{"id":62807,"date":"2023-09-03T15:07:00","date_gmt":"2023-09-03T09:37:00","guid":{"rendered":"https:\/\/www.kopykitab.com\/blog\/?p=62807"},"modified":"2023-11-01T11:09:31","modified_gmt":"2023-11-01T05:39:31","slug":"rd-sharma-solutions-class-11-maths-chapter-2-relations","status":"publish","type":"post","link":"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/","title":{"rendered":"RD Sharma Solutions Class 11 Math&#8217;s Chapter 2 &#8211; Relations (Updated For 2024)"},"content":{"rendered":"\n<p><img class=\"alignnone size-full wp-image-119177\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-2.jpg\" alt=\"RD Sharma Solutions Class 11 Maths Chapter 2 \" width=\"1200\" height=\"675\" srcset=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-2.jpg 1200w, https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/08\/RD-Sharma-Solutions-Class-11-Maths-Chapter-2-768x432.jpg 768w\" sizes=\"(max-width: 1200px) 100vw, 1200px\" \/><\/p>\n<p><span style=\"font-weight: 400;\"><strong>RD Sharma Solutions Class 11 Maths Chapter 2<\/strong>:\u00a0<\/span><span style=\"font-weight: 400;\">In this blog, we are going to talk about RD Sharma Solutions Class 11 Maths Chapter 2- Relations. For proper exam preparation, refer to <a href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions\/\">RD Sharma Solutions<\/a> Class 11 Maths Chapter 2. Get the exam ready with <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 2.\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-69e72c0c48a20\" 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-69e72c0c48a20\"><\/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-2-relations\/#download-rd-sharma-solutions-class-11-maths-chapter-2-%e2%80%93-relations-pdf\" title=\"Download RD Sharma Solutions Class 11 Maths Chapter 2 &#8211; Relations PDF\">Download RD Sharma Solutions Class 11 Maths Chapter 2 &#8211; Relations 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-2-relations\/#exercise-wise-rd-sharma-solutions-class-11-maths-chapter-2-relations\" title=\"Exercise-wise: RD Sharma Solutions Class 11 Maths Chapter 2 Relations\">Exercise-wise: RD Sharma Solutions Class 11 Maths Chapter 2 Relations<\/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-2-relations\/#rd-sharma-solutions-class-11-chapter-2-exercise-21\" title=\"RD Sharma Solutions Class 11 Chapter 2 Exercise 2.1\">RD Sharma Solutions Class 11 Chapter 2 Exercise 2.1<\/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-2-relations\/#rd-sharma-solutions-class-11-chapter-2-exercise-22\" title=\"RD Sharma Solutions Class 11 Chapter 2 Exercise 2.2\">RD Sharma Solutions Class 11 Chapter 2 Exercise 2.2<\/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-2-relations\/#rd-sharma-solutions-class-11-chapter-2-exercise-23\" title=\"RD Sharma Solutions Class 11 Chapter 2 Exercise 2.3\">RD Sharma Solutions Class 11 Chapter 2 Exercise 2.3<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#access-rd-sharma-solutions-class-11-maths-chapter-2\" title=\"Access RD Sharma Solutions Class 11 Maths Chapter 2\">Access RD Sharma Solutions Class 11 Maths Chapter 2<\/a><ul class='ez-toc-list-level-3'><li class='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-2-relations\/#exercise-21-page-no-28\" title=\"EXERCISE 2.1 PAGE NO: 2.8\">EXERCISE 2.1 PAGE NO: 2.8<\/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-2-relations\/#1-i-if-a3-1-b-%e2%80%93-23-53-13-find-the-values-of-a-and-b\" title=\"(1) (i) If (a\/3 + 1, b \u2013 2\/3) = (5\/3, 1\/3), find the values of a and b.\">(1) (i) If (a\/3 + 1, b \u2013 2\/3) = (5\/3, 1\/3), find the values of a and b.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#ii-if-x-1-1-3y-y-%e2%80%93-1-find-the-values-of-x-and-y\" title=\"(ii) If (x + 1, 1) = (3y, y \u2013 1), find the values of x and y.