {\u03d5}, {1, \u03d5}}\u00a0\u2282\u00a0A (ix) {{\u03d5}}\u00a0\u2282\u00a0A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-59\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-15-page-no-121\" title=\"EXERCISE 1.5 PAGE NO: 1.21\">EXERCISE 1.5 PAGE NO: 1.21<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-60\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-if-a-and-b-are-two-sets-such-that-a-%e2%8a%82-b-then-find-i-a-%e2%8b%82-b\" title=\"1. If A and B are two sets such that A\u00a0\u2282\u00a0B, then Find: (i) A\u00a0\u22c2\u00a0B\">1. If A and B are two sets such that A\u00a0\u2282\u00a0B, then Find: (i) A\u00a0\u22c2\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-61\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%8b%83-b\" title=\"(ii) A \u22c3 B\">(ii) A \u22c3 B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-62\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-if-a-1-2-3-4-5-b-4-5-6-7-8-c-7-8-9-10-11-and-d-10-11-12-13-14-find-i-a-%e2%88%aa-b-ii-a-%e2%88%aa-c-iii-b-%e2%88%aa-c-iv-b-%e2%88%aa-d-v-a-%e2%88%aa-b-%e2%88%aa-c-vi-a-%e2%88%aa-b-%e2%88%aa-d-vii-b-%e2%88%aa-c-%e2%88%aa-d-viii-a-%e2%88%a9-b-%e2%88%aa-c-ix-a-%e2%88%a9-b-%e2%88%a9-b-%e2%88%a9-c-x-a-%e2%88%aa-d-%e2%88%a9-b-%e2%88%aa-c\" title=\"2.\u00a0If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) A\u00a0\u222a\u00a0B (ii) A\u00a0\u222a\u00a0C (iii) B\u00a0\u222a\u00a0C (iv) B\u00a0\u222a\u00a0D (v) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0C (vi) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0D (vii) B\u00a0\u222a\u00a0C\u00a0\u222a\u00a0D (viii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) (ix) (A\u00a0\u2229\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u2229\u00a0C) (x) (A\u00a0\u222a\u00a0D)\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C).\">2.\u00a0If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find: (i) A\u00a0\u222a\u00a0B (ii) A\u00a0\u222a\u00a0C (iii) B\u00a0\u222a\u00a0C (iv) B\u00a0\u222a\u00a0D (v) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0C (vi) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0D (vii) B\u00a0\u222a\u00a0C\u00a0\u222a\u00a0D (viii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) (ix) (A\u00a0\u2229\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u2229\u00a0C) (x) (A\u00a0\u222a\u00a0D)\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-63\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-let-a-x-x-%e2%88%88-n-b-x-x-2n-n-%e2%88%88-n-c-x-x-2n-%e2%80%93-1-n-%e2%88%88-n-and-d-x-x-is-a-prime-natural-number-find-i-a-%e2%88%a9-b-ii-a-%e2%88%a9-c-iii-a-%e2%88%a9-d-iv-b-%e2%88%a9-c-v-b-%e2%88%a9-d-vi-c-%e2%88%a9-d\" title=\"3.\u00a0Let A = {x: x\u00a0\u2208\u00a0N}, B = {x: x = 2n, n\u00a0\u2208\u00a0N), C = {x: x = 2n \u2013 1, n\u00a0\u2208\u00a0N} and, D = {x: x is a prime natural number} Find: (i) A\u00a0\u2229\u00a0B (ii) A\u00a0\u2229\u00a0C (iii) A\u00a0\u2229\u00a0D (iv) B\u00a0\u2229\u00a0C (v) B\u00a0\u2229\u00a0D (vi) C\u00a0\u2229\u00a0D\">3.\u00a0Let A = {x: x\u00a0\u2208\u00a0N}, B = {x: x = 2n, n\u00a0\u2208\u00a0N), C = {x: x = 2n \u2013 1, n\u00a0\u2208\u00a0N} and, D = {x: x is a prime natural number} Find: (i) A\u00a0\u2229\u00a0B (ii) A\u00a0\u2229\u00a0C (iii) A\u00a0\u2229\u00a0D (iv) B\u00a0\u2229\u00a0C (v) B\u00a0\u2229\u00a0D (vi) C\u00a0\u2229\u00a0D<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-64\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-16-page-no-127\" title=\"EXERCISE 1.6 PAGE NO: 1.27\">EXERCISE 1.6 PAGE NO: 1.27<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-65\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-find-the-smallest-set-a-such-that-a-%e2%88%aa-1-2-1-2-3-5-9\" title=\"1. Find the smallest set A such that A\u00a0\u222a\u00a0{1, 2} = {1, 2, 3, 5, 9}.\">1. Find the smallest set A such that A\u00a0\u222a\u00a0{1, 2} = {1, 2, 3, 5, 9}.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-66\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-let-a-1-2-4-5-b-2-3-5-6-c-4-5-6-7-verify-the-following-identities-i-a-%e2%88%aa-b-%e2%88%a9-c-a-%e2%88%aa-b-%e2%88%a9-a-%e2%88%aa-c\" title=\"2.\u00a0Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A\u00a0\u222a\u00a0(B\u00a0\u2229\u00a0C) = (A\u00a0\u222a\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0C)\">2.\u00a0Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities: (i) A\u00a0\u222a\u00a0(B\u00a0\u2229\u00a0C) = (A\u00a0\u222a\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-67\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%88%a9-b-%e2%88%aa-c-a-%e2%88%a9-b-%e2%88%aa-a-%e2%88%a9-c\" title=\"(ii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0C)\">(ii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-68\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-a-%e2%88%a9-b-%e2%80%93-c-a-%e2%88%a9-b-%e2%80%93-a-%e2%88%a9-c\" title=\"(iii) A\u00a0\u2229\u00a0(B \u2013 C) = (A\u00a0\u2229\u00a0B) \u2013 (A\u00a0\u2229\u00a0C)\">(iii) A\u00a0\u2229\u00a0(B \u2013 C) = (A\u00a0\u2229\u00a0B) \u2013 (A\u00a0\u2229\u00a0C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-69\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-a-%e2%80%93-b-%e2%88%aa-c-a-%e2%80%93-b-%e2%88%a9-a-%e2%80%93-c\" title=\"(iv) A \u2013 (B\u00a0\u222a\u00a0C) = (A \u2013 B)\u00a0\u2229\u00a0(A \u2013 C)\">(iv) A \u2013 (B\u00a0\u222a\u00a0C) = (A \u2013 B)\u00a0\u2229\u00a0(A \u2013 C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-70\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#v-a-%e2%80%93-b-%e2%88%a9-c-a-%e2%80%93-b-%e2%88%aa-a-%e2%80%93-c\" title=\"(v) A \u2013 (B\u00a0\u2229\u00a0C) = (A \u2013 B)\u00a0\u222a\u00a0(A \u2013 C)\">(v) A \u2013 (B\u00a0\u2229\u00a0C) = (A \u2013 B)\u00a0\u222a\u00a0(A \u2013 C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-71\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#vi-a-%e2%88%a9-b-%e2%96%b3-c-a-%e2%88%a9-b-%e2%96%b3-a-%e2%88%a9-c\" title=\"(vi) A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u25b3\u00a0(A\u00a0\u2229\u00a0C)\">(vi) A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u25b3\u00a0(A\u00a0\u2229\u00a0C)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-72\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-if-u-2-3-5-7-9-is-the-universal-set-and-a-3-7-b-2-5-7-9-then-prove-that-i-a-%e2%88%aa-b%e2%80%99-a%e2%80%99-%e2%88%a9-b%e2%80%99\" title=\"3.\u00a0If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: (i) (A\u00a0\u222a\u00a0B)\u2019 = A\u2019\u00a0\u2229\u00a0B\u2019\">3.\u00a0If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that: (i) (A\u00a0\u222a\u00a0B)\u2019 = A\u2019\u00a0\u2229\u00a0B\u2019<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-73\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%88%a9-b%e2%80%99-a%e2%80%99-%e2%88%aa-b%e2%80%99\" title=\"(ii) (A\u00a0\u2229\u00a0B)\u2019 = A\u2019\u00a0\u222a\u00a0B\u2019\">(ii) (A\u00a0\u2229\u00a0B)\u2019 = A\u2019\u00a0\u222a\u00a0B\u2019<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-74\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-for-any-two-sets-a-and-b-prove-that-i-b-%e2%8a%82-a-%e2%88%aa-b\" title=\"4.\u00a0For any two sets A and B, prove that (i) B\u00a0\u2282\u00a0A\u00a0\u222a\u00a0B\">4.\u00a0For any two sets A and B, prove that (i) B\u00a0\u2282\u00a0A\u00a0\u222a\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-75\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%88%a9-b-%e2%8a%82-a\" title=\"(ii) A\u00a0\u2229\u00a0B\u00a0\u2282\u00a0A\">(ii) A\u00a0\u2229\u00a0B\u00a0\u2282\u00a0A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-76\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-a-%e2%8a%82-b-%e2%87%92-a-%e2%88%a9-b-a\" title=\"(iii) A\u00a0\u2282\u00a0B \u21d2\u00a0A\u00a0\u2229\u00a0B = A\">(iii) A\u00a0\u2282\u00a0B \u21d2\u00a0A\u00a0\u2229\u00a0B = A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-77\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#5-for-any-two-sets-a-and-b-show-that-the-following-statements-are-equivalent-i-a-%e2%8a%82-b\" title=\"5.\u00a0For any two sets A and B, show that the following statements are equivalent: (i) A\u00a0\u2282\u00a0B\">5.\u00a0For any two sets A and B, show that the following statements are equivalent: (i) A\u00a0\u2282\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-78\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%80%93-b-%cf%95\" title=\"(ii) A \u2013 B\u00a0=\u00a0\u03d5\">(ii) A \u2013 B\u00a0=\u00a0\u03d5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-79\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-a-%e2%88%aa-b-b\" title=\"(iii) A\u00a0\u222a\u00a0B = B\">(iii) A\u00a0\u222a\u00a0B = B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-80\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-a-%e2%88%a9-b-a\" title=\"(iv) A\u00a0\u2229\u00a0B = A\">(iv) A\u00a0\u2229\u00a0B = A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-81\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#6-for-three-sets-a-b-and-c-show-that-i-a-%e2%88%a9-b-a-%e2%88%a9-c-need-not-imply-b-c\" title=\"6. For three sets A, B, and C, show that (i) A\u00a0\u2229\u00a0B = A\u00a0\u2229\u00a0C need not imply B = C.\">6. For three sets A, B, and C, show that (i) A\u00a0\u2229\u00a0B = A\u00a0\u2229\u00a0C need not imply B = C.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-82\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%8a%82-b-%e2%87%92-c-%e2%80%93-b-%e2%8a%82-c-%e2%80%93-a\" title=\"(ii) A\u00a0\u2282\u00a0B \u21d2\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A\">(ii) A\u00a0\u2282\u00a0B \u21d2\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-83\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#7-for-any-two-sets-prove-that-i-a-%e2%88%aa-a-%e2%88%a9-b-a\" title=\"7.\u00a0For any two sets, prove that: (i) A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A\">7.\u00a0For any two sets, prove that: (i) A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-84\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%88%a9-a-%e2%88%aa-b-a\" title=\"(ii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B) = A\">(ii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B) = A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-85\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-17-page-no-134\" title=\"EXERCISE 1.7 PAGE NO: 1.34\">EXERCISE 1.7 PAGE NO: 1.34<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-86\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-for-any-two-sets-a-and-b-prove-that-a%e2%80%98-%e2%80%93-b%e2%80%98-b-%e2%80%93-a\" title=\"1. For any two sets A and B, prove that: A\u2018\u00a0\u2013 B\u2018\u00a0= B \u2013 A\">1. For any two sets A and B, prove that: A\u2018\u00a0\u2013 B\u2018\u00a0= B \u2013 A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-87\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-for-any-two-sets-a-and-b-prove-the-following-i-a-%e2%88%a9-a%e2%80%98-%e2%88%aa-b-a-%e2%88%a9-b\" title=\"2. For any two sets A and B, prove the following: (i) A\u00a0\u2229\u00a0(A\u2018\u00a0\u222a\u00a0B) = A\u00a0\u2229\u00a0B\">2. For any two sets A and B, prove the following: (i) A\u00a0\u2229\u00a0(A\u2018\u00a0\u222a\u00a0B) = A\u00a0\u2229\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-88\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%80%93-a-%e2%80%93-b-a-%e2%88%a9-b\" title=\"(ii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B\">(ii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-89\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-a-%e2%88%a9-a-%e2%88%aa-b%e2%80%99-%cf%95\" title=\"(iii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B\u2019) =\u00a0\u03d5\">(iii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B\u2019) =\u00a0\u03d5<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-90\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-a-%e2%80%93-b-a-%ce%94-a-%e2%88%a9-b\" title=\"(iv) A \u2013 B = A \u0394 (A\u00a0\u2229\u00a0B)\">(iv) A \u2013 B = A \u0394 (A\u00a0\u2229\u00a0B)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-91\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-if-a-b-c-are-three-sets-such-that-a-%e2%8a%82-b-then-prove-that-c-%e2%80%93-b-%e2%8a%82-c-%e2%80%93-a\" title=\"3. If A, B, C are three sets such that A\u00a0\u2282\u00a0B, then prove that C \u2013 B\u00a0\u2282\u00a0C \u2013 A.\">3. If A, B, C are three sets such that A\u00a0\u2282\u00a0B, then prove that C \u2013 B\u00a0\u2282\u00a0C \u2013 A.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-92\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-for-any-two-sets-a-and-b-prove-that-i-a-%e2%88%aa-b-%e2%80%93-b-a-%e2%80%93-b\" title=\"4.\u00a0For any two sets A and B, prove that (i) (A\u00a0\u222a\u00a0B) \u2013 B = A \u2013 B\">4.\u00a0For any two sets A and B, prove that (i) (A\u00a0\u222a\u00a0B) \u2013 B = A \u2013 B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-93\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-a-%e2%80%93-a-%e2%88%a9-b-a-%e2%80%93-b\" title=\"(ii) A \u2013 (A\u00a0\u2229\u00a0B) = A \u2013 B\">(ii) A \u2013 (A\u00a0\u2229\u00a0B) = A \u2013 B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-94\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-a-%e2%80%93-a-%e2%80%93-b-a-%e2%88%a9-b\" title=\"(iii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B\">(iii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-95\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iv-a-%e2%88%aa-b-%e2%80%93-a-a-%e2%88%aa-b\" title=\"(iv) A\u00a0\u222a\u00a0(B \u2013 A) = A\u00a0\u222a\u00a0B\">(iv) A\u00a0\u222a\u00a0(B \u2013 A) = A\u00a0\u222a\u00a0B<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-96\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#v-a-%e2%80%93-b-%e2%88%aa-a-%e2%88%a9-b-a\" title=\"(v) (A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A\">(v) (A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-97\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#exercise-18-page-no-146\" title=\"EXERCISE 1.8 PAGE NO: 1.46\">EXERCISE 1.8 PAGE NO: 1.46<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-98\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#1-if-a-and-b-are-two-sets-such-that-n-a-%e2%88%aa-b-50-n-a-28-and-n-b-32-find-n-a-%e2%88%a9-b\" title=\"1. If A and B are two sets such that n (A\u00a0\u222a\u00a0B) = 50, n (A) = 28 and n (B) = 32, find n (A\u00a0\u2229\u00a0B).\">1. If A and B are two sets such that n (A\u00a0\u222a\u00a0B) = 50, n (A) = 28 and n (B) = 32, find n (A\u00a0\u2229\u00a0B).<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-99\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#2-if-p-and-q-are-two-sets-such-that-p-has-40-elements-p-%e2%88%aa-q-has-60-elements-and-p-%e2%88%a9-q-has-10-elements-how-many-elements-does-q-have\" title=\"2. If P and Q are two sets such that P has 40 elements, P\u00a0\u222a\u00a0Q has 60 elements and P\u00a0\u2229\u00a0Q has 10 elements, how many elements does Q have?