\">(ii) If (x + 1, 1) = (3y, y \u2013 1), find the values of x and y.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#2-if-the-ordered-pairs-x-%e2%80%93-1-and-5-y-belong-to-the-set-a-b-b-2a-%e2%80%93-3-find-the-values-of-x-and-y\" title=\"2. If the ordered pairs (x, \u2013 1) and (5, y) belong to the set {(a, b): b = 2a \u2013 3}, find the values of x and y.\">2. If the ordered pairs (x, \u2013 1) and (5, y) belong to the set {(a, b): b = 2a \u2013 3}, find the values of x and y.<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#3-if-a-%e2%88%88-1-2-3-4-5-and-b-%e2%88%88-0-3-6-write-the-set-of-all-ordered-pairs-a-b-such-that-a-b-5\" title=\"3. If a\u00a0\u2208\u00a0{- 1, 2, 3, 4, 5} and b\u00a0\u2208\u00a0{0, 3, 6}, write the set of all ordered pairs (a, b) such that a + b = 5.\">3. If a\u00a0\u2208\u00a0{- 1, 2, 3, 4, 5} and b\u00a0\u2208\u00a0{0, 3, 6}, write the set of all ordered pairs (a, b) such that a + b = 5.<\/a><\/li><li class='ez-toc-page-1 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-2-relations\/#4-if-a-%e2%88%88-2-4-6-9-and-b-%e2%88%88-4-6-18-27-then-form-the-set-of-all-ordered-pairs-a-b-such-that-a-divides-b-and-a\" title=\"4. If a\u00a0\u2208\u00a0{2, 4, 6, 9} and b\u00a0\u2208 {4, 6, 18, 27}, then form the set of all ordered pairs (a, b) such that a divides b and a&lt;b.\">4. If a\u00a0\u2208\u00a0{2, 4, 6, 9} and b\u00a0\u2208 {4, 6, 18, 27}, then form the set of all ordered pairs (a, b) such that a divides b and a&lt;b.<\/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-2-relations\/#5-if-a-1-2-and-b-1-3-find-a-x-b-and-b-x-a\" title=\"5. If A = {1, 2} and B = {1, 3}, find A x B and B x A.\">5. If A = {1, 2} and B = {1, 3}, find A x B and B x A.<\/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-2-relations\/#6-let-a-1-2-3-and-b-3-4-find-a-x-b-and-show-it-graphically\" title=\"6. Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically\">6. Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically<\/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-2-relations\/#7-if-a-1-2-3-and-b-2-4-what-are-a-x-b-b-x-a-a-x-a-b-x-b-and-a-x-b-%e2%88%a9-b-x-a\" title=\"7. If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B)\u00a0\u2229\u00a0(B x A)?\">7. If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B)\u00a0\u2229\u00a0(B x A)?<\/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-2-relations\/#1-given-a-1-2-3-b-3-4-c-4-5-6-find-a-x-b-%e2%88%a9-b-x-c\" title=\"1. Given A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, find (A x B)\u00a0\u2229\u00a0(B x C).\">1. Given A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, find (A x B)\u00a0\u2229\u00a0(B x C).<\/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-2-relations\/#2-if-a-2-3-b-4-5-c-5-6-find-a-x-b-%e2%88%aa-c-a-x-b-%e2%88%aa-a-x-c\" title=\"2. If A = {2, 3}, B = {4, 5}, C = {5, 6} find A x (B\u00a0\u222a\u00a0C), (A x B)\u00a0\u222a\u00a0(A x C).\">2. If A = {2, 3}, B = {4, 5}, C = {5, 6} find A x (B\u00a0\u222a\u00a0C), (A x B)\u00a0\u222a\u00a0(A x C).<\/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-2-relations\/#3-if-a-1-2-3-b-4-c-5-then-verify-that-i-a-x-b-%e2%88%aa-c-a-x-b-%e2%88%aa-a-x-c-ii-a-x-b-%e2%88%a9-c-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%80%93-c-a-x-b-%e2%80%93-a-x-c\" title=\"3.\u00a0If A = {1, 2, 3}, B = {4}, C = {5}, then verify that: (i) A x (B\u00a0\u222a\u00a0C) = (A x B)\u00a0\u222a\u00a0(A x C) (ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B \u2013 C) = (A x B) \u2013 (A x C)\">3.\u00a0If A = {1, 2, 3}, B = {4}, C = {5}, then verify that: (i) A x (B\u00a0\u222a\u00a0C) = (A x B)\u00a0\u222a\u00a0(A x C) (ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B \u2013 C) = (A x B) \u2013 (A x C)<\/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-2-relations\/#4-let-a-1-2-b-1-2-3-4-c-5-6-and-d-5-6-7-8-verify-that-i-a-x-c-%e2%8a%82-b-x-d-ii-a-x-b-%e2%88%a9-c-a-x-b-%e2%88%a9-a-x-c\" title=\"4. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that: (i) A x C \u2282\u00a0B x D (ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C)\">4. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that: (i) A x C \u2282\u00a0B x D (ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C)<\/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-2-relations\/#5-if-a-1-2-3-b-3-4-and-c-4-5-6-find-i-a-x-b-%e2%88%a9-c-ii-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%88%aa-c-iv-a-x-b-%e2%88%aa-a-x-c5-if-a-1-2-3-b-3-4-and-c-4-5-6-find-i-a-x-b-%e2%88%a9-c-ii-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%88%aa-c-iv-a-x-b-%e2%88%aa-a-x-c\" title=\"5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find (i) A x (B\u00a0\u2229\u00a0C) (ii) (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B\u00a0\u222a\u00a0C) (iv) (A x B)\u00a0\u222a\u00a0(A x C)5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find (i) A x (B\u00a0\u2229\u00a0C) (ii) (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B\u00a0\u222a\u00a0C) (iv) (A x B)\u00a0\u222a\u00a0(A x C)\">5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find (i) A x (B\u00a0\u2229\u00a0C) (ii) (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B\u00a0\u222a\u00a0C) (iv) (A x B)\u00a0\u222a\u00a0(A x C)5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find (i) A x (B\u00a0\u2229\u00a0C) (ii) (A x B)\u00a0\u2229\u00a0(A x C) (iii) A x (B\u00a0\u222a\u00a0C) (iv) (A x B)\u00a0\u222a\u00a0(A x C)<\/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-2-relations\/#6-prove-that-i-a-%e2%88%aa-b-x-c-a-x-c-a-x-c-%e2%88%aa-b-x-c\" title=\"6. Prove that: (i) (A\u00a0\u222a\u00a0B) x C = (A x C) = (A x C)\u00a0\u222a\u00a0(B x C)\">6. Prove that: (i) (A\u00a0\u222a\u00a0B) x C = (A x C) = (A x C)\u00a0\u222a\u00a0(B x C)<\/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-2-relations\/#ii-a-%e2%88%a9-b-x-c-a-x-c-%e2%88%a9-b-x-c\" title=\"(ii) (A\u00a0\u2229\u00a0B) x C = (A x C)\u00a0\u2229\u00a0(B x C)\">(ii) (A\u00a0\u2229\u00a0B) x C = (A x C)\u00a0\u2229\u00a0(B x C)<\/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-2-relations\/#7-if-a-x-b-%e2%8a%86-c-x-d-and-a-%e2%88%a9-b-%e2%88%88-%e2%88%85-prove-that-a-%e2%8a%86-c-and-b-%e2%8a%86-d\" title=\"7. If A x B\u00a0\u2286\u00a0C x D and A\u00a0\u2229\u00a0B\u00a0\u2208\u00a0\u2205, Prove that A\u00a0\u2286\u00a0C and B\u00a0\u2286\u00a0D.\">7. If A x B\u00a0\u2286\u00a0C x D and A\u00a0\u2229\u00a0B\u00a0\u2208\u00a0\u2205, Prove that A\u00a0\u2286\u00a0C and B\u00a0\u2286\u00a0D.<\/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-2-relations\/#exercise-23-page-no-220\" title=\"EXERCISE 2.3 PAGE NO: 2.20\">EXERCISE 2.3 PAGE NO: 2.20<\/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-2-relations\/#1-if-a-1-2-3-b-4-5-6-which-of-the-following-are-relations-from-a-to-b-give-reasons-in-support-of-your-answer\" title=\"1. If A = {1, 2, 3}, B = {4, 5, 6}, which of the following are relations from A to B? Give reasons in support of your answer.\">1. If A = {1, 2, 3}, B = {4, 5, 6}, which of the following are relations from A to B? Give reasons in support of your answer.<\/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-2-relations\/#i-1-6-3-4-5-2\" title=\"(i) {(1, 6), (3, 4), (5, 2)}\">(i) {(1, 6), (3, 4), (5, 2)}<\/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-2-relations\/#ii-1-5-2-6-3-4-3-6\" title=\"(ii) {(1, 5), (2, 6), (3, 4), (3, 6)}\">(ii) {(1, 5), (2, 6), (3, 4), (3, 6)}<\/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-2-relations\/#iii-4-2-4-3-5-1\" title=\"(iii) {(4, 2), (4, 3), (5, 1)}\">(iii) {(4, 2), (4, 3), (5, 1)}<\/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-2-relations\/#iv-a-%c3%97-b\" title=\"(iv) A \u00d7 B\">(iv) A \u00d7 B<\/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-2-relations\/#2-a-relation-r-is-defined-from-a-set-a-2-3-4-5-to-a-set-b-3-6-7-10-as-follows-x-y-r-x-is-relatively-prime-to-y-express-r-as-a-set-of-ordered-pairs-and-determine-its-domain-and-range\" title=\"2. A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows: (x, y)\u00a0R\u00a0x is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.