\">2. If P and Q are two sets such that P has 40 elements, P\u00a0\u222a\u00a0Q has 60 elements and P\u00a0\u2229\u00a0Q has 10 elements, how many elements does Q have?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-100\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#3-in-a-school-there-are-20-teachers-who-teach-mathematics-or-physics-of-these-12-teach-mathematics-and-4-teach-physics-and-mathematics-how-many-teach-physics\" title=\"3. In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?\">3. In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-101\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#4-in-a-group-of-70-people-37-like-coffee-52-like-tea-and-each-person-likes-at-least-one-of-the-two-drinks-how-many-like-both-coffee-and-tea\" title=\"4. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?\">4. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many like both coffee and tea?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-102\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#5-let-a-and-b-be-two-sets-such-that-n-a-20-n-a-%e2%88%aa-b-42-and-n-a-%e2%88%a9-b-4-find\" title=\"5. Let A and B be two sets such that: n (A) = 20, n (A\u00a0\u222a\u00a0B) = 42 and n (A\u00a0\u2229\u00a0B) = 4. Find\">5. Let A and B be two sets such that: n (A) = 20, n (A\u00a0\u222a\u00a0B) = 42 and n (A\u00a0\u2229\u00a0B) = 4. Find<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-103\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#i-n-b\" title=\"(i) n (B)\">(i) n (B)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-104\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#ii-n-a-%e2%80%93-b\" title=\"(ii) n (A \u2013 B)\">(ii) n (A \u2013 B)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-105\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#iii-n-b-%e2%80%93-a\" title=\"(iii) n (B \u2013 A)\">(iii) n (B \u2013 A)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-106\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#6-a-survey-shows-that-76-of-the-indians-like-oranges-whereas-62-like-bananas-what-percentage-of-the-indians-like-both-oranges-and-bananas\" title=\"6. A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?\">6. A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-107\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#7-in-a-group-of-950-persons-750-can-speak-hindi-and-460-can-speak-english-find-i-how-many-can-speak-both-hindi-and-english-ii-how-many-can-speak-hindi-only-iii-how-many-can-speak-english-only\" title=\"7. In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: (i) How many can speak both Hindi and English. (ii) How many can speak Hindi only. (iii) how many can speak English only.\">7. In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: (i) How many can speak both Hindi and English. (ii) How many can speak Hindi only. (iii) how many can speak English only.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-108\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#important-topics-from-rd-sharma-solutions-class-11-maths-chapter-1-sets\" title=\"Important Topics from RD Sharma Solutions Class 11 Maths Chapter 1- Sets\">Important Topics from RD Sharma Solutions Class 11 Maths Chapter 1- Sets<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-109\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#access-other-important-chapters-of-rd-sharma-solutions-class-11-maths\" title=\"Access Other Important Chapters of RD Sharma Solutions Class 11 Maths\">Access Other Important Chapters of RD Sharma Solutions Class 11 Maths<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-110\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#faqs-on-rd-sharma-solutions-class-11-maths-chapter-1\" title=\"FAQs on RD Sharma Solutions Class 11 Maths Chapter 1\">FAQs on RD Sharma Solutions Class 11 Maths Chapter 1<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-111\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#from-where-can-i-find-the-solutions-of-maths-chapter-1-exercise-17\" title=\"From where can I find the solutions of Maths Chapter 1 Exercise 1.7?\">From where can I find the solutions of Maths Chapter 1 Exercise 1.7?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-112\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#is-it-necessary-to-solve-all-the-questions-of-rd-sharma-chapter-1\" title=\"Is it necessary to solve all the questions of RD Sharma Chapter 1?\">Is it necessary to solve all the questions of RD Sharma Chapter 1?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-113\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/#how-much-does-it-cost-to-download-the-pdf-of-rd-sharma-solutions-class-11-maths-chapter-1\" title=\"How much does it cost to download the PDF of RD Sharma Solutions Class 11 Maths Chapter 1?\">How much does it cost to download the PDF of RD Sharma Solutions Class 11 Maths Chapter 1?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-solutions-class-11-maths-chapter-1-%e2%80%93-sets-pdf\"><\/span>Download RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets PDF<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><a href=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-1-1-1.pdf\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 11 Maths Chapter 1<\/a><\/p>\n<div id=\"example1\" style=\"text-align: justify;\">\u00a0<\/div>\n<p style=\"text-align: justify;\"><style>\n.pdfobject-container { height: 500px;}<br \/>\n.pdfobject { border: 1px solid #666; }<br \/>\n<\/style><\/p>\n<p style=\"text-align: justify;\"><script src=\"https:\/\/www.kopykitab.com\/_utility\/js\/pdfobject.min.js\"><\/script><br \/><script>PDFObject.embed(\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-1-1-1.pdf\", \"#example1\");<\/script><\/p>\n<h2><span class=\"ez-toc-section\" id=\"exercises-rd-sharma-solutions-class-11-maths-chapter-1\"><\/span>Exercises: RD Sharma Solutions Class 11 Maths Chapter 1\u00a0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-1\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1A<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-2\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1B<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-3\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1C<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-4\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1D<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-5\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1E<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-6\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1F<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-7\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1G<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\"><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-exercise-1-8\/\" target=\"_blank\" rel=\"noopener\">RD Sharma Solution Class 11 Chapter 1 Exercise1H<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span style=\"font-size: 24px; font-weight: bold; background-color: initial;\">RD Sharma Class 11 Chapter 1 Exercise 1.1 Solution<\/span><\/p>\n<p><span style=\"font-weight: 400;\">RD Sharma Solutions Exercise-1.1 covers the basics of sets. It includes the basics of sets, definitions, and how to form a set. <\/span><span style=\"font-weight: 400;\">The Exercise-1.1 solutions are so basic that it won\u2019t take much of your time. You can easily understand this concept and implement it later.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-12-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.2 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-1.2 covers some more information about sets. It helps students to learn various other concepts of sets such as Roster Form, Set Builder Form, Describing Roster or Tabular, and Set Builder form by explaining with the help of examples.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In RD Sharma Solutions you will get step-by-step solutions to every exercise. Go through these initial concepts carefully because every exercise is linked to others.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-13-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.3 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-1.3 covers the different types of sets. It helps students understand the meaning of an empty set, singleton set, infinite sets, and equivalent sets. The concepts are explained with the help of illustrations and examples. Also, every exercise in RD Sharma Class 11 Solutions comes up in the next exercise along with the new concepts to make the previous concept stronger. In this way, by the end of the chapter, you will get a particular topic more clearly.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-14-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.4 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">By the time a student reaches Exercise-1.4, he will get a clear understanding of sets, its types, and how to solve questions using these concepts. Exercise-1.4 includes a whole new topic i.e Subset. The subsets are explained with illustrations and theorems too. The exercise also includes the introduction to elements, universal and power sets too.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-15-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.5 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-1.5 solutions of RD Sharma is the most interesting exercise among all other exercises. It covers all the topics that you have studies till Exercise-1.4 along with a new concept of Venn Diagrams. Venn diagram is the pictorial representation that helps in making sets fun and easily understandable by the students.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-16-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.6 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-1.6 solutions cover some laws of algebra that set follow. You can download the exercise-wise solutions to get through with this exercise.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-class-11-chapter-1-exercise-17-ex-18-solution\"><\/span>RD Sharma Class 11 Chapter 1 Exercise 1.7 &amp; Ex 1.8 Solution<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-1.7 &amp; 1.8 covers some advanced questions and answers. These exercises are a blend of all the concepts studied in chapter 1-sets. These Exercise-1.7 and 1.8 solutions along with some new theorems and formulae brush up all the concepts. Therefore, students are advised not to miss these exercises. Do all the exercises of the chapter-1 carefully. The exercises are very simple, small and fun too. It gives you a different angle on how you look at things.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"access-rd-sharma-solutions-class-11-maths-chapter-1\"><\/span>Access RD Sharma Solutions Class 11 Maths Chapter 1\u00a0<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2><span class=\"ez-toc-section\" id=\"exercise-11-page-no-12\"><\/span>EXERCISE 1.1 PAGE NO: 1.2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h2><span class=\"ez-toc-section\" id=\"1-what-is-the-difference-between-a-collection-and-a-set-give-reasons-to-support-your-answer\"><\/span>1. What is the difference between a collection and a set? Give reasons to support your answer.<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><strong>Solution:<\/strong><\/p>\n<p>Well defined collections are sets.<\/p>\n<p>Examples: The collection of good cricket players of India is not a set. However, the collection of all good player is a set.<\/p>\n<p>The collection of vowels in English alphabets is a set.<\/p>\n<p>Thus, we can say that every set is a collection, but every collection is not necessarily a set. Only well defined collections are sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-which-of-the-following-collections-are-sets-justify-your-answer-i-a-collection-of-all-natural-numbers-less-than-50-ii-the-collection-of-good-hockey-players-in-india-iii-the-collection-of-all-the-girls-in-your-class-iv-the-collection-of-all-talented-writers-of-india-v-the-collection-of-difficult-topics-in-mathematics-vi-the-collection-of-novels-written-by-munshi-prem-chand-vii-the-collection-of-all-questions-of-this-chapter-viii-the-collection-of-all-months-of-a-year-beginning-with-the-letter-j-ix-a-collection-of-most-dangerous-animals-of-the-world-x-the-collection-of-prime-integers\"><\/span>2.\u00a0Which of the following collections are sets? Justify your answer:<br \/>(i) A collection\u00a0of all natural numbers less than 50.<br \/>(ii) The collection\u00a0of good hockey players in India.<br \/>(iii)\u00a0The collection of all the girls in your class.<br \/>(iv)\u00a0The collection\u00a0of all talented writers of India.<br \/>(v) The collection\u00a0of difficult topics in Mathematics.<br \/>(vi)\u00a0The collection of novels written by Munshi Prem Chand.<br \/>(vii)\u00a0The collection\u00a0of all questions of this chapter.<br \/>(viii)\u00a0The collection of all months of a year beginning with the letter J.<br \/>(ix) A collection\u00a0of most dangerous animals of the world.<br \/>(x) The collection\u00a0of prime integers.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0It is a set. Since, collection of all natural numbers less than 50 forms a set as it is well defined.<\/p>\n<p><strong>(ii)<\/strong>\u00a0It is not a set. Since, the term \u2018good\u2019 is not well defined.<\/p>\n<p><strong>(iii)<\/strong>\u00a0It is a set. Since, a\u00a0collection\u00a0of all the girls in your class\u00a0it is a definite quantity hence it is a set.<\/p>\n<p><strong>(iv)<\/strong>\u00a0It is not a set. Since, the term \u2018most\u2019 is not well defined. A writer may be talented in the eye of one person, but he may not be talented in the eye of some other person.<\/p>\n<p><strong>(v)<\/strong>\u00a0It is not a set. Since, the term \u2018difficult\u2019 is not well defined. A topic may be difficult for one person, but may not be difficult for another person.<\/p>\n<p><strong>(vi)<\/strong>\u00a0It is a set. Since, a\u00a0collection\u00a0of novels written by Munshi Prem Chand it is a definite quantity hence it is a set.<\/p>\n<p><strong>(vii)<\/strong>\u00a0It is a set. Since, a\u00a0collection\u00a0of all the questions in this chapter.\u00a0It is a definite quantity hence it is a set.<\/p>\n<p><strong>(viii)<\/strong>\u00a0It is a set. Since, a\u00a0collection\u00a0of all months of a year beginning with the letter J.\u00a0It is a definite quantity hence it is a set.<\/p>\n<p><strong>(ix)<\/strong>\u00a0It is not a set. Since, the term \u2018most dangerous\u2019 is not well defined. The notion of dangerous animals differs from person to person.<\/p>\n<p><strong>(x)<\/strong>\u00a0It is a set. Since, a\u00a0collection\u00a0of prime integers.\u00a0It is a definite quantity hence it is a set.