\">2. A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows: (x, y)\u00a0R\u00a0x is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.<\/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-2-relations\/#3-let-a-be-the-set-of-the-first-five-natural-and-let-r-be-a-relation-on-a-defined-as-follows-x-y-r-x-%e2%89%a4-y-express-r-and-r-1-as-sets-of-ordered-pairs-determine-also-i-the-domain-of-r%e2%80%911-ii-the-range-of-r\" title=\"3. Let A be the set of the first five natural and let R be a relation on A defined as follows: (x, y) R x \u2264 y Express R and R-1\u00a0as sets of ordered pairs. Determine also (i) The domain of R\u20111 (ii) The Range of R.\">3. Let A be the set of the first five natural and let R be a relation on A defined as follows: (x, y) R x \u2264 y Express R and R-1\u00a0as sets of ordered pairs. Determine also (i) The domain of R\u20111 (ii) The Range of R.<\/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-2-relations\/#4-find-the-inverse-relation-r-1-in-each-of-the-following-cases-i-r-1-2-1-3-2-3-3-2-5-6-ii-r-x-y-x-y-%e2%88%88-n-x-2y-8-iii-r-is-a-relation-from-11-12-13-to-8-10-12-defined-by-y-x-%e2%80%93-3\" title=\"4. Find the inverse relation R-1\u00a0in each of the following cases: (i) R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)} (ii) R= {(x, y) : x, y\u00a0\u2208\u00a0N; x + 2y = 8} (iii) R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x \u2013 3\">4. Find the inverse relation R-1\u00a0in each of the following cases: (i) R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)} (ii) R= {(x, y) : x, y\u00a0\u2208\u00a0N; x + 2y = 8} (iii) R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x \u2013 3<\/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-2-relations\/#5-write-the-following-relations-as-sets-of-ordered-pairs-i-a-relation-r-from-the-set-2-3-4-5-6-to-the-set-1-2-3-defined-by-x-2y\" title=\"5. Write the following relations as sets of ordered pairs: (i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.\">5. Write the following relations as sets of ordered pairs: (i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.<\/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-2-relations\/#ii-a-relation-r-on-the-set-1-2-3-4-5-6-7-defined-by-x-y-%e2%88%88-r-%e2%87%94-x-is-relatively-prime-to-y\" title=\"(ii) A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y)\u00a0\u2208\u00a0R \u21d4 x is relatively prime to y.\">(ii) A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y)\u00a0\u2208\u00a0R \u21d4 x is relatively prime to y.<\/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-2-relations\/#iii-a-relation-r-on-the-set-0-1-2%e2%80%a610-defined-by-2x-3y-12\" title=\"(iii) A relation R on the set {0, 1, 2,\u2026,10} defined by 2x + 3y = 12.\">(iii) A relation R on the set {0, 1, 2,\u2026,10} defined by 2x + 3y = 12.<\/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-2-relations\/#iv-a-relation-r-form-a-set-a-5-6-7-8-to-the-set-b-10-12-15-16-18-defined-by-x-y-r-x-divides-y\" title=\"(iv) A relation R form a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by (x, y)\u00a0R\u00a0x divides y.\">(iv) A relation R form a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by (x, y)\u00a0R\u00a0x divides y.