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-if-a012345678910-then-insert-the-appropriate-symbol-or-in-each-of-the-following-blank-spaces-i-4-%e2%80%a6%e2%80%a6a\"><\/span>3. If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol\u00a0or\u00a0in each of the following blank spaces:<br \/>(i) 4 \u2026\u2026A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-4%e2%80%a6%e2%80%a6a-iii-12%e2%80%a6a\"><\/span>(ii) -4\u2026\u2026A<br \/>(iii) 12\u2026..A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-9%e2%80%a6a-v-0%e2%80%a6%e2%80%a6a\"><\/span>(iv) 9\u2026..A<br \/>(v) 0\u2026\u2026A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"vi-2%e2%80%a6%e2%80%a6a\"><\/span>(vi) -2\u2026\u2026A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>The symbol \u2018\u2208\u2019 means belongs to. \u2018\u2209\u2019 means does not belong to.<\/p>\n<p><strong>(i)\u00a0<\/strong>4 \u2026\u2026A<\/p>\n<p>4 \u2208 A (since, 4 is present in set A)<\/p>\n<p><strong>(ii)\u00a0<\/strong>-4\u2026\u2026A<\/p>\n<p>-4 \u2209 A (since, -4 is not present in set A)<\/p>\n<p><strong>(iii)\u00a0<\/strong>12\u2026..A<\/p>\n<p>12 \u2209 A (since, 12 is not present in set A)<\/p>\n<p><strong>(iv)\u00a0<\/strong>9\u2026..A<\/p>\n<p>9 \u2208 A (since, 9 is present in set A)<\/p>\n<p><strong>(v)\u00a0<\/strong>0\u2026\u2026A<\/p>\n<p>0 \u2208 A (since, 0 is present in set A)<\/p>\n<p><strong>(vi)\u00a0<\/strong>-2\u2026\u2026A<\/p>\n<p>-2 \u2209 A (since, -2 is not present in set A)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-12-page-no-16\"><\/span>EXERCISE 1.2 PAGE NO: 1.6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-describe-the-following-sets-in-roster-form\"><\/span><strong>1. Describe the following sets in Roster form:<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"i-x-x-is-a-letter-before-e-in-the-english-alphabet\"><\/span><strong>(i)<\/strong>\u00a0<strong>{x : x is a letter before e in the English alphabet}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-x-%e2%88%88-n-x2-%3c-25\"><\/span><strong>(ii) {x\u00a0\u2208 N: x<sup>2<\/sup>\u00a0&lt; 25}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-x-%e2%88%88-n-x-is-a-prime-number-10-%3c-x-%3c-20\"><\/span><strong>(iii) {x \u2208\u00a0N: x is a prime number, 10 &lt; x &lt; 20}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-x-%e2%88%88-n-x-2n-n-%e2%88%88-n\"><\/span><strong>(iv) {x \u2208\u00a0N: x = 2n, n \u2208\u00a0N}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"v-x-%e2%88%88-r-x-%3e-x\"><\/span><strong>(v) {x \u2208\u00a0R: x &gt; x}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"vi-x-x-is-a-prime-number-which-is-a-divisor-of-60\"><\/span><strong>(vi) {x : x is a prime number which is a divisor of 60}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"vii-x-x-is-a-two-digit-number-such-that-the-sum-of-its-digits-is-8\"><\/span><strong>(vii) {x : x is a two digit number such that the sum of its digits is 8}<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"viii-the-set-of-all-letters-in-the-word-%e2%80%98trigonometry%e2%80%99\"><\/span><strong>(viii) The set of all letters in the word \u2018Trigonometry\u2019<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ix-the-set-of-all-letters-in-the-word-%e2%80%98better%e2%80%99\"><\/span><strong>(ix) The set of all letters in the word \u2018Better.&#8217;<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>{x : x is a letter before e in the English alphabet}<\/p>\n<p>So, when we read whole sentence it becomes x is such that x is a letter before \u2018e\u2019 in the English alphabet. Now letters before \u2018e\u2019 are a,b,c,d.<\/p>\n<p>\u2234\u00a0Roster form will be {a,b,c,d}.<\/p>\n<p><strong>(ii)<\/strong>\u00a0{x\u00a0\u2208 N: x<sup>2<\/sup>\u00a0&lt; 25}<\/p>\n<p>x\u00a0\u2208\u00a0N that implies x is a natural number.<\/p>\n<p>x<sup>2<\/sup>\u00a0&lt; 25<\/p>\n<p>x &lt;\u00a0\u00b15<\/p>\n<p>As x belongs to the natural number that means x &lt; 5.<\/p>\n<p>All numbers less than 5 are 1,2,3,4.<\/p>\n<p>\u2234\u00a0Roster form will be {1,2,3,4}.<\/p>\n<p><strong>(iii)\u00a0<\/strong>{x \u2208\u00a0N: x is a prime number, 10 &lt; x &lt; 20}<\/p>\n<p>X is a natural number and is between 10 and 20.<\/p>\n<p>X is such that X is a prime number between 10 and 20.<\/p>\n<p>Prime numbers between 10 and 20 are 11,13,17,19.<\/p>\n<p>\u2234\u00a0Roster form will be {11,13,17,19}.<\/p>\n<p><strong>(iv)<\/strong>\u00a0{x \u2208\u00a0N: x = 2n, n \u2208\u00a0N}<\/p>\n<p>X is a natural number also x = 2n<\/p>\n<p>\u2234\u00a0Roster form will be {2,4,6,8\u2026..}.<\/p>\n<p>This an infinite set.<\/p>\n<p><strong>(v)\u00a0<\/strong>{x \u2208\u00a0R: x &gt; x}<\/p>\n<p>Any real number is equal to its value it is neither less nor greater.<\/p>\n<p>So, Roster form of such real numbers which has value less than itself has no such numbers.<\/p>\n<p>\u2234\u00a0Roster form will be\u00a0\u03d5. This is called a null set.<\/p>\n<p><strong>(vi)\u00a0<\/strong>{x : x is a prime number which is a divisor of 60}<\/p>\n<p>All numbers which are divisor of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.<\/p>\n<p>Now, prime numbers are = 2, 3, 5.<\/p>\n<p>\u2234\u00a0Roster form will be {2, 3, 5}.<\/p>\n<p><strong>(vii)\u00a0<\/strong>{x : x is a two digit number such that the sum of its digits is 8}<\/p>\n<p>Numbers which have sum of its digits as 8 are = 17, 26, 35, 44, 53, 62, 71, 80<\/p>\n<p>\u2234 Roster form will be {17, 26, 35, 44, 53, 62, 71, 80}.<\/p>\n<p><strong>(viii) The set of all letters in the word \u2018Trigonometry\u2019<\/strong><\/p>\n<p>As repetition is not allowed in a set, then the distinct letters are<\/p>\n<p>Trigonometry = t, r, i, g, o, n, m, e, y<\/p>\n<p>\u2234\u00a0Roster form will be {t, r, i, g, o, n, m, e, y}<\/p>\n<p><strong>(ix) The set of all letters in the word \u2018Better.\u2019<\/strong><\/p>\n<p>As repetition is not allowed in a set, then the distinct letters are<\/p>\n<p>Better = b, e, t, r<\/p>\n<p>\u2234\u00a0Roster form will be {b, e, t, r}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-describe-the-following-sets-in-set-builder-form-i-a-1-2-3-4-5-6-ii-b-1-12-13-14-15-%e2%80%a6-iii-c-0-3-6-9-12%e2%80%a6-iv-d-10-11-12-13-14-15-v-e-0-vi-1-4-9-16%e2%80%a6100-vii-2-4-6-8%e2%80%a6-viii-5-25-125-625\"><\/span>2.\u00a0Describe the following sets in set-builder form:<br \/>(i) A = {1, 2, 3, 4, 5, 6}<br \/>(ii) B = {1, 1\/2, 1\/3, 1\/4, 1\/5, \u2026..}<br \/>(iii) C = {0, 3, 6, 9, 12,\u2026.}<br \/>(iv) D = {10, 11, 12, 13, 14, 15}<br \/>(v) E = {0}<br \/>(vi) {1, 4, 9, 16,\u2026,100}<br \/>(vii) {2, 4, 6, 8,\u2026.}<br \/>(viii) {5, 25, 125, 625}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0A = {1, 2, 3, 4, 5, 6}<\/p>\n<p>{x : x\u00a0\u2208\u00a0N, x&lt;7}<\/p>\n<p>This is read as x is such that x belongs to natural number and x is less than 7. It satisfies all condition of roster form.<\/p>\n<p><strong>(ii)\u00a0<\/strong>B = {1, 1\/2, 1\/3, 1\/4, 1\/5, \u2026}<\/p>\n<p>{x\u00a0: x = 1\/n, n\u00a0\u2208\u00a0N}<\/p>\n<p>This is read as x is such that x =1\/n, where n \u2208\u00a0N.<\/p>\n<p><strong>(iii)<\/strong>\u00a0C = {0, 3, 6, 9, 12, \u2026.}<strong><br \/><\/strong><br \/>{x : x = 3n, n \u2208\u00a0Z<sup>+<\/sup>, the set of positive integers}<\/p>\n<p>This is read as x is such that C is the set of multiples of 3 including 0.<\/p>\n<p><strong>(iv)<\/strong>\u00a0D = {10, 11, 12, 13, 14, 15}<\/p>\n<p>{x : x \u2208\u00a0N, 9&lt;x&lt;16}<\/p>\n<p>This is read as x is such that D is the set of natural numbers which are more than 9 but less than 16.<\/p>\n<p><strong>(v)<\/strong>\u00a0E = {0}<\/p>\n<p>{x : x = 0}<\/p>\n<p>This is read as x is such that E is an integer equal to 0.<\/p>\n<p><strong>(vi)<\/strong>\u00a0{1, 4, 9, 16,\u2026, 100}<\/p>\n<p>Where,<\/p>\n<p>1<sup>2<\/sup>\u00a0= 1<\/p>\n<p>2<sup>2<\/sup>\u00a0= 4<\/p>\n<p>3<sup>2<\/sup>\u00a0= 9<\/p>\n<p>4<sup>2<\/sup>\u00a0= 16<\/p>\n<p>.<\/p>\n<p>.<\/p>\n<p>.<\/p>\n<p>10<sup>2<\/sup>\u00a0= 100<\/p>\n<p>So, above set can be expressed in set-builder form as {x<sup>2<\/sup>: x\u00a0\u2208\u00a0N, 1\u2264 x \u226410}<\/p>\n<p><strong>(vii)\u00a0<\/strong>{2, 4, 6, 8,\u2026.}<\/p>\n<p>{x: x = 2n, n \u2208\u00a0N}<\/p>\n<p>This is read as x is such that the given set are multiples of 2.<\/p>\n<p><strong>(viii)<\/strong>\u00a0{5, 25, 125, 625}<\/p>\n<p>Where,<\/p>\n<p>5<sup>1<\/sup>\u00a0= 5<\/p>\n<p>5<sup>2<\/sup>\u00a0= 25<\/p>\n<p>5<sup>3<\/sup>\u00a0= 125<\/p>\n<p>5<sup>4<\/sup>\u00a0= 625<\/p>\n<p>So, above set can be expressed in set-builder form as {5<sup>n<\/sup>: n \u2208 N, 1\u2264 n \u2264 4}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-list-all-the-elements-of-the-following-sets\"><\/span>3.\u00a0List all the elements of the following sets:<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"i-ax-x2%e2%89%a4-10-x-%e2%88%88-z\"><\/span>(i) A={x : x<sup>2<\/sup>\u2264\u00a010, x\u00a0\u2208\u00a0Z}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-b-x-x-12n-1-1-%e2%89%a4-n-%e2%89%a4-5\"><\/span>(ii) B = {x : x = 1\/(2n-1), 1 \u2264 n\u00a0\u2264 5}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-c-x-x-is-an-integer-12-%3c-x-%3c-92\"><\/span>(iii) C = {x : x is an integer, -1\/2 &lt; x &lt; 9\/2}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-dx-x-is-a-vowel-in-the-word-%e2%80%9cequation%e2%80%9d\"><\/span>(iv) D={x : x is a vowel in the word \u201cEQUATION\u201d}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"v-e-x-x-is-a-month-of-a-year-not-having-31-days\"><\/span>(v) E = {x : x is a month of a year not having 31 days}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"vi-fx-x-is-a-letter-of-the-word-%e2%80%9cmississippi%e2%80%9d\"><\/span>(vi) F={x : x is a letter of the word \u201cMISSISSIPPI\u201d}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>A={x : x<sup>2<\/sup>\u2264\u00a010, x \u2208 Z}<\/p>\n<p>First of all, x is an integer hence it can be positive and negative also.<\/p>\n<p>x<sup>2<\/sup>\u00a0\u2264 10<\/p>\n<p>(-3)<sup>2<\/sup>\u00a0= 9 &lt; 10<\/p>\n<p>(-2)<sup>2<\/sup>\u00a0= 4 &lt; 10<\/p>\n<p>(-1)<sup>2<\/sup>\u00a0= 1 &lt; 10<\/p>\n<p>0<sup>2<\/sup>\u00a0= 0 &lt; 10<\/p>\n<p>1<sup>2<\/sup>\u00a0= 1 &lt; 10<\/p>\n<p>2<sup>2<\/sup>\u00a0= 4 &lt; 10<\/p>\n<p>3<sup>2<\/sup>\u00a0= 9 &lt; 10<\/p>\n<p>Square root of next integers are greater than 10.<\/p>\n<p>x \u2264 \u221a10<\/p>\n<p>x = 0, \u00b11, \u00b12, \u00b13<\/p>\n<p>A = {0, \u00b11, \u00b12, \u00b13}<\/p>\n<p><strong>(ii)\u00a0<\/strong>B = {x : x = 1\/(2n-1), 1 \u2264 n \u2264 5}<\/p>\n<p>Let us substitute the value of n to find the values of x.<\/p>\n<p>At n=1, x = 1\/(2(1)-1) = 1\/1<\/p>\n<p>At n=2, x = 1\/(2(2)-1) = 1\/3<\/p>\n<p>At n=3, x = 1\/(2(3)-1) = 1\/5<\/p>\n<p>At n=4, x = 1\/(2(4)-1) = 1\/7<\/p>\n<p>At n=5, x = 1\/(2(5)-1) = 1\/9<\/p>\n<p>x = 1, 1\/3, 1\/5, 1\/7, 1\/9<\/p>\n<p>\u2234 B = {1, 1\/3, 1\/5, 1\/7, 1\/9}<\/p>\n<p><strong>(iii)\u00a0<\/strong>C = {x : x is an integer, -1\/2 &lt; x &lt; 9\/2}<\/p>\n<p>Given, x is an integer between -1\/2 and 9\/2<\/p>\n<p>So all integers between -0.5&lt;x&lt;4.5 are = 0, 1, 2, 3, 4<\/p>\n<p>\u2234\u00a0C = {0, 1, 2, 3, 4}<\/p>\n<p><strong>(iv)\u00a0<\/strong>D={x : x is a vowel in the word \u201cEQUATION\u201d}<\/p>\n<p>All vowels in the word \u2018EQUATION\u2019 are E, U, A, I, O<\/p>\n<p>\u2234\u00a0D = {A, E, I, O, U}<\/p>\n<p><strong>(v)\u00a0<\/strong>E = {x : x is a month of a year not having 31 days}<\/p>\n<p>A month has either 28, 29, 30, 31 days.<\/p>\n<p>Out of 12 months in a year which are not having 31 days are:<\/p>\n<p>February, April, June, September, November.<\/p>\n<p>\u2234 E: {February, April, June, September, November}<\/p>\n<p><strong>(vi)\u00a0<\/strong>F = {x : x is a letter of the word \u201cMISSISSIPPI\u201d}<\/p>\n<p>Letters in word \u2018MISSISSIPPI\u2019 are M, I, S, P.<\/p>\n<p>\u2234 F = {M, I, S, P}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-match-each-of-the-sets-on-the-left-in-the-roster-form-with-the-same-set-on-the-right-described-in-the-set-builder-form-i-aple-i-x-x55-x-%e2%88%88-z-ii-5-5-ii-x-x-is-a-prime-natural-number-and-a-divisor-of-10-iii-0-iii-x-x-is-a-letter-of-the-word-%e2%80%9crajasthan%e2%80%9d-iv-1-2-5-10-iv-x-x-is-a-natural-and-divisor-of-10-v-a-h-j-r-s-t-n-v-x-x2-%e2%80%93-25-0-vi-25-vi-x-x-is-a-letter-of-word-%e2%80%9capple%e2%80%9d\"><\/span>4.\u00a0Match each of the sets on the left in the roster form with the same set on the right described in the set-builder form:<br \/>(i) {A,P,L,E} (i) {x : x+5=5, x\u00a0\u2208\u00a0z}<br \/>(ii) {5,-5} (ii) {x : x is a prime natural number and a divisor of 10}<br \/>(iii) {0} (iii) {x : x is a letter of the word \u201cRAJASTHAN\u201d}<br \/>(iv) {1, 2, 5, 10} (iv) {x : x is a natural and divisor of 10}<br \/>(v) {A, H, J, R, S, T, N} (v) {x : x<sup>2<\/sup>\u00a0\u2013 25 =0}<br \/>(vi) {2,5} (vi) {x : x is a letter of word \u201cAPPLE\u201d}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>{A, P, L, E}<strong>\u00a0\u21d4\u00a0<\/strong>{x: x is a letter of word \u201cAPPLE\u201d}<\/p>\n<p><strong>(ii)<\/strong>\u00a0{5,-5}\u00a0<strong>\u21d4\u00a0<\/strong>{x: x<sup>2<\/sup>\u00a0\u2013 25 =0}<\/p>\n<p>The solution set of x<sup>2<\/sup>\u00a0\u2013 25 = 0 is x = \u00b15<\/p>\n<p><strong>(iii)<\/strong>\u00a0{0}<strong>\u00a0\u21d4\u00a0<\/strong>{x: x+5=5, x \u2208\u00a0z}<\/p>\n<p>The solution set of x + 5 = 5 is x = 0.<\/p>\n<p><strong>(iv)\u00a0<\/strong>{1, 2, 5, 10}\u00a0<strong>\u21d4\u00a0<\/strong>{x: x is a natural and divisor of 10}<\/p>\n<p>The natural numbers which are divisor of 10 are 1, 2, 5, 10.<\/p>\n<p><strong>(v)<\/strong>\u00a0{A, H, J, R, S, T, N}\u00a0<strong>\u21d4\u00a0<\/strong>{x: x is a letter of the word \u201cRAJASTHAN\u201d}<\/p>\n<p>The distinct letters of word \u201cRAJASTHAN\u201d are A, H, J, R, S, T, N.<\/p>\n<p><strong>(vi)<\/strong>\u00a0{2, 5}\u00a0<strong>\u21d4\u00a0<\/strong>{x: x is a prime natural number and a divisor of 10}<\/p>\n<p>The prime natural numbers which are divisor of 10 are 2, 5.<\/p>\n<p><strong>5. Write the set of all vowels in the English alphabet which precede q.<\/strong><\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Set of all vowels which precede q are<\/p>\n<p>A, E, I, O these are the vowels they come before q.<\/p>\n<p>\u2234 B = {A, E, I, O}.\u00a0<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-write-the-set-of-all-positive-integers-whose-cube-is-odd\"><\/span>6. Write the set of all positive integers whose cube is odd.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Every odd number has an odd cube<\/p>\n<p>Odd numbers can be represented as 2n+1.<\/p>\n<p>{2n+1: n \u2208 Z, n&gt;0} or<\/p>\n<p>{1,3,5,7,\u2026\u2026}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-write-the-set-12-25-310-417-526-637-750-in-the-set-builder-form\"><\/span>7. Write the set {1\/2, 2\/5, 3\/10, 4\/17, 5\/26, 6\/37, 7\/50}\u00a0in the set-builder form.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Where,<\/p>\n<p>2 = 1<sup>2<\/sup>\u00a0+ 1<\/p>\n<p>5 = 2<sup>2<\/sup>\u00a0+ 1<\/p>\n<p>10 = 3<sup>2<\/sup>\u00a0+ 1<\/p>\n<p>.<\/p>\n<p>.<\/p>\n<p>50 = 7<sup>2<\/sup>\u00a0+ 1<\/p>\n<p>Here we can see denominator is square of numerator +1.<\/p>\n<p>So, we can write the set builder form as<\/p>\n<p>{n\/(n<sup>2<\/sup>+1): n \u2208 N, 1\u2264 n\u2264 7}<\/p>\n<p>EXERCISE 1.3 PAGE NO: 1.9<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1-which-of-the-following-are-examples-of-empty-set-i-set-of-all-even-natural-numbers-divisible-by-5-ii-set-of-all-even-prime-numbers-iii-x-x2%e2%80%9320-and-x-is-rational-iv-x-x-is-a-natural-number-x-%3c-8-and-simultaneously-x-%3e-12-v-x-x-is-a-point-common-to-any-two-parallel-lines\"><\/span>1. Which of the following are examples of empty set?<br \/>(i) Set of all even natural numbers divisible by 5.<br \/>(ii) Set of all even prime numbers.<br \/>(iii) {x: x<sup>2<\/sup>\u20132=0 and x is rational}.<br \/>(iv) {x: x is a natural number, x &lt; 8 and simultaneously x &gt; 12}.<br \/>(v) {x: x is a point common to any two parallel lines}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0All numbers ending with 0. Except 0 is divisible by 5 and are even natural number. Hence it is not an example of empty set.<\/p>\n<p><strong>(ii)<\/strong>\u00a02 is a prime number and is even, and it is the only prime which is even. So this not an example of the empty set.<\/p>\n<p><strong>(iii)<\/strong>\u00a0x<sup>2<\/sup>\u00a0\u2013 2 = 0, x<sup>2<\/sup>\u00a0= 2, x = \u00b1 \u221a2 \u2208 N. There is not natural number whose square is 2. So it is an example of empty set.<\/p>\n<p><strong>(iv)<\/strong>\u00a0There is no natural number less than 8 and greater than 12. Hence it is an example of the empty set.<\/p>\n<p><strong>(v)<\/strong>\u00a0No two parallel lines intersect at each other. Hence it is an example of empty set.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-which-of-the-following-sets-are-finite-and-which-are-infinite-i-set-of-concentric-circles-in-a-plane-ii-set-of-letters-of-the-english-alphabets-iii-x-%e2%88%88-n-x-%3e-5-iv-x-%e2%88%88-n-x-%3c-200-v-x-%e2%88%88-z-x-%3c-5-vi-x-%e2%88%88-r-0-%3c-x-%3c-1\"><\/span>2. Which of the following sets are finite and which are infinite?<br \/>(i) Set of concentric circles in a plane.<br \/>(ii) Set of letters of the English Alphabets.<br \/>(iii) {x\u00a0\u2208\u00a0N: x &gt; 5}<br \/>(iv) {x\u00a0\u2208\u00a0N: x &lt; 200}<br \/>(v) {x\u00a0\u2208\u00a0Z: x &lt; 5}<br \/>(vi) {x\u00a0\u2208\u00a0R: 0 &lt; x &lt; 1}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0Infinite concentric circles can be drawn in a plane. Hence it is an infinite set.<\/p>\n<p><strong>(ii)<\/strong>\u00a0There are just 26 letters in English Alphabets. Hence it is finite set.<\/p>\n<p><strong>(iii)<\/strong>\u00a0It is an infinite set because, natural numbers greater than 5 is infinite.<\/p>\n<p><strong>(iv)<\/strong>\u00a0It is a finite set. Since, natural numbers start from 1 and there are 199 numbers less than 200. Hence it is a finite set.<\/p>\n<p><strong>(v)<\/strong>\u00a0It is an infinite set. Because integers less than 5 are infinite so it is an infinite set.<\/p>\n<p><strong>(vi)<\/strong>\u00a0It is an infinite set. Because between two real numbers, there are infinite real numbers.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-which-of-the-following-sets-are-equal-i-a-1-2-3-ii-b-x-%e2%88%88-r-x2%e2%80%932x10-iii-c-1-2-2-3-iv-d-x-%e2%88%88-r-x3-%e2%80%93-6x211x-%e2%80%93-6-0\"><\/span>3. Which of the following sets are equal?<br \/>(i) A = {1, 2, 3}<br \/>(ii) B = {x\u00a0\u2208\u00a0R:x<sup>2<\/sup>\u20132x+1=0}<br \/>(iii) C = (1, 2, 2, 3}<br \/>(iv) D = {x\u00a0\u2208 R : x<sup>3<\/sup>\u00a0\u2013 6x<sup>2<\/sup>+11x \u2013 6 = 0}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A set is said to be equal with another set if all elements of both the sets are equal and same.<\/p>\n<p>A = {1, 2, 3}<\/p>\n<p>B ={x\u00a0\u2208\u00a0R: x<sup>2<\/sup>\u20132x+1=0}<\/p>\n<p>x<sup>2<\/sup>\u20132x+1 = 0<\/p>\n<p>(x\u20131)<sup>2<\/sup>\u00a0= 0<\/p>\n<p>\u2234\u00a0x = 1.<\/p>\n<p>B = {1}<\/p>\n<p>C= {1, 2, 2, 3}<\/p>\n<p>In sets we do not repeat elements hence C can be written as {1, 2, 3}<\/p>\n<p>D = {x\u00a0\u2208 R: x<sup>3<\/sup>\u00a0\u2013 6x<sup>2<\/sup>+11x \u2013 6 = 0}<\/p>\n<p>For x = 1, x<sup>2<\/sup>\u20132x+1=0<\/p>\n<p>= (1)<sup>3<\/sup>\u20136(1)<sup>2<\/sup>+11(1)\u20136<\/p>\n<p>= 1\u20136+11\u20136<\/p>\n<p>= 0<\/p>\n<p>For x =2,<\/p>\n<p>= (2)<sup>3<\/sup>\u20136(2)<sup>2<\/sup>+11(2)\u20136<\/p>\n<p>= 8\u201324+22\u20136<\/p>\n<p>= 0<\/p>\n<p>For x =3,<\/p>\n<p>= (3)<sup>3<\/sup>\u20136(3)<sup>2<\/sup>+11(3)\u20136<\/p>\n<p>= 27\u201354+33\u20136<\/p>\n<p>= 0<\/p>\n<p>\u2234\u00a0D = {1, 2, 3}<\/p>\n<p>Hence, the set A, C and D are equal.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-are-the-following-sets-equal-ax-x-is-a-letter-in-the-word-reap\"><\/span>4. Are the following sets equal?<br \/>A={x: x is a letter in the word reap},<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"bx-x-is-a-letter-in-the-word-paper-cx-x-is-a-letter-in-the-word-rope\"><\/span>B={x: x is a letter in the word paper},<br \/>C={x: x is a letter in the word rope}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>For A<\/p>\n<p>Letters in word reap<\/p>\n<p>A ={R, E, A, P} = {A, E, P, R}<\/p>\n<p>For B<\/p>\n<p>Letters in word paper<\/p>\n<p>B = {P, A, E, R} = {A, E, P, R}<\/p>\n<p>For C<\/p>\n<p>Letters in word rope<\/p>\n<p>C = {R, O, P, E} = {E, O, P, R}.<\/p>\n<p>Set A = Set B<\/p>\n<p>Because every element of set A is present in set B<\/p>\n<p>But Set C is not equal to either of them because all elements are not present.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-from-the-sets-given-below-pair-the-equivalent-sets-a-1-2-3-b-t-p-q-r-s-c-%ce%b1-%ce%b2-%ce%b3-d-a-e-i-o-u\"><\/span>5. From the sets given below, pair the equivalent sets:<br \/>A= {1, 2, 3}, B = {t, p, q, r, s}, C = {\u03b1,\u00a0\u03b2,\u00a0\u03b3}, D = {a, e, i, o, u}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Equivalent set are different from equal sets, Equivalent sets are those which have equal number of elements they do not have to be same.<\/p>\n<p>A = {1, 2, 3}<\/p>\n<p>Number of elements = 3<\/p>\n<p>B = {t, p, q, r, s}<\/p>\n<p>Number of elements = 5<\/p>\n<p>C = {\u03b1,\u00a0\u03b2,\u00a0\u03b3}<\/p>\n<p>Number of elements = 3<\/p>\n<p>D = {a, e, i, o, u}<\/p>\n<p>Number of elements = 5<\/p>\n<p>\u2234 Set A is equivalent with Set C.<\/p>\n<p>Set B is equivalent with Set D.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-are-the-following-pairs-of-sets-equal-give-reasons-i-a-2-3-b-x-x-is-a-solution-of-x2-5x-6-0\"><\/span>6. Are the following pairs of sets equal? Give reasons.<br \/>(i) A = {2, 3}, B = {x: x is a solution of x<sup>2<\/sup>\u00a0+ 5x + 6= 0}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-ax-x-is-a-letter-of-the-word-%e2%80%9cwolf%e2%80%9d-bx-x-is-letter-of-word-%e2%80%9cfollow%e2%80%9d\"><\/span>(ii) A={x: x is a letter of the word \u201cWOLF\u201d}<br \/>B={x: x is letter of word \u201cFOLLOW\u201d}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0A = {2, 3}<\/p>\n<p>B = x<sup>2<\/sup>\u00a0+ 5x + 6 = 0<\/p>\n<p>x<sup>2<\/sup>\u00a0+ 3x + 2x + 6 = 0<\/p>\n<p>x(x+3) + 2(x+3) = 0<\/p>\n<p>(x+3) (x+2) = 0<\/p>\n<p>x = -2 and -3<\/p>\n<p>= {\u20132, \u20133}<\/p>\n<p>Since, A and B do not have exactly same elements hence they are not equal.<\/p>\n<p><strong>(ii)<\/strong>\u00a0Every letter in WOLF<\/p>\n<p>A = {W, O, L, F} = {F, L, O, W}<\/p>\n<p>Every letter in FOLLOW<\/p>\n<p>B = {F, O, L, W} = {F, L, O, W}<\/p>\n<p>Since, A and B have same number of elements which are exactly same, hence they are equal sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-from-the-sets-given-below-select-equal-sets-and-equivalent-sets-a-0-a-b-1-2-3-4-c-4-8-12-d-3-1-2-4-e-1-0-f-8-4-12-g-1-5-7-11-h-a-b\"><\/span>7. From the sets given below, select equal sets and equivalent sets.<br \/>A = {0, a}, B = {1, 2, 3, 4}, C = {4, 8, 12},<br \/>D = {3, 1, 2, 4}, E = {1, 0}, F = {8, 4, 12},<br \/>G = {1, 5, 7, 11}, H = {a, b}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A = {0, a}<\/p>\n<p>B = {1, 2, 3, 4}<\/p>\n<p>C = {4, 8, 12}<\/p>\n<p>D = {3, 1, 2, 4} = {1, 2, 3, 4}<\/p>\n<p>E = {1, 0}<\/p>\n<p>F = {8, 4, 12} = {4, 8, 12}<\/p>\n<p>G = {1, 5, 7, 11}<\/p>\n<p>H = {a, b}<\/p>\n<p>Equivalent sets:<\/p>\n<p>i. A, E, H (all of them have exactly two elements in them)<\/p>\n<p>ii. B, D, G (all of them have exactly four elements in them)<\/p>\n<p>iii. C, F (all of them have exactly three elements in them)<\/p>\n<p>Equal sets:<\/p>\n<p>i. B, D (all of them have exactly the same elements, so they are equal)<\/p>\n<p>ii. C, F (all of them have exactly the same elements, so they are equal)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"8-which-of-the-following-sets-are-equal-a-x-x-%e2%88%88-n-x-%3c-3-b-1-2-c-3-1-d-x-x-%e2%88%88-n-x-is-odd-x-%3c-5-e-1-2-1-1\"><\/span>8. Which of the following sets are equal?<br \/>A = {x: x\u00a0\u2208\u00a0N, x &lt; 3}<br \/>B = {1, 2}, C= {3, 1}<br \/>D = {x: x\u00a0\u2208\u00a0N, x is odd, x &lt; 5}<br \/>E = {1, 2, 1, 1}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"f-1-1-3\"><\/span>F = {1, 1, 3}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A = {1, 2}<\/p>\n<p>B = {1, 2}<\/p>\n<p>C = {3, 1}<\/p>\n<p>D = {1, 3} (since, the odd natural numbers less than 5 are 1 and 3)<\/p>\n<p>E = {1, 2} (since, repetition is not allowed)<\/p>\n<p>F = {1, 3} (since, repetition is not allowed)<\/p>\n<p>\u2234 Sets A, B and E are equal.<\/p>\n<p>C, D and F are equal.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"9-show-that-the-set-of-letters-needed-to-spell-%e2%80%9ccataract%e2%80%9d-and-the-set-of-letters-needed-to-spell-%e2%80%9ctract%e2%80%9d-are-equal\"><\/span>9. Show that the set of letters needed to spell \u201cCATARACT\u201d and the set of letters needed to spell \u201cTRACT\u201d are equal.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>For \u201cCATARACT\u201d<\/p>\n<p>Distinct letters are<\/p>\n<p>{C, A, T, R} = {A, C, R, T}<\/p>\n<p>For \u201cTRACT\u201d<\/p>\n<p>Distinct letters are<\/p>\n<p>{T, R, A, C} = {A, C, R, T}<\/p>\n<p>As we see letters need to spell cataract is equal to set of letters need to spell tract.<\/p>\n<p>Hence the two sets are equal.<\/p>\n<p>EXERCISE 1.4 PAGE NO: 1.16<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1-which-of-the-following-statements-are-true-give-a-reason-to-support-your-answer-i-for-any-two-sets-a-and-b-either-a-b-or-b-a-ii-every-subset-of-an-infinite-set-is-infinite-iii-every-subset-of-a-finite-set-is-finite-iv-every-set-has-a-proper-subset-v-a-b-a-b-a-b%e2%80%a6-is-an-infinite-set-vi-a-b-c-and-1-2-3-are-equivalent-sets-vii-a-set-can-have-infinitely-many-subsets\"><\/span>1. Which of the following statements are true? Give a reason to support your answer.<br \/>(i) For any two sets A and B either\u00a0A\u00a0B or B\u00a0A.<br \/>(ii) Every subset of an infinite set is infinite.<br \/>(iii) Every subset of a finite set is finite.<br \/>(iv) Every set has a proper subset.<br \/>(v) {a, b, a, b, a, b,\u2026.} is an infinite set.<br \/>(vi) {a, b, c} and {1, 2, 3} are equivalent sets.<br \/>(vii) A set can have infinitely many subsets.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0False<\/p>\n<p>No,\u00a0it is not necessary for any two set A and B to be either A\u00a0B or B\u00a0A.<\/p>\n<p><strong>(ii)<\/strong>\u00a0False<\/p>\n<p>A = {1,2,3} It is finite subset of infinite set N of natural numbers.<\/p>\n<p><strong>(iii)<\/strong>\u00a0True<\/p>\n<p>Logically smaller part of something finite can never be infinite.<\/p>\n<p>So, every subset of a finite set is finite.<\/p>\n<p><strong>(iv)<\/strong>\u00a0False<\/p>\n<p>Null set or empty set does not have a proper subset.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>We do not repeat elements in a set, so the given set becomes {a, b} which is a finite set.<\/p>\n<p><strong>(vi)<\/strong>\u00a0True<\/p>\n<p>Equivalent sets have same number of elements.<\/p>\n<p><strong>(vii)<\/strong>\u00a0False<\/p>\n<p>In A = {1}<\/p>\n<p>The subsets can be\u00a0\u03d5\u00a0and {1} which are finite.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-state-whether-the-following-statements-are-true-or-false-i-1-%e2%88%88-123-ii-a-%e2%8a%82-bca-iii-a-%e2%88%88-abc-iv-a-b-a-a-b-b-a-v-the-set-x-x-8-8-is-the-null-set\"><\/span>2.\u00a0State whether the following statements are true or false:<br \/>(i) 1 \u2208\u00a0{ 1,2,3}<br \/>(ii) a\u00a0\u2282\u00a0{b,c,a}<br \/>(iii) {a} \u2208\u00a0{a,b,c}<br \/>(iv) {a, b} = {a, a, b, b, a}<br \/>(v) The set {x: x + 8 = 8} is the null set.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0True<\/p>\n<p>1 belongs to the given set {1, 2, 3} as it is present in it.<\/p>\n<p><strong>(ii)<\/strong>\u00a0False<\/p>\n<p>Since, a is an element and not a subset of a set {b, c, a}<\/p>\n<p><strong>(iii)<\/strong>\u00a0False<\/p>\n<p>Since, {a} is a subset of set {b, c, a} and not an element.<\/p>\n<p><strong>(iv)<\/strong>\u00a0True<\/p>\n<p>We do not repeat same elements in a given set.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>Given, x+8 = 8<\/p>\n<p>i.e. x = 0<\/p>\n<p>So, the given set is a singleton set {0}. Where it is not a null set.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-decide-among-the-following-sets-which-are-subsets-of-which-a-x-x-satisfies-x2-%e2%80%93-8x-120-b-246-c-2468%e2%80%a6-d-6\"><\/span>3. Decide among the following sets, which are subsets of which:<br \/>A = {x: x satisfies x<sup>2<\/sup>\u00a0\u2013 8x + 12=0}, B = {2,4,6}, C = {2,4,6,8,\u2026.}, D = {6}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A = x<sup>2<\/sup>\u00a0\u2013 8x + 12=0<\/p>\n<p>\u21d2\u00a0(x\u20136) (x\u20132) =0<\/p>\n<p>\u21d2\u00a0x = 2 or x = 6<\/p>\n<p>A = {2, 6}<\/p>\n<p>B = {2, 4, 6}<\/p>\n<p>C = {2, 4, 6, 8}<\/p>\n<p>D = {6}<\/p>\n<p>So we can say<\/p>\n<p>D\u2282A\u2282B\u2282C<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-write-which-of-the-following-statements-are-true-justify-your-answer-i-the-set-of-all-integers-is-contained-in-the-set-of-all-rational-numbers-ii-the-set-of-all-crows-is-contained-in-the-set-of-all-birds-iii-the-set-of-all-rectangles-is-contained-in-the-set-of-all-squares-iv-the-set-of-all-rectangle-is-contained-in-the-set-of-all-squares-v-the-sets-p-a-and-b-a-are-equal-vi-the-sets-ax-x-is-a-letter-of-word-%e2%80%9clittle%e2%80%9d-and-b-x-x-is-a-letter-of-the-word-%e2%80%9ctitle%e2%80%9d-are-equal\"><\/span>4.\u00a0Write which of the following statements are true? Justify your answer.<br \/>(i) The set of all integers is contained in the set of all rational numbers.<br \/>(ii) The set of all crows is contained in the set of all birds.<br \/>(iii) The set of all rectangles is contained in the set of all squares.<br \/>(iv) The set of all rectangle is contained in the set of all squares.<br \/>(v) The sets P = {a} and B = {{a}} are equal.<br \/>(vi) The sets A={x: x is a letter of word \u201cLITTLE\u201d} AND, b = {x: x is a letter of the word \u201cTITLE\u201d} are equal.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0True<\/p>\n<p>A rational number is represented by the form p\/q where p and q are integers and (q not equal to 0) keeping q = 1 we can place any number as p. Which then will be an integer.