<\/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-2-relations\/#6-let-r-be-a-relation-in-n-defined-by-x-y-%e2%88%88-r-%e2%87%94-x-2y-8-express-r-and-r-1-as-sets-of-ordered-pairs\" title=\"6. Let R be a relation in N defined by (x, y)\u00a0\u2208\u00a0R\u00a0\u21d4\u00a0x + 2y = 8. Express R and R-1\u00a0as sets of ordered pairs.\">6. Let R be a relation in N defined by (x, y)\u00a0\u2208\u00a0R\u00a0\u21d4\u00a0x + 2y = 8. Express R and R-1\u00a0as sets of ordered pairs.<\/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-2-relations\/#7-let-a-3-5-and-b-7-11-let-r-a-b-a-%e2%88%88-a-b-%e2%88%88-b-a-b-is-odd-show-that-r-is-an-empty-relation-from-a-to-b\" title=\"7. Let A = {3, 5} and B = {7, 11}. Let R = {(a, b): a \u2208 A, b \u2208 B, a-b is odd}. Show that R is an empty relation from A to B.\">7. Let A = {3, 5} and B = {7, 11}. Let R = {(a, b): a \u2208 A, b \u2208 B, a-b is odd}. Show that R is an empty relation from A to B.<\/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-2-relations\/#8-let-a-1-2-and-b-3-4-find-the-total-number-of-relations-from-a-to-b\" title=\"8. Let A = {1, 2} and B = {3, 4}. Find the total number of relations from A to B.\">8. Let A = {1, 2} and B = {3, 4}. Find the total number of relations from A to B.<\/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-2-relations\/#9-determine-the-domain-and-range-of-the-relation-r-defined-by-i-r-x-x5-x-%e2%88%88-0-1-2-3-4-5\" title=\"9. Determine the domain and range of the relation R defined by (i) R = {(x, x+5): x\u00a0\u2208\u00a0{0, 1, 2, 3, 4, 5}\">9. Determine the domain and range of the relation R defined by (i) R = {(x, x+5): x\u00a0\u2208\u00a0{0, 1, 2, 3, 4, 5}<\/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-2-relations\/#ii-r-x-x3-x-is-a-prime-number-less-than-10\" title=\"(ii) R= {(x, x3): x is a prime number less than 10}\">(ii) R= {(x, x3): x is a prime number less than 10}<\/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-2-relations\/#10-determine-the-domain-and-range-of-the-following-relations-i-r-a-b-a-%e2%88%88-n-a-%3c-5-b-4\" title=\"10. Determine the domain and range of the following relations: (i) R= {a, b): a\u00a0\u2208\u00a0N, a &lt; 5, b = 4}\">10. Determine the domain and range of the following relations: (i) R= {a, b): a\u00a0\u2208\u00a0N, a &lt; 5, b = 4}<\/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-2-relations\/#ii-s-a-b-b-a-1-a-%e2%88%88-z-and-a-%e2%89%a4-3\" title=\"(ii) S= {a, b): b = |a-1|, a \u2208\u00a0Z and |a|\u00a0\u2264\u00a03}\">(ii) S= {a, b): b = |a-1|, a \u2208\u00a0Z and |a|\u00a0\u2264\u00a03}<\/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-2-relations\/#11-let-a-a-b-list-all-relations-on-a-and-find-their-number\" title=\"11. Let A = {a, b}. List all relations on A and find their number.\">11. Let A = {a, b}. List all relations on A and find their number.<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-45\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#important-topics-from-rd-sharma-solutions-class-11-maths-chapter-2\" title=\"Important Topics From RD Sharma Solutions Class 11 Maths Chapter 2\">Important Topics From RD Sharma Solutions Class 11 Maths Chapter 2<\/a><ul class='ez-toc-list-level-3'><li class='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-2-relations\/#key-take-away-from-rd-sharma-solutions-class-11-maths-chapter-2\" title=\"Key Take Away From RD Sharma Solutions Class 11 Maths Chapter 2\">Key Take Away From RD Sharma Solutions Class 11 Maths Chapter 2<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-47\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#access-other-important-chapters-of-rd-sharma-solutions-class-11-maths\" title=\"Access Other Important Chapters of RD Sharma Solutions Class 11 Maths\u00a0\">Access Other Important Chapters of RD Sharma Solutions Class 11 Maths\u00a0<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-48\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/#faqs-on-rd-sharma-solutions-class-11-maths-chapter-2\" title=\"FAQs on RD Sharma Solutions Class 11 Maths Chapter 2\">FAQs on RD Sharma Solutions Class 11 Maths Chapter 2<\/a><ul class='ez-toc-list-level-3'><li class='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-2-relations\/#are-the-solutions-of-rd-sharma-solutions-class-11-maths-chapter-2-reliable\" title=\"Are the solutions of RD Sharma Solutions Class 11 Maths Chapter 2 reliable?