<\/p>\n<p><strong>(ii)<\/strong>\u00a0True<\/p>\n<p>Crows are also birds, so they are contained in the set of all birds.<\/p>\n<p><strong>(iii)<\/strong>\u00a0False<\/p>\n<p>Every square can be a rectangle, but every rectangle cannot be a square.<\/p>\n<p><strong>(iv)<\/strong>\u00a0False<\/p>\n<p>Every square can be a rectangle, but every rectangle cannot be a square.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>P = {a}<\/p>\n<p>B = {{a}}<\/p>\n<p>But {a} = P<\/p>\n<p>B = {P}<\/p>\n<p>Hence they are not equal.<\/p>\n<p><strong>(vi)<\/strong>\u00a0True<\/p>\n<p>A = For \u201cLITTLE\u201d<\/p>\n<p>A = {L, I, T, E} = {E, I, L, T}<\/p>\n<p>B = For \u201cTITLE\u201d<\/p>\n<p>B = {T, I, L, E} = {E, I, L, T}<\/p>\n<p>\u2234 A = B<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-which-of-the-following-statements-are-correct-write-a-correct-form-of-each-of-the-incorrect-statements-i-a-%e2%8a%82-a-b-c-ii-a-a-b-c-iii-a-a-b-iv-a-%e2%8a%82-a-b-v-b-c-%e2%8a%82-ab-c-vi-a-b-%e2%8a%82-ab-c-vii-%cf%95-a-b-viii-%cf%95-%e2%8a%82-a-b-c-ix-x-x-3-3-%cf%95\"><\/span>5.\u00a0Which of the following statements are correct? Write a correct form of each of the incorrect statements.<br \/>(i) a\u00a0\u2282\u00a0{a, b, c}<br \/>(ii) {a}\u00a0{a, b, c}<br \/>(iii) a\u00a0{{a}, b}<br \/>(iv) {a}\u00a0\u2282\u00a0{{a}, b}<br \/>(v) {b, c}\u00a0\u2282\u00a0{a,{b, c}}<br \/>(vi) {a, b}\u00a0\u2282\u00a0{a,{b, c}}<br \/>(vii)\u00a0\u03d5\u00a0{a, b}<br \/>(viii)\u00a0\u03d5\u00a0\u2282\u00a0{a, b, c}<br \/>(ix) {x: x + 3 = 3}=\u00a0\u03d5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0This isn\u2019t subset of given set but belongs to the given set.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>a\u00a0\u2208{a,b,c}<\/p>\n<p><strong>(ii)<\/strong>\u00a0In this {a} is subset of {a, b, c}<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{a}\u00a0\u2282\u00a0{a, b, c}<\/p>\n<p><strong>(iii)<\/strong>\u00a0\u2018a\u2019 is not the element of the set.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{a}\u00a0\u2208 {{a}, b}<\/p>\n<p><strong>(iv)<\/strong>\u00a0{a} is not a subset of given set.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{a}\u00a0\u2208\u00a0{{a}, b}<\/p>\n<p><strong>(v)<\/strong>\u00a0{b, c} is not a subset of given set. But it belongs to the given set.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{b, c}\u00a0\u2208\u00a0{a,{b, c}}<\/p>\n<p><strong>(vi)<\/strong>\u00a0{a, b} is not a subset of given set.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{a, b}\u2284{a,{b, c}}<\/p>\n<p><strong>(vii)<\/strong>\u00a0\u03d5\u00a0does not belong to the given set but it is subset.<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>\u03d5\u00a0\u2282\u00a0{a, b}<\/p>\n<p><strong>(viii)<\/strong>\u00a0It is the correct form.\u00a0\u03d5\u00a0is subset of every set.<\/p>\n<p><strong>(ix)<\/strong>\u00a0x + 3 = 3<\/p>\n<p>x = 0 = {0}<\/p>\n<p>It is not\u00a0\u03d5<\/p>\n<p>\u2234\u00a0The correct form would be<\/p>\n<p>{x: x + 3 = 3} \u2260 \u03d5<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-let-a-a-bc-d-e-which-of-the-following-statements-are-false-and-why-i-c-d-%e2%8a%82-a-ii-c-d-a-iii-c-d-%e2%8a%82-a-iv-a-a-v-a-%e2%8a%82-a-vi-a-b-e-%e2%8a%82-a-vii-a-b-e-a-viii-a-b-c-%e2%8a%82-a-ix-%cf%95-a-x-%cf%95-%e2%8a%82-a\"><\/span>6.\u00a0Let A = {a, b,{c, d}, e}. Which of the following statements are false and why?<br \/>(i) {c, d}\u00a0\u2282\u00a0A<br \/>(ii) {c, d}\u00a0A<br \/>(iii) {{c, d}}\u00a0\u2282\u00a0A<br \/>(iv) a\u00a0A<br \/>(v) a\u00a0\u2282\u00a0A.<br \/>(vi) {a, b, e}\u00a0\u2282\u00a0A<br \/>(vii) {a, b, e}\u00a0A<br \/>(viii) {a, b, c}\u00a0\u2282\u00a0A<br \/>(ix)\u00a0\u03d5\u00a0A<br \/>(x) {\u03d5}\u00a0\u2282\u00a0A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0False<\/p>\n<p>{c, d} is not a subset of A but it belong to A.<\/p>\n<p>{c, d}\u00a0\u2208\u00a0A<\/p>\n<p><strong>(ii)<\/strong>\u00a0True<\/p>\n<p>{c, d}\u00a0\u2208\u00a0A<\/p>\n<p><strong>(iii)<\/strong>\u00a0True<\/p>\n<p>{c, d} is a subset of A.<\/p>\n<p><strong>(iv)<\/strong>\u00a0It is true that a belongs to A.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>a is not a subset of A but it belongs to A<\/p>\n<p><strong>(vi)<\/strong>\u00a0True<\/p>\n<p>{a, b, e} is a subset of A.<\/p>\n<p><strong>(vii)<\/strong>\u00a0False<\/p>\n<p>{a, b, e}\u00a0does not belong to A, {a, b, e}\u00a0\u2282\u00a0A this is the correct form.<\/p>\n<p><strong>(viii)<\/strong>\u00a0False<\/p>\n<p>{a, b, c} is not a subset of A<\/p>\n<p><strong>(ix)<\/strong>\u00a0False<\/p>\n<p>\u03d5\u00a0is a subset of A.<\/p>\n<p>\u03d5\u00a0\u2282\u00a0A.<\/p>\n<p><strong>(x)<\/strong>\u00a0False<\/p>\n<p>{\u03d5} is not subset of A, \u03d5 is a subset of A.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-let-a-1-2-3-4-5-6-7-8-determine-which-of-the-following-is-true-or-false-i-1-%e2%88%88-a-ii-1-2-3-%e2%8a%82-a-iii-6-7-8-%e2%88%88-a-iv-4-5-%e2%8a%82-a-v-%cf%95-%e2%88%88-a-vi-%cf%95-%e2%8a%82-a\"><\/span>7.\u00a0Let A = {{1, 2, 3}, {4, 5}, {6, 7, 8}}. Determine which of the following is true or false:<br \/>(i) 1 \u2208\u00a0A<br \/>(ii) {1, 2, 3}\u00a0\u2282\u00a0A<br \/>(iii) {6, 7, 8} \u2208\u00a0A<br \/>(iv) {4,\u00a05}\u00a0\u2282\u00a0A<br \/>(v) \u03d5\u00a0\u2208\u00a0A<br \/>(vi) \u03d5 \u2282\u00a0A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0False<\/p>\n<p>1 is not an element of A.<\/p>\n<p><strong>(ii)<\/strong>\u00a0True<\/p>\n<p>{1,2,3}\u00a0\u2208\u00a0A. this is correct form.<\/p>\n<p><strong>(iii)<\/strong>\u00a0True.<\/p>\n<p>{6, 7, 8}\u00a0\u2208 A.<\/p>\n<p><strong>(iv)<\/strong>\u00a0True<\/p>\n<p>{{4, 5}} is a subset of A.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>\u03a6 is a subset of A, not an element of A.<\/p>\n<p><strong>(vi)<\/strong>\u00a0True<\/p>\n<p>\u03a6 is a subset of every set, so it is a subset of A.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"8-let-a-%cf%95-%cf%95-1-1-%cf%95-2-which-of-the-following-are-true-i-%cf%95-%e2%88%88-a-ii-%cf%95-%e2%88%88-a-iii-1-%e2%88%88-a-iv-2-%cf%95-%e2%8a%82-a-v-2-%e2%8a%82-a-vi-2-1-%e2%8a%84a-vii-2-1-%e2%8a%84-a-viii-%cf%95-%cf%95-1-%cf%95-%e2%8a%82-a-ix-%cf%95-%e2%8a%82-a\"><\/span>8.\u00a0Let A = {\u03d5, {\u03d5}, 1, {1, \u03d5}, 2}. Which of the following are true?<br \/>(i)\u00a0\u03d5\u00a0\u2208\u00a0A<br \/>(ii) {\u03d5}\u00a0\u2208\u00a0A<br \/>(iii) {1}\u00a0\u2208\u00a0A<br \/>(iv) {2,\u00a0\u03d5}\u00a0\u2282\u00a0A<br \/>(v) 2\u00a0\u2282\u00a0A<br \/>(vi) {2, {1}}\u00a0\u2284A<br \/>(vii) {{2}, {1}}\u00a0\u2284 A<br \/>(viii) {\u03d5, {\u03d5}, {1, \u03d5}}\u00a0\u2282\u00a0A<br \/>(ix) {{\u03d5}}\u00a0\u2282\u00a0A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0True<\/p>\n<p>\u03a6 belongs to set A. Hence, true.<\/p>\n<p><strong>(ii)<\/strong>\u00a0True<\/p>\n<p>{\u03a6} is an element of set A. Hence, true.<\/p>\n<p><strong>(iii)<\/strong>\u00a0False<\/p>\n<p>1 is not an element of A. Hence, false.<\/p>\n<p><strong>(iv)<\/strong>\u00a0True<\/p>\n<p>{2,\u00a0\u03a6} is a subset of A. Hence, true.<\/p>\n<p><strong>(v)<\/strong>\u00a0False<\/p>\n<p>2 is not a subset of set A, it is an element of set A. Hence, false.<\/p>\n<p><strong>(vi)<\/strong>\u00a0True<\/p>\n<p>{2, {1}} is not a subset of set A. Hence, true.<\/p>\n<p><strong>(vii)<\/strong>\u00a0True<\/p>\n<p>Neither {2} and nor {1} is a subset of set A. Hence, true.<\/p>\n<p><strong>(viii)<\/strong>\u00a0True<\/p>\n<p>All three {\u03d5, {\u03d5}, {1, \u03d5}}<strong>\u00a0<\/strong>are subset of set A. Hence, true.<\/p>\n<p><strong>(ix)<\/strong>\u00a0True<\/p>\n<p>{{\u03d5}} is a subset of set A. Hence, true.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-15-page-no-121\"><\/span>EXERCISE 1.5 PAGE NO: 1.21<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-if-a-and-b-are-two-sets-such-that-a-%e2%8a%82-b-then-find-i-a-%e2%8b%82-b\"><\/span><strong>1. If A and B are two sets such that A\u00a0\u2282\u00a0B, then Find:<br \/>(i) A\u00a0\u22c2\u00a0B<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%8b%83-b\"><\/span><strong>(ii) A \u22c3 B<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0A\u00a0\u2229\u00a0B<\/p>\n<p>A\u00a0\u2229\u00a0B denotes A intersection B. Common elements of A and B consists in this set.<\/p>\n<p>Given A\u00a0\u2282\u00a0B, every element of A are already an element of B.<\/p>\n<p>\u2234\u00a0A\u00a0\u2229\u00a0B = A<\/p>\n<p><strong>(ii)\u00a0<\/strong>A \u22c3 B<\/p>\n<p>A\u00a0\u222a\u00a0B denotes A union B. Elements of either A or B or in both A and B consist in this set.<\/p>\n<p>Given A\u00a0\u2282\u00a0B, B is having all elements including elements of A.<\/p>\n<p>\u2234\u00a0A\u00a0\u222a\u00a0B = B<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-if-a-1-2-3-4-5-b-4-5-6-7-8-c-7-8-9-10-11-and-d-10-11-12-13-14-find-i-a-%e2%88%aa-b-ii-a-%e2%88%aa-c-iii-b-%e2%88%aa-c-iv-b-%e2%88%aa-d-v-a-%e2%88%aa-b-%e2%88%aa-c-vi-a-%e2%88%aa-b-%e2%88%aa-d-vii-b-%e2%88%aa-c-%e2%88%aa-d-viii-a-%e2%88%a9-b-%e2%88%aa-c-ix-a-%e2%88%a9-b-%e2%88%a9-b-%e2%88%a9-c-x-a-%e2%88%aa-d-%e2%88%a9-b-%e2%88%aa-c\"><\/span>2.\u00a0If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10, 11} and D = {10, 11, 12, 13, 14}. Find:<br \/>(i) A\u00a0\u222a\u00a0B<br \/>(ii) A\u00a0\u222a\u00a0C<br \/>(iii) B\u00a0\u222a\u00a0C<br \/>(iv) B\u00a0\u222a\u00a0D<br \/>(v) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0C<br \/>(vi) A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0D<br \/>(vii) B\u00a0\u222a\u00a0C\u00a0\u222a\u00a0D<br \/>(viii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C)<br \/>(ix) (A\u00a0\u2229\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u2229\u00a0C)<strong><br \/><\/strong>(x) (A\u00a0\u222a\u00a0D)\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C).<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>In general X\u00a0\u222a\u00a0Y = {a: a\u00a0<strong>\u2208<\/strong>\u00a0X or a\u00a0<strong>\u2208<\/strong>\u00a0Y}<\/p>\n<p>X\u00a0\u2229\u00a0Y = {a: a\u00a0<strong>\u2208<\/strong>\u00a0X and a\u00a0<strong>\u2208<\/strong>\u00a0Y}.<\/p>\n<p><strong>(i)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>A\u00a0\u222a\u00a0B = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n<p><strong>(ii)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>A\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}<\/p>\n<p><strong>(iii)<\/strong>\u00a0B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5, 6, 7, 8, 9, 10, 11}<\/p>\n<p><strong>(iv)<\/strong>\u00a0B = {4, 5, 6, 7, 8}<\/p>\n<p>D = {10, 11, 12, 13, 14}<\/p>\n<p>B\u00a0\u222a\u00a0D = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0D}<\/p>\n<p>= {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}<\/p>\n<p><strong>(v)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>A\u00a0\u222a\u00a0B = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n<p>A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A\u00a0\u222a\u00a0B\u00a0or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}<\/p>\n<p><strong>(vi)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>D = {10, 11, 12, 13, 14}<\/p>\n<p>A\u00a0\u222a\u00a0B = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {1, 2, 3, 4, 5, 6, 7, 8}<\/p>\n<p>A\u00a0\u222a\u00a0B\u00a0\u222a\u00a0D = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A\u00a0\u222a\u00a0B\u00a0or x\u00a0<strong>\u2208<\/strong>\u00a0D}<\/p>\n<p>= {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}<\/p>\n<p><strong>(vii)<\/strong>\u00a0B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>D = {10, 11, 12, 13, 14}<\/p>\n<p>B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5, 6, 7, 8, 9, 10, 11}<\/p>\n<p>B\u00a0\u222a\u00a0C\u00a0\u222a\u00a0D = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B\u00a0\u222a\u00a0C\u00a0or x\u00a0<strong>\u2208<\/strong>\u00a0D}<\/p>\n<p>= {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}<\/p>\n<p><strong>(viii)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5, 6, 7, 8, 9, 10, 11}<\/p>\n<p>A\u00a0\u2229\u00a0B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0B\u00a0\u222a\u00a0C}<\/p>\n<p>= {4, 5}<\/p>\n<p><strong>(ix)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>(A\u00a0\u2229\u00a0B) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {4, 5}<\/p>\n<p>(B\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B and x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {7, 8}<\/p>\n<p>(A\u00a0\u2229\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A\u00a0\u2229\u00a0B) and x\u00a0<strong>\u2208<\/strong>\u00a0(B\u00a0\u2229\u00a0C)}<\/p>\n<p>=\u00a0\u03d5<\/p>\n<p><strong>(x)<\/strong>\u00a0A = {1, 2, 3, 4, 5}<\/p>\n<p>B = {4, 5, 6, 7, 8}<\/p>\n<p>C = {7, 8, 9, 10, 11}<\/p>\n<p>D = {10, 11, 12, 13, 14}<\/p>\n<p>A\u00a0\u222a\u00a0D = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0D}<\/p>\n<p>= {1, 2, 3, 4, 5, 10, 11, 12, 13, 14}<\/p>\n<p>B\u00a0\u222a\u00a0C = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5, 6, 7, 8, 9, 10, 11}<\/p>\n<p>(A\u00a0\u222a\u00a0D)\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A\u00a0\u222a\u00a0D) and x\u00a0<strong>\u2208<\/strong>\u00a0(B\u00a0\u222a\u00a0C)}<\/p>\n<p>= {4, 5, 10, 11}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-let-a-x-x-%e2%88%88-n-b-x-x-2n-n-%e2%88%88-n-c-x-x-2n-%e2%80%93-1-n-%e2%88%88-n-and-d-x-x-is-a-prime-natural-number-find-i-a-%e2%88%a9-b-ii-a-%e2%88%a9-c-iii-a-%e2%88%a9-d-iv-b-%e2%88%a9-c-v-b-%e2%88%a9-d-vi-c-%e2%88%a9-d\"><\/span>3.\u00a0Let A = {x: x\u00a0\u2208\u00a0N}, B = {x: x = 2n, n\u00a0\u2208\u00a0N), C = {x: x = 2n \u2013 1, n\u00a0\u2208\u00a0N} and, D = {x: x is a prime natural number} Find:<br \/>(i) A\u00a0\u2229\u00a0B<br \/>(ii) A\u00a0\u2229\u00a0C<br \/>(iii) A\u00a0\u2229\u00a0D<br \/>(iv) B\u00a0\u2229\u00a0C<br \/>(v) B\u00a0\u2229\u00a0D<br \/>(vi) C\u00a0\u2229\u00a0D<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A = All natural numbers i.e. {1, 2, 3\u2026..}<\/p>\n<p>B = All even natural numbers i.e. {2, 4, 6, 8\u2026}<\/p>\n<p>C = All odd natural numbers i.e. {1, 3, 5, 7\u2026\u2026}<\/p>\n<p>D = All prime natural numbers i.e. {1, 2, 3, 5, 7, 11, \u2026}<\/p>\n<p><strong>(i)<\/strong>\u00a0A\u00a0\u2229\u00a0B<\/p>\n<p>A contains all elements of B.<\/p>\n<p>\u2234\u00a0B\u00a0\u2282\u00a0A = {2, 4, 6, 8\u2026}<\/p>\n<p>\u2234\u00a0A\u00a0\u2229\u00a0B = B<\/p>\n<p><strong>(ii)<\/strong>\u00a0A\u00a0\u2229\u00a0C<\/p>\n<p>A contains all elements of C.<\/p>\n<p>\u2234\u00a0C\u00a0\u2282\u00a0A = {1, 3, 5\u2026}<\/p>\n<p>\u2234\u00a0A\u00a0\u2229\u00a0C = C<\/p>\n<p><strong>(iii)<\/strong>\u00a0A\u00a0\u2229\u00a0D<\/p>\n<p>A contains all elements of D.<\/p>\n<p>\u2234\u00a0D\u00a0\u2282\u00a0A = {2, 3, 5, 7..}<\/p>\n<p>\u2234\u00a0A\u00a0\u2229\u00a0D = D<\/p>\n<p><strong>(iv)<\/strong>\u00a0B\u00a0\u2229\u00a0C<\/p>\n<p>B\u00a0\u2229\u00a0C =\u00a0\u03d5<\/p>\n<p>There is no natural number which is both even and odd at same time.