\">Are the solutions of RD Sharma Solutions Class 11 Maths Chapter 2 reliable?<\/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-2-relations\/#can-i-access-the-rd-sharma-solutions-class-11-maths-chapter-2-pdf-offline\" title=\"Can I access the RD Sharma Solutions Class 11 Maths Chapter 2 PDF offline?\">Can I access the RD Sharma Solutions Class 11 Maths Chapter 2 PDF offline?<\/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-2-relations\/#from-where-can-i-download-the-rd-sharma-solutions-class-11-maths-chapter-2-pdf\" title=\"From where can I download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?\">From where can I download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?<\/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-2-relations\/#how-much-does-it-cost-to-download-the-rd-sharma-solutions-class-11-maths-chapter-2-pdf\" title=\"How much does it cost to download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?\">How much does it cost to download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"download-rd-sharma-solutions-class-11-maths-chapter-2-%e2%80%93-relations-pdf\"><\/span>Download RD Sharma Solutions Class 11 Maths Chapter 2 &#8211; Relations 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-2-1.pdf\" target=\"_blank\" rel=\"noopener\">RD Sharma Solutions Class 11 Maths Chapter 2<\/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-2-1.pdf\", \"#example1\");<\/script><\/p>\n<h2><span class=\"ez-toc-section\" id=\"exercise-wise-rd-sharma-solutions-class-11-maths-chapter-2-relations\"><\/span>Exercise-wise: RD Sharma Solutions Class 11 Maths Chapter 2 Relations<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-2-exercise-2-1\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">RD Sharma Solutions Class 11 Chapter 2A<\/span><\/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-2-exercise-2-2\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">RD Sharma Solutions Class 11 Chapter 2B<\/span><\/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-2-exercise-2-3\/\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">RD Sharma Solutions Class 11 Chapter 2C<\/span><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-class-11-chapter-2-exercise-21\"><\/span>RD Sharma Solutions Class 11 Chapter 2 Exercise 2.1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">The RD Sharma Solutions for Chapter 2 Exercise 2.1 is a basic exercise that starts with the introduction of relations, ordered pairs, Cartesian of products, and graphical representations of the same.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Exercise 2.1 has some illustrative examples to help students understand each concept and then apply it together. The solutions provide a detailed step-by-step process of solving a question.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-class-11-chapter-2-exercise-22\"><\/span>RD Sharma Solutions Class 11 Chapter 2 Exercise 2.2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-2.2 is all about using some useful results to move ahead in the chapter. It includes working with theorems and their results for getting solutions quickly.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The exercise is very simple and small. Once you understand some basic terminologies, theorems, and results you can easily solve these questions.