<\/p>\n<p><strong>(v)<\/strong>\u00a0B\u00a0\u2229\u00a0D<\/p>\n<p>B\u00a0\u2229\u00a0D = 2<\/p>\n<p>{2} is the only natural number which is even and a prime number.<\/p>\n<p><strong>(vi)<\/strong>\u00a0C\u00a0\u2229\u00a0D<\/p>\n<p>C\u00a0\u2229\u00a0D = {1, 3, 5, 7\u2026}<\/p>\n<p>= D \u2013 {2}<\/p>\n<p>Every prime number is odd except {2}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-16-page-no-127\"><\/span>EXERCISE 1.6 PAGE NO: 1.27<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-find-the-smallest-set-a-such-that-a-%e2%88%aa-1-2-1-2-3-5-9\"><\/span>1. Find the smallest set A such that A\u00a0\u222a\u00a0{1, 2} = {1, 2, 3, 5, 9}.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A\u00a0\u222a\u00a0{1, 2} = {1, 2, 3, 5, 9}<\/p>\n<p>Elements of A and {1, 2} together give us the result<\/p>\n<p>So smallest set of A can be<\/p>\n<p>A = {1, 2, 3, 5, 9} \u2013 {1, 2}<\/p>\n<p>A = {3, 5, 9}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-let-a-1-2-4-5-b-2-3-5-6-c-4-5-6-7-verify-the-following-identities-i-a-%e2%88%aa-b-%e2%88%a9-c-a-%e2%88%aa-b-%e2%88%a9-a-%e2%88%aa-c\"><\/span>2.\u00a0Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the following identities:<br \/>(i) A\u00a0\u222a\u00a0(B\u00a0\u2229\u00a0C) = (A\u00a0\u222a\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%88%a9-b-%e2%88%aa-c-a-%e2%88%a9-b-%e2%88%aa-a-%e2%88%a9-c\"><\/span>(ii) A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-%e2%88%a9-b-%e2%80%93-c-a-%e2%88%a9-b-%e2%80%93-a-%e2%88%a9-c\"><\/span>(iii) A\u00a0\u2229\u00a0(B \u2013 C) = (A\u00a0\u2229\u00a0B) \u2013 (A\u00a0\u2229\u00a0C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-%e2%80%93-b-%e2%88%aa-c-a-%e2%80%93-b-%e2%88%a9-a-%e2%80%93-c\"><\/span>(iv) A \u2013 (B\u00a0\u222a\u00a0C) = (A \u2013 B)\u00a0\u2229\u00a0(A \u2013 C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"v-a-%e2%80%93-b-%e2%88%a9-c-a-%e2%80%93-b-%e2%88%aa-a-%e2%80%93-c\"><\/span>(v) A \u2013 (B\u00a0\u2229\u00a0C) = (A \u2013 B)\u00a0\u222a\u00a0(A \u2013 C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"vi-a-%e2%88%a9-b-%e2%96%b3-c-a-%e2%88%a9-b-%e2%96%b3-a-%e2%88%a9-c\"><\/span>(vi) A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u25b3\u00a0(A\u00a0\u2229\u00a0C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0A\u00a0\u222a\u00a0(B\u00a0\u2229\u00a0C) = (A\u00a0\u222a\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0C)<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(B\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B and x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {5, 6}<\/p>\n<p>A\u00a0\u222a\u00a0(B\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0(B\u00a0\u2229\u00a0C)}<\/p>\n<p>= {1, 2, 4, 5, 6}<\/p>\n<p>Now, RHS<\/p>\n<p>(A\u00a0\u222a\u00a0B) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {1, 2, 4, 5, 6}.<\/p>\n<p>(A\u00a0\u222a\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {1, 2, 4, 5, 6, 7}<\/p>\n<p>(A\u00a0\u222a\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A\u00a0\u222a\u00a0B) and x\u00a0<strong>\u2208<\/strong>\u00a0(A\u00a0\u222a\u00a0C)}<\/p>\n<p>= {1, 2, 4, 5, 6}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence Verified.<\/p>\n<p><strong>(ii)<\/strong>\u00a0A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0C)<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(B\u00a0\u222a\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {2, 3, 4, 5, 6, 7}<\/p>\n<p>(A\u00a0\u2229\u00a0(B\u00a0\u222a\u00a0C)) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0(B\u00a0\u222a\u00a0C)}<\/p>\n<p>= {2, 4, 5}<\/p>\n<p>Now, RHS<\/p>\n<p>(A\u00a0\u2229\u00a0B) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {2, 5}<\/p>\n<p>(A\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5}<\/p>\n<p>(A\u00a0\u2229\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A\u2229B) and x\u00a0<strong>\u2208<\/strong>\u00a0(A\u2229C)}<\/p>\n<p>= {2, 4, 5}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence verified.<\/p>\n<p><strong>(iii)\u00a0<\/strong>A\u00a0\u2229\u00a0(B \u2013 C) = (A\u00a0\u2229\u00a0B) \u2013 (A\u00a0\u2229\u00a0C)<\/p>\n<p>B\u2013C is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0B: x\u00a0\u2209\u00a0C}<\/p>\n<p>B = {2, 3, 5, 6}<\/p>\n<p>C = {4, 5, 6, 7}<\/p>\n<p>B\u2013C = {2, 3}<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(A\u00a0\u2229\u00a0(B \u2013 C)) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0(B \u2013 C)}<\/p>\n<p>= {2}<\/p>\n<p>Now, RHS<\/p>\n<p>(A\u00a0\u2229\u00a0B) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {2, 5}<\/p>\n<p>(A\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {4, 5}<\/p>\n<p>(A\u00a0\u2229\u00a0B) \u2013 (A\u00a0\u2229\u00a0C) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0(A\u00a0\u2229\u00a0B): x\u00a0\u2209\u00a0(A\u00a0\u2229\u00a0C)}<\/p>\n<p>= {2}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence Verified.<\/p>\n<p><strong>(iv)<\/strong>\u00a0A \u2013 (B\u00a0\u222a\u00a0C) = (A \u2013 B)\u00a0\u2229\u00a0(A \u2013 C)<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(B\u00a0\u222a\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B or x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {2, 3, 4, 5, 6, 7}.<\/p>\n<p>A \u2013 (B\u00a0\u222a\u00a0C) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209\u00a0(B\u00a0\u222a\u00a0C)}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>(B\u00a0\u222a\u00a0C)\u00a0= {2, 3, 4, 5, 6, 7}<\/p>\n<p>A \u2013 (B\u00a0\u222a\u00a0C) = {1}<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u2013 B)<\/p>\n<p>A \u2013 B is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209\u00a0B}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>B = {2, 3, 5, 6}<\/p>\n<p>A \u2013 B = {1, 4}<\/p>\n<p>(A \u2013 C)<\/p>\n<p>A \u2013 C is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209\u00a0C}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>C = {4, 5, 6, 7}<\/p>\n<p>A \u2013 C = {1, 2}<\/p>\n<p>(A \u2013 B)\u00a0\u2229\u00a0(A \u2013 C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A \u2013 B) and x\u00a0<strong>\u2208<\/strong>\u00a0(A \u2013 C)}.<\/p>\n<p>= {1}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence verified.<\/p>\n<p><strong>(v)<\/strong>\u00a0A \u2013 (B\u00a0\u2229\u00a0C) = (A \u2013 B)\u00a0\u222a\u00a0(A \u2013 C)<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(B\u00a0\u2229\u00a0C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0B and x\u00a0<strong>\u2208<\/strong>\u00a0C}<\/p>\n<p>= {5, 6}<\/p>\n<p>A \u2013 (B\u00a0\u2229\u00a0C) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209 (B\u00a0\u2229\u00a0C)}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>(B\u00a0\u2229\u00a0C) = {5, 6}<\/p>\n<p>(A \u2013 (B\u00a0\u2229\u00a0C)) = {1, 2, 4}<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u2013 B)<\/p>\n<p>A \u2013 B is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209\u00a0B}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>B = {2, 3, 5, 6}<\/p>\n<p>A\u2013B = {1, 4}<\/p>\n<p>(A \u2013 C)<\/p>\n<p>A \u2013 C is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0A: x\u00a0\u2209\u00a0C}<\/p>\n<p>A = {1, 2, 4, 5}<\/p>\n<p>C = {4, 5, 6, 7}<\/p>\n<p>A \u2013 C = {1, 2}<\/p>\n<p>(A \u2013 B)\u00a0\u222a\u00a0(A \u2013 C) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0(A \u2013 B) OR x\u00a0<strong>\u2208<\/strong>\u00a0(A \u2013 C)}.<\/p>\n<p>= {1, 2, 4}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence verified.<\/p>\n<p><strong>(vi)<\/strong>\u00a0A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C) = (A\u00a0\u2229\u00a0B)\u00a0\u25b3\u00a0(A\u00a0\u2229\u00a0C)<\/p>\n<p>A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}.<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C)<\/p>\n<p>B\u00a0\u25b3\u00a0C = (B \u2013 C) \u222a (C \u2013 B) = {2, 3} \u222a {4, 7} = {2, 3, 4, 7}<\/p>\n<p>A\u00a0\u2229\u00a0(B\u00a0\u25b3\u00a0C) = {2, 4}<\/p>\n<p>Now, RHS<\/p>\n<p>A\u00a0\u2229\u00a0B = {2, 5}<\/p>\n<p>A\u00a0\u2229\u00a0C = {4, 5}<\/p>\n<p>(A\u00a0\u2229\u00a0B)\u00a0\u25b3\u00a0(A\u00a0\u2229\u00a0C) = [(A\u00a0\u2229\u00a0B)\u00a0\u2013 (A\u00a0\u2229\u00a0C)] \u222a [(A\u00a0\u2229\u00a0C) \u2013 (A\u00a0\u2229\u00a0B)]<\/p>\n<p>= {2} \u222a {4}<\/p>\n<p>= {2, 4}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence, Verified.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-if-u-2-3-5-7-9-is-the-universal-set-and-a-3-7-b-2-5-7-9-then-prove-that-i-a-%e2%88%aa-b%e2%80%99-a%e2%80%99-%e2%88%a9-b%e2%80%99\"><\/span>3.\u00a0If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:<br \/>(i) (A\u00a0\u222a\u00a0B)\u2019 = A\u2019\u00a0\u2229\u00a0B\u2019<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%88%a9-b%e2%80%99-a%e2%80%99-%e2%88%aa-b%e2%80%99\"><\/span>(ii) (A\u00a0\u2229\u00a0B)\u2019 = A\u2019\u00a0\u222a\u00a0B\u2019<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>(A\u00a0\u222a\u00a0B)\u2019 = A\u2019\u00a0\u2229\u00a0B\u2019<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>A\u00a0\u222a\u00a0B = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {2, 3, 5, 7, 9}<\/p>\n<p>(A\u222aB)\u2019 means Complement of (A\u222aB) with respect to universal set U.<\/p>\n<p>So, (A\u222aB)\u2019 = U \u2013 (A\u222aB)\u2019<\/p>\n<p>U \u2013 (A\u222aB)\u2019 is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0(A\u222aB)\u2019}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>(A\u222aB)\u2019 = {2, 3, 5, 7, 9}<\/p>\n<p>U \u2013 (A\u222aB)\u2019 =\u00a0\u03d5<\/p>\n<p>Now, RHS<\/p>\n<p>A\u2019 means Complement of A with respect to universal set U.<\/p>\n<p>So, A\u2019 = U \u2013 A<\/p>\n<p>(U \u2013 A) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0A}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>A = {3, 7}<\/p>\n<p>A\u2019 = U \u2013 A = {2, 5, 9}<\/p>\n<p>B\u2019 means Complement of B with respect to universal set U.<\/p>\n<p>So, B\u2019 = U \u2013 B<\/p>\n<p>(U \u2013 B) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0B}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>B = {2, 5, 7, 9}<\/p>\n<p>B\u2019 = U \u2013 B = {3}<\/p>\n<p>A\u2019\u00a0\u2229\u00a0B\u2019 = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A\u2019 and x\u00a0<strong>\u2208<\/strong>\u00a0C\u2019}.<\/p>\n<p>=\u00a0\u03d5<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence verified.<\/p>\n<p><strong>(ii)\u00a0<\/strong>(A\u00a0\u2229\u00a0B)\u2019 = A\u2019\u00a0\u222a\u00a0B\u2019<\/p>\n<p>Firstly let us consider the LHS<\/p>\n<p>(A\u00a0\u2229\u00a0B)\u2019<\/p>\n<p>(A\u00a0\u2229\u00a0B) = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A and x\u00a0<strong>\u2208<\/strong>\u00a0B}.<\/p>\n<p>= {7}<\/p>\n<p>(A\u2229B)\u2019 means Complement of (A \u2229 B) with respect to universal set U.<\/p>\n<p>So, (A\u2229B)\u2019 = U \u2013 (A \u2229 B)<\/p>\n<p>U \u2013 (A \u2229 B) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0(A \u2229 B)\u2019}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>(A \u2229 B) = {7}<\/p>\n<p>U \u2013 (A \u2229 B) = {2, 3, 5, 9}<\/p>\n<p>(A \u2229 B)\u2019 = {2, 3, 5, 9}<\/p>\n<p>Now, RHS<\/p>\n<p>A\u2019 means Complement of A with respect to universal set U.<\/p>\n<p>So, A\u2019 = U \u2013 A<\/p>\n<p>(U \u2013 A) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0A}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>A = {3, 7}<\/p>\n<p>A\u2019 = U \u2013 A = {2, 5, 9}<\/p>\n<p>B\u2019 means Complement of B with respect to universal set U.<\/p>\n<p>So, B\u2019 = U \u2013 B<\/p>\n<p>(U \u2013 B) is defined as {x\u00a0<strong>\u2208<\/strong>\u00a0U: x\u00a0\u2209\u00a0B}<\/p>\n<p>U = {2, 3, 5, 7, 9}<\/p>\n<p>B = {2, 5, 7, 9}<\/p>\n<p>B\u2019 = U \u2013 B = {3}<\/p>\n<p>A\u2019\u00a0\u222a\u00a0B\u2019 = {x: x\u00a0<strong>\u2208<\/strong>\u00a0A or x\u00a0<strong>\u2208<\/strong>\u00a0B}<\/p>\n<p>= {2, 3, 5, 9}<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence verified.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-for-any-two-sets-a-and-b-prove-that-i-b-%e2%8a%82-a-%e2%88%aa-b\"><\/span>4.\u00a0For any two sets A and B, prove that<br \/>(i) B\u00a0\u2282\u00a0A\u00a0\u222a\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%88%a9-b-%e2%8a%82-a\"><\/span>(ii) A\u00a0\u2229\u00a0B\u00a0\u2282\u00a0A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-%e2%8a%82-b-%e2%87%92-a-%e2%88%a9-b-a\"><\/span>(iii) A\u00a0\u2282\u00a0B \u21d2\u00a0A\u00a0\u2229\u00a0B = A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>B\u00a0\u2282\u00a0A\u00a0\u222a\u00a0B<\/p>\n<p>Let us consider an element \u2018p\u2019 such that it belongs to B<\/p>\n<p>\u2234\u00a0p\u00a0\u2208\u00a0B<\/p>\n<p>p\u00a0\u2208\u00a0B\u00a0\u222a\u00a0A<\/p>\n<p>B\u00a0\u2282\u00a0A\u00a0\u222a\u00a0B<\/p>\n<p><strong>(ii)\u00a0<\/strong>A\u00a0\u2229\u00a0B\u00a0\u2282\u00a0A<\/p>\n<p>Let us consider an element \u2018p\u2019 such that it belongs to B<\/p>\n<p>\u2234\u00a0p\u00a0\u2208\u00a0A\u00a0\u2229\u00a0B<\/p>\n<p>p\u00a0\u2208\u00a0A and p\u00a0\u2208\u00a0B<\/p>\n<p>A\u00a0\u2229\u00a0B\u00a0\u2282\u00a0A<\/p>\n<p><strong>(iii)\u00a0<\/strong>A\u00a0\u2282\u00a0B \u21d2\u00a0A\u00a0\u2229\u00a0B = A<\/p>\n<p>Let us consider an element \u2018p\u2019 such that it belongs to A\u00a0\u2282\u00a0B.<\/p>\n<p>p\u00a0\u2208\u00a0A\u00a0\u2282\u00a0B<\/p>\n<p>Then, x\u00a0\u2208\u00a0B<\/p>\n<p>Let and p\u00a0\u2208\u00a0A\u00a0\u2229\u00a0B<\/p>\n<p>x\u00a0\u2208\u00a0A and x\u00a0\u2208\u00a0B<\/p>\n<p>x\u00a0\u2208\u00a0A and x\u00a0\u2208\u00a0A (since, A\u00a0\u2282\u00a0B)<\/p>\n<p>\u2234\u00a0(A\u00a0\u2229\u00a0B) = A<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-for-any-two-sets-a-and-b-show-that-the-following-statements-are-equivalent-i-a-%e2%8a%82-b\"><\/span>5.\u00a0For any two sets A and B, show that the following statements are equivalent:<br \/>(i) A\u00a0\u2282\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%80%93-b-%cf%95\"><\/span>(ii) A \u2013 B\u00a0=\u00a0\u03d5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-%e2%88%aa-b-b\"><\/span>(iii) A\u00a0\u222a\u00a0B = B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-%e2%88%a9-b-a\"><\/span>(iv) A\u00a0\u2229\u00a0B = A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0A\u00a0\u2282\u00a0B<\/p>\n<p>To\u00a0show that the following four statements are equivalent, we need to prove (i)=(ii),\u00a0(ii)=(iii),\u00a0(iii)=(iv), (iv)=(v)<\/p>\n<p>Firstly let us prove (i)=(ii)<\/p>\n<p>We know, A\u2013B = {x \u2208\u00a0A: x\u00a0\u2209\u00a0B} as A\u00a0\u2282\u00a0B,<\/p>\n<p>So, Each element of A is an element of B,<\/p>\n<p>\u2234\u00a0A\u2013B =\u00a0\u03d5<\/p>\n<p>Hence, (i)=(ii)<\/p>\n<p><strong>(ii)<\/strong>\u00a0A \u2013 B\u00a0=\u00a0\u03d5<\/p>\n<p>We need to show that (ii)=(iii)<\/p>\n<p>By assuming A\u2013B =\u00a0\u03d5<\/p>\n<p>To show: A\u222aB = B<\/p>\n<p>\u2234\u00a0Every element of A is an element of B<\/p>\n<p>So, A\u00a0\u2282\u00a0B and so A\u222aB = B<\/p>\n<p>Hence, (ii)=(iii)<\/p>\n<p><strong>(iii)<\/strong>\u00a0A\u00a0\u222a\u00a0B = B<\/p>\n<p>We need to show that (iii)=(iv)<\/p>\n<p>By assuming A\u00a0\u222a\u00a0B = B<\/p>\n<p>To show: A\u00a0\u2229\u00a0B = A.<\/p>\n<p>\u2234\u00a0A\u2282\u00a0B\u00a0and so A\u00a0\u2229\u00a0B = A<\/p>\n<p>Hence, (iii)=(iv)<\/p>\n<p><strong>(iv)<\/strong>\u00a0A\u00a0\u2229\u00a0B = A<\/p>\n<p>Finally,\u00a0now we need to show (iv)=(i)<\/p>\n<p>By assuming A\u00a0\u2229\u00a0B = A<\/p>\n<p>To show: A\u00a0\u2282\u00a0B<\/p>\n<p>Since, A\u00a0\u2229\u00a0B = A, so A\u2282B<\/p>\n<p>Hence, (iv)=(i)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-for-three-sets-a-b-and-c-show-that-i-a-%e2%88%a9-b-a-%e2%88%a9-c-need-not-imply-b-c\"><\/span>6. For three sets A, B, and C, show that<br \/>(i) A\u00a0\u2229\u00a0B = A\u00a0\u2229\u00a0C need not imply B = C.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%8a%82-b-%e2%87%92-c-%e2%80%93-b-%e2%8a%82-c-%e2%80%93-a\"><\/span>(ii) A\u00a0\u2282\u00a0B \u21d2\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>A\u00a0\u2229\u00a0B = A\u00a0\u2229\u00a0C need not imply B = C.<\/p>\n<p>Let us consider, A = {1, 2}<\/p>\n<p>B = {2, 3}<\/p>\n<p>C = {2, 4}<\/p>\n<p>Then,<\/p>\n<p>A\u00a0\u2229\u00a0B = {2}<\/p>\n<p>A\u00a0\u2229\u00a0C = {2}<\/p>\n<p>Hence, A\u00a0\u2229\u00a0B = A\u00a0\u2229\u00a0C, where, B is not equal to C<\/p>\n<p><strong>(ii)\u00a0<\/strong>A\u00a0\u2282\u00a0B \u21d2\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A<\/p>\n<p>Given:\u00a0A\u00a0\u2282\u00a0B<\/p>\n<p>To show: C\u2013B\u00a0\u2282\u00a0C\u2013A<\/p>\n<p>Let us consider x\u00a0\u2208\u00a0C\u2013 B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C and x\u00a0\u2209\u00a0B [by definition C\u2013B]<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C and x\u00a0\u2209\u00a0A<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C\u2013A<\/p>\n<p>Thus x\u00a0\u2208\u00a0C\u2013B\u00a0\u21d2\u00a0x\u00a0\u2208\u00a0C\u2013A. This is true for all x\u00a0\u2208\u00a0C\u2013B.<\/p>\n<p>\u2234 A\u00a0\u2282\u00a0B \u21d2\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-for-any-two-sets-prove-that-i-a-%e2%88%aa-a-%e2%88%a9-b-a\"><\/span>7.\u00a0For any two sets, prove that:<br \/>(i) A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%88%a9-a-%e2%88%aa-b-a\"><\/span>(ii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B) = A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<\/p>\n<p>We know union is distributive over intersection<\/p>\n<p>So, A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>(A\u00a0\u222a\u00a0A)\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B) [Since,\u00a0A\u00a0\u222a\u00a0A = A]<\/p>\n<p>A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B)<\/p>\n<p>A<\/p>\n<p><strong>(ii)\u00a0<\/strong>A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B) = A<\/p>\n<p>We know union is distributive over intersection<\/p>\n<p>So, (A\u00a0\u2229\u00a0A)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>A\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) [Since,\u00a0A\u00a0\u2229\u00a0A = A]<\/p>\n<p>A<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-17-page-no-134\"><\/span>EXERCISE 1.7 PAGE NO: 1.34<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-for-any-two-sets-a-and-b-prove-that-a%e2%80%98-%e2%80%93-b%e2%80%98-b-%e2%80%93-a\"><\/span>1. For any two sets A and B, prove that: A\u2018\u00a0\u2013 B\u2018\u00a0= B \u2013 A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>To prove, A\u2019 \u2013 B\u2019 = B \u2013 A<\/p>\n<p>Firstly we need to show<\/p>\n<p>A\u2019 \u2013 B\u2019\u00a0\u2286\u00a0B \u2013 A<\/p>\n<p>Let,\u00a0x\u00a0\u2208\u00a0A\u2019 \u2013 B\u2019<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A\u2019 and x\u00a0\u2209\u00a0B\u2019<\/p>\n<p>\u21d2\u00a0x\u00a0\u2209\u00a0A and x\u00a0\u2208\u00a0B (since, A\u00a0\u2229\u00a0A\u2019 = \u03d5 )<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0B \u2013 A<\/p>\n<p>It is true for all x\u00a0\u2208\u00a0A\u2019 \u2013 B\u2019<\/p>\n<p>\u2234\u00a0A\u2019 \u2013 B\u2019 = B \u2013 A<\/p>\n<p>Hence Proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-for-any-two-sets-a-and-b-prove-the-following-i-a-%e2%88%a9-a%e2%80%98-%e2%88%aa-b-a-%e2%88%a9-b\"><\/span>2. For any two sets A and B, prove the following:<br \/>(i) A\u00a0\u2229\u00a0(A\u2018\u00a0\u222a\u00a0B) = A\u00a0\u2229\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%80%93-a-%e2%80%93-b-a-%e2%88%a9-b\"><\/span>(ii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-%e2%88%a9-a-%e2%88%aa-b%e2%80%99-%cf%95\"><\/span>(iii) A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B\u2019) =\u00a0\u03d5<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-%e2%80%93-b-a-%ce%94-a-%e2%88%a9-b\"><\/span>(iv) A \u2013 B = A \u0394 (A\u00a0\u2229\u00a0B)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>A\u00a0\u2229\u00a0(A\u2019\u00a0\u222a\u00a0B) = A\u00a0\u2229\u00a0B<\/p>\n<p>Let us consider LHS A\u00a0\u2229\u00a0(A\u2019\u00a0\u222a\u00a0B)<\/p>\n<p>Expanding<\/p>\n<p>(A\u00a0\u2229\u00a0A\u2019)\u00a0\u222a\u00a0(A \u2229\u00a0B)<\/p>\n<p>We know, (A\u00a0\u2229\u00a0A\u2019) =\u03d5<\/p>\n<p>\u21d2\u00a0\u03d5\u00a0\u222a\u00a0(A\u2229\u00a0B)<\/p>\n<p>\u21d2\u00a0(A \u2229\u00a0B)<\/p>\n<p>\u2234 LHS = RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(ii)\u00a0<\/strong>A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<\/p>\n<p>For any sets A and B we have De-Morgan\u2019s law<\/p>\n<p>(A\u00a0\u222a B)\u2019 = A\u2019 \u2229 B\u2019, (A\u00a0\u2229\u00a0B) \u2018 = A\u2019\u00a0\u222a\u00a0B\u2019<\/p>\n<p>Consider LHS<\/p>\n<p>= A \u2013 (A\u2013B)<\/p>\n<p>= A\u00a0\u2229\u00a0(A\u2013B)\u2019<\/p>\n<p>= A \u2229 (A\u2229B\u2019)\u2019<\/p>\n<p>= A \u2229 (A\u2019 \u222a\u00a0B\u2019)\u2019) (since, (B\u2019)\u2019 = B)<\/p>\n<p>= A\u00a0\u2229\u00a0(A\u2019 \u222a B)<\/p>\n<p>= (A\u00a0\u2229\u00a0A\u2019)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>=\u00a0\u03d5\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) (since, A\u00a0\u2229\u00a0A\u2019 = \u03d5)<\/p>\n<p>= (A\u00a0\u2229\u00a0B) (since, \u03d5\u00a0\u222a\u00a0x = x, for any set)<\/p>\n<p>= RHS<\/p>\n<p>\u2234\u00a0LHS=RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(iii)\u00a0<\/strong>A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B\u2019) =\u00a0\u03d5<\/p>\n<p>Let us consider LHS A\u00a0\u2229\u00a0(A\u00a0\u222a\u00a0B\u2019)<\/p>\n<p>= A \u2229\u00a0(A\u00a0\u222a\u00a0B\u2019)<\/p>\n<p>= A \u2229 (A\u2019\u2229\u00a0B\u2019) (By De\u2013Morgan\u2019s law)<\/p>\n<p>= (A\u00a0\u2229\u00a0A\u2019)\u00a0\u2229\u00a0B\u2019 (since,\u00a0A\u00a0\u2229\u00a0A\u2019 =\u00a0\u03d5)<\/p>\n<p>=\u00a0\u03d5\u00a0\u2229\u00a0B\u2019<\/p>\n<p>=\u00a0\u03d5 (since, \u03d5\u00a0\u2229\u00a0B\u2019 = \u03d5)<\/p>\n<p>= RHS<\/p>\n<p>\u2234\u00a0LHS=RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(iv)\u00a0<\/strong>A \u2013 B = A \u0394 (A\u00a0\u2229\u00a0B)<\/p>\n<p>Let us consider RHS A \u0394 (A\u00a0\u2229\u00a0B)<\/p>\n<p>A \u0394 (A\u00a0\u2229\u00a0B) (since,\u00a0E \u0394 F = (E\u2013F)\u00a0\u222a\u00a0(F\u2013E))<\/p>\n<p>= (A \u2013 (A\u00a0\u2229\u00a0B))\u00a0\u222a\u00a0(A\u00a0\u2229 B \u2013A) (since, E \u2013 F = E\u00a0\u2229\u00a0F\u2019)<\/p>\n<p>= (A\u00a0\u2229\u00a0(A\u00a0\u2229\u00a0B)\u2019)\u00a0\u222a\u00a0(A \u2229 B \u2229 A\u2019)<\/p>\n<p>= (A\u00a0\u2229\u00a0(A\u2019 \u222a B\u2019))\u00a0\u222a\u00a0(A \u2229 A\u2019 \u2229 B) (by using De-Morgan\u2019s law and associative law)<\/p>\n<p>= (A\u00a0\u2229\u00a0A\u2019) \u222a (A \u2229 B\u2019) \u222a (\u03d5 \u2229 B) (by using distributive law)<\/p>\n<p>=\u00a0\u03d5\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B\u2019)\u00a0\u222a\u00a0\u03d5<\/p>\n<p>= A\u00a0\u2229\u00a0B\u2019 (since, A\u00a0\u2229\u00a0B\u2019 = A\u2013B)<\/p>\n<p>= A \u2013 B<\/p>\n<p>= LHS<\/p>\n<p>\u2234\u00a0LHS=RHS<\/p>\n<p>Hence Proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-if-a-b-c-are-three-sets-such-that-a-%e2%8a%82-b-then-prove-that-c-%e2%80%93-b-%e2%8a%82-c-%e2%80%93-a\"><\/span>3. If A, B, C are three sets such that A\u00a0\u2282\u00a0B, then prove that C \u2013 B\u00a0\u2282\u00a0C \u2013 A.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given, ACB<\/p>\n<p>To prove: C \u2013 B\u00a0\u2282\u00a0C \u2013 A<\/p>\n<p>Let us consider, x\u00a0\u2208\u00a0C\u2013B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C and x \u2209\u00a0B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C and x \u2209\u00a0A<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0C \u2013 A<\/p>\n<p>Thus, x\u00a0\u2208\u00a0C\u2013B\u00a0\u21d2\u00a0x\u00a0\u2208\u00a0C \u2013 A<\/p>\n<p>This is true for all x\u00a0\u2208\u00a0C\u2013B<\/p>\n<p>\u2234\u00a0C \u2013 B\u00a0\u2282\u00a0C \u2013 A<\/p>\n<p>Hence proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-for-any-two-sets-a-and-b-prove-that-i-a-%e2%88%aa-b-%e2%80%93-b-a-%e2%80%93-b\"><\/span>4.\u00a0For any two sets A and B, prove that<br \/>(i) (A\u00a0\u222a\u00a0B) \u2013 B = A \u2013 B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%80%93-a-%e2%88%a9-b-a-%e2%80%93-b\"><\/span>(ii) A \u2013 (A\u00a0\u2229\u00a0B) = A \u2013 B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-%e2%80%93-a-%e2%80%93-b-a-%e2%88%a9-b\"><\/span>(iii) A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-%e2%88%aa-b-%e2%80%93-a-a-%e2%88%aa-b\"><\/span>(iv) A\u00a0\u222a\u00a0(B \u2013 A) = A\u00a0\u222a\u00a0B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"v-a-%e2%80%93-b-%e2%88%aa-a-%e2%88%a9-b-a\"><\/span>(v) (A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>(A\u00a0\u222a\u00a0B) \u2013 B = A \u2013 B<\/p>\n<p>Let us consider LHS (A\u00a0\u222a\u00a0B) \u2013 B<\/p>\n<p>= (A\u2013B)\u00a0\u222a\u00a0(B\u2013B)<\/p>\n<p>= (A\u2013B)\u00a0\u222a\u00a0\u03d5 (since, B\u2013B = \u03d5)<\/p>\n<p>= A\u2013B (since, x\u00a0\u222a\u00a0\u03d5 = x for any set)<\/p>\n<p>= RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(ii)\u00a0<\/strong>A \u2013 (A\u00a0\u2229\u00a0B) = A \u2013 B<\/p>\n<p>Let us consider LHS A \u2013 (A\u00a0\u2229\u00a0B)<\/p>\n<p>= (A\u2013A)\u00a0\u2229 (A\u2013B)<\/p>\n<p>= \u03d5\u00a0\u2229\u00a0(A \u2013 B) (since, A-A = \u03d5)<\/p>\n<p>= A \u2013 B<\/p>\n<p>= RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(iii)\u00a0<\/strong>A \u2013 (A \u2013 B) = A\u00a0\u2229\u00a0B<\/p>\n<p>Let us consider LHS A \u2013 (A \u2013 B)<\/p>\n<p>Let, x\u00a0\u2208\u00a0A \u2013 (A\u2013B)\u00a0=\u00a0x\u00a0\u2208\u00a0A and x\u00a0\u2209\u00a0(A\u2013B)<\/p>\n<p>x\u00a0\u2208\u00a0A and x\u00a0\u2209\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>=\u00a0x\u00a0\u2208\u00a0A \u2229 (A\u00a0\u2229\u00a0B)<\/p>\n<p>= x\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>=\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>= RHS<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(iv)\u00a0<\/strong>A\u00a0\u222a\u00a0(B \u2013 A) = A\u00a0\u222a\u00a0B<\/p>\n<p>Let us consider LHS A\u00a0\u222a\u00a0(B \u2013 A)<\/p>\n<p>Let, x\u00a0\u2208\u00a0A\u00a0\u222a\u00a0(B \u2013A)\u00a0\u21d2\u00a0x\u00a0\u2208\u00a0A or x\u00a0\u2208\u00a0(B \u2013 A)<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A or x\u00a0\u2208\u00a0B and x\u00a0\u2209\u00a0A<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B) (since,\u00a0B\u00a0\u2282\u00a0(A\u00a0\u222a\u00a0B))<\/p>\n<p>This is true for all x\u00a0\u2208\u00a0A\u00a0\u222a\u00a0(B\u2013A)<\/p>\n<p>\u2234\u00a0A \u222a (B\u2013A) \u2282 (A \u222a B)\u2026\u2026 (1)<\/p>\n<p>Conversely,<\/p>\n<p>Let x\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B)\u00a0\u21d2\u00a0x\u00a0\u2208\u00a0A or x\u00a0\u2208\u00a0B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A or x\u00a0\u2208\u00a0(B\u2013A) (since,\u00a0B\u00a0\u2282\u00a0(A\u00a0\u222a\u00a0B))<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A\u00a0\u222a\u00a0(B\u2013A)<\/p>\n<p>\u2234\u00a0(A \u222a B) \u2282\u00a0A \u222a (B\u2013A)\u2026\u2026 (2)<\/p>\n<p>From 1 and 2 we get,<\/p>\n<p>A\u00a0\u222a\u00a0(B \u2013 A) = A\u00a0\u222a\u00a0B<\/p>\n<p>Hence proved.<\/p>\n<p><strong>(v)\u00a0<\/strong>(A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<\/p>\n<p>Let us consider LHS (A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>Let, x\u00a0\u2208\u00a0A<\/p>\n<p>Then either x\u00a0\u2208\u00a0(A\u2013B) or x\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0(A\u2013B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>\u2234\u00a0A\u00a0\u2282\u00a0(A \u2013 B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)\u2026. (1)<\/p>\n<p>Conversely,<\/p>\n<p>Let x\u00a0\u2208\u00a0(A\u2013B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0(A\u2013B) or x\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A and x\u00a0\u2209\u00a0B or x\u00a0\u2208\u00a0B<\/p>\n<p>\u21d2\u00a0x\u00a0\u2208\u00a0A<\/p>\n<p>(A\u2013B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B)\u00a0\u2282\u00a0A\u2026\u2026\u2026. (2)<\/p>\n<p>\u2234\u00a0From (1) and (2), We get<\/p>\n<p>(A\u2013B)\u00a0\u222a\u00a0(A\u00a0\u2229\u00a0B) = A<\/p>\n<p>Hence proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-18-page-no-146\"><\/span>EXERCISE 1.8 PAGE NO: 1.46<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-if-a-and-b-are-two-sets-such-that-n-a-%e2%88%aa-b-50-n-a-28-and-n-b-32-find-n-a-%e2%88%a9-b\"><\/span>1. If A and B are two sets such that n (A\u00a0\u222a\u00a0B) = 50, n (A) = 28 and n (B) = 32, find n (A\u00a0\u2229\u00a0B).<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>n (A\u00a0\u222a\u00a0B) = 50<\/p>\n<p>n (A) = 28<\/p>\n<p>n (B) = 32<\/p>\n<p>We know, n (A\u00a0\u222a\u00a0B) = n (A) + n (B) \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>Substituting the values we get<\/p>\n<p>50 = 28 + 32 \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>50 = 60 \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>\u201310 = \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>\u2234\u00a0n (A\u00a0\u2229\u00a0B) = 10<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-if-p-and-q-are-two-sets-such-that-p-has-40-elements-p-%e2%88%aa-q-has-60-elements-and-p-%e2%88%a9-q-has-10-elements-how-many-elements-does-q-have\"><\/span>2. If P and Q are two sets such that P has 40 elements, P\u00a0\u222a\u00a0Q has 60 elements and P\u00a0\u2229\u00a0Q has 10 elements, how many elements does Q have?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>n (P) = 40<\/p>\n<p>n (P\u00a0\u222a\u00a0Q) = 60<\/p>\n<p>n (P\u00a0\u2229\u00a0Q) =10<\/p>\n<p>We know, n (P\u00a0\u222a\u00a0Q) = n (P) + n (Q) \u2013 n (P\u00a0\u2229\u00a0Q)<\/p>\n<p>Substituting the values we get<\/p>\n<p>60 = 40 + n (Q)\u201310<\/p>\n<p>60 = 30 + n (Q)<\/p>\n<p>N (Q) = 30<\/p>\n<p>\u2234\u00a0Q has 30 elements.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-in-a-school-there-are-20-teachers-who-teach-mathematics-or-physics-of-these-12-teach-mathematics-and-4-teach-physics-and-mathematics-how-many-teach-physics\"><\/span>3. In a school, there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics, and 4 teach physics and mathematics. How many teach physics?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>Teachers teaching physics or math = 20<\/p>\n<p>Teachers teaching physics and math = 4<\/p>\n<p>Teachers teaching maths = 12<\/p>\n<p>Let teachers who teach physics be \u2018n (P)\u2019 and for Maths be \u2018n (M)\u2019<\/p>\n<p>Now,<\/p>\n<p>20 teachers who teach physics or math = n (P\u00a0\u222a\u00a0M) = 20<\/p>\n<p>4 teachers who teach physics and math = n (P\u00a0\u2229\u00a0M) = 4<\/p>\n<p>12 teachers who teach maths = n (M) = 12<\/p>\n<p>We know,<\/p>\n<p>n (P\u00a0\u222a\u00a0M) = n (M) + n (P) \u2013 n (P\u00a0\u2229\u00a0M)<\/p>\n<p>Substituting the values we get,<\/p>\n<p>20 = 12 + n (P) \u2013 4<\/p>\n<p>20 = 8 + n (P)<\/p>\n<p>n (P) =12<\/p>\n<p>\u2234\u00a0There are 12 physics teachers.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-in-a-group-of-70-people-37-like-coffee-52-like-tea-and-each-person-likes-at-least-one-of-the-two-drinks-how-many-like-both-coffee-and-tea\"><\/span><strong>4. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks.<\/strong> How<strong> many like both coffee and tea?<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>A total\u00a0number of people = 70<\/p>\n<p>Number of people who like Coffee = n (C) = 37<\/p>\n<p>Number of people who like Tea = n (T) = 52<\/p>\n<p>Total number = n (C\u00a0\u222a\u00a0T) = 70<\/p>\n<p>Person who likes both would be n (C\u00a0\u2229\u00a0T)<\/p>\n<p>We know,<\/p>\n<p>n (C\u00a0\u222a\u00a0T) = n (C) + n (T) \u2013 n (C\u00a0\u2229\u00a0T)<\/p>\n<p>Substituting the values we get<\/p>\n<p>70 = 37 + 52 \u2013 n (C\u00a0\u2229\u00a0T)<\/p>\n<p>70 = 89 \u2013 n (C\u00a0\u2229\u00a0T)<\/p>\n<p>n (C\u00a0\u2229\u00a0T) =19<\/p>\n<p>\u2234\u00a0There are 19 persons who like both coffee and tea.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-let-a-and-b-be-two-sets-such-that-n-a-20-n-a-%e2%88%aa-b-42-and-n-a-%e2%88%a9-b-4-find\"><\/span>5. Let A and B be two sets such that: n (A) = 20, n (A\u00a0\u222a\u00a0B) = 42 and n (A\u00a0\u2229\u00a0B) = 4. Find<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"i-n-b\"><\/span>(i) n (B)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-n-a-%e2%80%93-b\"><\/span>(ii) n (A \u2013 B)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-n-b-%e2%80%93-a\"><\/span>(iii) n (B \u2013 A)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>n (B)<\/p>\n<p>We know,<\/p>\n<p>n (A\u00a0\u222a\u00a0B) = n (A) + n (B) \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>Substituting the values we get<\/p>\n<p>42 = 20 + n (B) \u2013 4<\/p>\n<p>42 = 16 + n (B)<\/p>\n<p>n (B) = 26<\/p>\n<p>\u2234 n (B) = 26<\/p>\n<p><strong>(ii)\u00a0<\/strong>n (A \u2013 B)<\/p>\n<p>We know,<\/p>\n<p>n (A \u2013 B) = n (A\u00a0\u222a\u00a0B) \u2013 n (B)<\/p>\n<p>Substituting the values we get<\/p>\n<p>n (A \u2013 B) = 42 \u2013 26<\/p>\n<p>= 16<\/p>\n<p>\u2234 n (A \u2013 B) = 16<\/p>\n<p><strong>(iii)<\/strong>\u00a0n (B \u2013 A)<\/p>\n<p>We know,<\/p>\n<p>n (B \u2013 A) = n (B) \u2013 n (A\u00a0\u2229\u00a0B)<\/p>\n<p>Substituting the values we get<\/p>\n<p>n (B \u2013 A) = 26 \u2013 4<\/p>\n<p>= 22<\/p>\n<p>\u2234 n (B \u2013 A) = 22<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-a-survey-shows-that-76-of-the-indians-like-oranges-whereas-62-like-bananas-what-percentage-of-the-indians-like-both-oranges-and-bananas\"><\/span>6. A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>People who like oranges = 76%<\/p>\n<p>People who like bananas = 62%<\/p>\n<p>Let people who like oranges be n (O)<\/p>\n<p>Let people who like bananas be n (B)<\/p>\n<p>Total number of people who like oranges or bananas = n (O\u00a0\u222a\u00a0B) = 100<\/p>\n<p>People who like both oranges and bananas = n (O\u00a0\u2229\u00a0B)<\/p>\n<p>We know,<\/p>\n<p>n (O\u00a0\u222a\u00a0B) = n (O) + n (B) \u2013 n (O\u00a0\u2229\u00a0B)<\/p>\n<p>Substituting the values we get<\/p>\n<p>100 = 76 + 62 \u2013 n (O\u00a0\u2229\u00a0B)<\/p>\n<p>100 = 138 \u2013 n (O\u00a0\u2229\u00a0B)<\/p>\n<p>n (O\u00a0\u2229\u00a0B) = 38<\/p>\n<p>\u2234 People who like both oranges and banana is 38%.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-in-a-group-of-950-persons-750-can-speak-hindi-and-460-can-speak-english-find-i-how-many-can-speak-both-hindi-and-english-ii-how-many-can-speak-hindi-only-iii-how-many-can-speak-english-only\"><\/span>7. In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:<br \/>(i) How many can speak both Hindi and English.<br \/>(ii) How many can speak Hindi only.<br \/>(iii) how many can speak English only.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>We have,<\/p>\n<p>Let, total number of people be n (P) = 950<\/p>\n<p>People who can speak English n (E) = 460<\/p>\n<p>People who can speak Hindi n (H) = 750<\/p>\n<p><strong>(i)<\/strong>\u00a0How many can speak both Hindi and English.<\/p>\n<p>People who can speak both Hindi and English = n (H\u00a0\u2229\u00a0E)<\/p>\n<p>We know,<\/p>\n<p>n (P) = n (E) + n (H) \u2013 n (H\u00a0\u2229\u00a0E)<\/p>\n<p>Substituting the values we get<\/p>\n<p>950 = 460 + 750 \u2013 n (H\u00a0\u2229\u00a0E)<\/p>\n<p>950 = 1210 \u2013 n (H\u00a0\u2229\u00a0E)<\/p>\n<p>n (H\u00a0\u2229\u00a0E) = 260<\/p>\n<p>\u2234 Number\u00a0of people who can speak both English and Hindi are 260.<\/p>\n<p><strong>(ii)<\/strong>\u00a0How many can speak Hindi only.<\/p>\n<p>We can see that H is disjoint union of n (H\u2013E) and n (H\u00a0\u2229\u00a0E).<\/p>\n<p>(If A and B are disjoint then n (A\u00a0\u222a\u00a0B) = n (A) + n (B))<\/p>\n<p>\u2234\u00a0H = n (H\u2013E)\u00a0\u222a\u00a0n (H\u00a0\u2229\u00a0E)<\/p>\n<p>n (H) = n (H\u2013E) + n (H\u00a0\u2229\u00a0E)<\/p>\n<p>750 = n (H \u2013 E) + 260<\/p>\n<p>n (H\u2013E) = 490<\/p>\n<p>\u2234 490 people can speak only Hindi.<\/p>\n<p><strong>(iii)<\/strong>\u00a0How many can speak English only.<\/p>\n<p>We can see that E is disjoint union of n (E\u2013H) and n (H\u00a0\u2229\u00a0E)<\/p>\n<p>(If A and B are disjoint then n (A\u00a0\u222a\u00a0B) = n (A) + n (B))<\/p>\n<p>\u2234\u00a0E = n (E\u2013H)\u00a0\u222a\u00a0n (H\u00a0\u2229\u00a0E).<\/p>\n<p>n (E) = n (E\u2013H) + n (H\u00a0\u2229\u00a0E).<\/p>\n<p>460 = n (H \u2013 E) + 260<\/p>\n<p>n (H\u2013E) = 460 \u2013 260 = 200<\/p>\n<p>\u2234 200 people can speak only English.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"important-topics-from-rd-sharma-solutions-class-11-maths-chapter-1-sets\"><\/span>Important Topics from RD Sharma Solutions Class 11 Maths Chapter 1- Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The RD Sharma Solutions Class 11 Maths Chapter 1 has eight exercises, and you may wonder that the chapter is too big. But this is not the case. The exercises are very small, covering one concept at a time so that you get every topic. All the topics in the exercise are of equal importance, the main topics of the chapter 1-sets are:<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Understanding sets and its types<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Subsets<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Universal and power sets<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Venn Diagrams<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">De-Morgan Theorem<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Elements and some important theorems in the chapter<\/span><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"access-other-important-chapters-of-rd-sharma-solutions-class-11-maths\"><\/span>Access Other Important Chapters of RD Sharma Solutions Class 11 Maths<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 2 &#8211; Relations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 3 &#8211; Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 4 &#8211; Measurement of Angles<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 5 &#8211; Trigonometric Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 6 &#8211; Graphs of Trigonometric Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 7 &#8211; Trigonometric Ratios of Compound Angles<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 8 &#8211; Transformation Formulae<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 9 &#8211; Trigonometric Ratios of Multiple and Sub Multiple Angles<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 10 &#8211; Sine and Cosine Formulae and Their Applications<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 11 &#8211; Trigonometric Equations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 12 &#8211; Mathematical Induction<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 13 &#8211; Complex Numbers<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 14 &#8211; Quadratic Equations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-15-linear-inequations\/\">Chapter 15 &#8211; Linear Inequations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-16-permutations\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 16 &#8211; Permutations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-17-combinations\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 17 &#8211; Combinations<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-18-binomial-theorem\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 18 &#8211; Binomial Theorem<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-19-arithmetic-progressions\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 19 &#8211; Arithmetic Progressions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-20-geometric-progressions\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 20 &#8211; Geometric Progressions<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-21-some-special-series\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 21 &#8211; Some Special Series<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-22-brief-review-of-cartesian-system-of-rectangular-coordinates\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 22 &#8211; Brief Review of Cartesian System of Rectangular Coordinates<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-23-the-straight-lines\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 23 &#8211; The Straight Lines<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-24-the-circle\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 24 &#8211; The Circle<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-25-parabola\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 25 &#8211; Parabola<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-26-ellipse\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 26 &#8211; Ellipse<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-27-hyperbola\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 27 &#8211; Hyperbola<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-28-introduction-to-3d-coordinate-geometry\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 28 &#8211; Introduction To 3D Coordinate Geometry<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-29-limits\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 29 &#8211; Limits<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-30-derivatives\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 30 &#8211; Derivatives<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-31-mathematical-reasoning\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 31 &#8211; Mathematical Reasoning<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-32-statistics\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 32 &#8211; Statistics<\/a><\/li>\n<li><a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-33-probability\/\" target=\"_blank\" rel=\"noopener noreferrer\">Chapter 33 &#8211; Probability<\/a><\/li>\n<\/ul>\n<p>All the best for your <a style=\"font-size: inherit; background-color: initial;\" href=\"http:\/\/cbse.nic.in\" target=\"_blank\" rel=\"noopener noreferrer\">CBSE<\/a><span style=\"font-size: inherit; background-color: initial;\"> class 11 examinations! If you have any doubts regarding the Class 11 Maths Exam, mention in the comments.\u00a0<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-solutions-class-11-maths-chapter-1\"><\/span>FAQs on RD Sharma Solutions Class 11 Maths Chapter 1<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n<div id=\"rank-math-faq\" class=\"rank-math-block\">\n<div class=\"rank-math-list \">\n<div id=\"faq-question-1629708527662\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"from-where-can-i-find-the-solutions-of-maths-chapter-1-exercise-17\"><\/span>From where can I find the solutions of Maths Chapter 1 Exercise 1.7?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can find the solutions in the above blog. <\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629708710084\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"is-it-necessary-to-solve-all-the-questions-of-rd-sharma-chapter-1\"><\/span>Is it necessary to solve all the questions of RD Sharma Chapter 1?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>To better understand the topics, you must solve all the questions of RD Sharma Chapter 1.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629709804736\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"how-much-does-it-cost-to-download-the-pdf-of-rd-sharma-solutions-class-11-maths-chapter-1\"><\/span>How much does it cost to download the PDF of RD Sharma Solutions Class 11 Maths Chapter 1?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download the PDF for free. <\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 11 Maths Chapter 1: \u00a0If you want to ace your Class 11 Maths exams then RD Sharma Solutions Class 11 Maths Chapter 1 is the key to unlock your success in class 11 Maths. It offers a detailed and stepwise solution to every problem no matter how hard or easy the &#8230; <a title=\"RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets (Updated For 2021-22)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-1-sets\/\" aria-label=\"More on RD Sharma Solutions Class 11 Maths Chapter 1 &#8211; Sets (Updated For 2021-22)\">Read more<\/a><\/p>\n","protected":false},"author":243,"featured_media":119190,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"fifu_image_url":"","fifu_image_alt":""},"categories":[73411,2985,73410],"tags":[3428,73334],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/62829"}],"collection":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/users\/243"}],"replies":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/comments?post=62829"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/62829\/revisions"}],"predecessor-version":[{"id":119679,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/62829\/revisions\/119679"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/119190"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=62829"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=62829"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=62829"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}