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"rd-sharma-solutions-class-11-chapter-2-exercise-23\"><\/span>RD Sharma Solutions Class 11 Chapter 2 Exercise 2.3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><span style=\"font-weight: 400;\">Exercise-2.3 covers the relations with some practical application questions. It all covers all the concepts covered in the previous exercises to let students think and use all the concepts together when needed. Exercise-2.3 solutions talk about Domain, Domain Range, and the inverse of a relation.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"access-rd-sharma-solutions-class-11-maths-chapter-2\"><\/span>Access <strong>RD Sharma Solutions Class 11 Maths Chapter 2<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3><span class=\"ez-toc-section\" id=\"exercise-21-page-no-28\"><\/span>EXERCISE 2.1 PAGE NO: 2.8<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-i-if-a3-1-b-%e2%80%93-23-53-13-find-the-values-of-a-and-b\"><\/span><strong>(1) (i) If (a\/3 + 1, b \u2013 2\/3) = (5\/3, 1\/3), find the values of a and b.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-if-x-1-1-3y-y-%e2%80%93-1-find-the-values-of-x-and-y\"><\/span><strong>(ii) If (x + 1, 1) = (3y, y \u2013 1), find the values of x and y.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>(a\/3 + 1, b \u2013 2\/3) = (5\/3, 1\/3)<\/p>\n<p>By the definition of equality of ordered pairs,<\/p>\n<p>Let us solve for a and b<\/p>\n<p>a\/3 + 1 = 5\/3 and b \u2013 2\/3 = 1\/3<\/p>\n<p>a\/3 = 5\/3 \u2013 1 and b = 1\/3 + 2\/3<\/p>\n<p>a\/3 = (5-3)\/3 and b = (1+2)\/3<\/p>\n<p>a\/3 = 2\/3 and b = 3\/3<\/p>\n<p>a = 2(3)\/3 and b = 1<\/p>\n<p>a = 2 and b = 1<\/p>\n<p>\u2234 Values of a and b are, a = 2 and b = 1<\/p>\n<p><strong>(ii)\u00a0<\/strong>If (x + 1, 1) = (3y, y \u2013 1), find the values of x and y.<\/p>\n<p>Given:<\/p>\n<p>(x + 1, 1) = (3y, y \u2013 1)<\/p>\n<p>By the definition of equality of ordered pairs,<\/p>\n<p>Let us solve for x and y<\/p>\n<p>x + 1 = 3y and\u00a01 = y \u2013 1<\/p>\n<p>x = 3y \u2013 1 and\u00a0y = 1 + 1<\/p>\n<p>x = 3y \u2013 1 and y = 2<\/p>\n<p>Since, y = 2 we can substitute in<\/p>\n<p>x = 3y \u2013 1<\/p>\n<p>= 3(2) \u2013 1<\/p>\n<p>= 6 \u2013 1<\/p>\n<p>= 5<\/p>\n<p>\u2234 Values of x and y are, x = 5 and y = 2<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-if-the-ordered-pairs-x-%e2%80%93-1-and-5-y-belong-to-the-set-a-b-b-2a-%e2%80%93-3-find-the-values-of-x-and-y\"><\/span>2. If the ordered pairs (x, \u2013 1) and (5, y) belong to the set {(a, b): b = 2a \u2013 3}, find the values of x and y.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>The ordered pairs (x, \u2013 1) and (5, y) belong to the set {(a, b): b = 2a \u2013 3}<\/p>\n<p>Solving for first-order pair<\/p>\n<p>(x, \u2013 1) = {(a, b): b = 2a \u2013 3}<\/p>\n<p>x = a and -1 = b<\/p>\n<p>By taking b = 2a \u2013 3<\/p>\n<p>If b = \u2013 1 then 2a = \u2013 1 + 3<\/p>\n<p>= 2<\/p>\n<p>a = 2\/2<\/p>\n<p>= 1<\/p>\n<p>So, a = 1<\/p>\n<p>Since x = a, x = 1<\/p>\n<p>Similarly, solving for second-order pair<\/p>\n<p>(5, y) = {(a, b): b = 2a \u2013 3}<\/p>\n<p>5 = a and y = b<\/p>\n<p>By taking b = 2a \u2013 3<\/p>\n<p>If a = 5 then b = 2\u00d75 \u2013 3<\/p>\n<p>= 10 \u2013 3<\/p>\n<p>= 7<\/p>\n<p>So, b = 7<\/p>\n<p>Since y = b, y = 7<\/p>\n<p>\u2234 Values of x and y are, x = 1 and y = 7<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-if-a-%e2%88%88-1-2-3-4-5-and-b-%e2%88%88-0-3-6-write-the-set-of-all-ordered-pairs-a-b-such-that-a-b-5\"><\/span>3. If a\u00a0\u2208\u00a0{- 1, 2, 3, 4, 5} and b\u00a0\u2208\u00a0{0, 3, 6}, write the set of all ordered pairs (a, b) such that a + b = 5.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given: a\u00a0\u2208\u00a0{- 1, 2, 3, 4, 5} and b\u00a0\u2208\u00a0{0, 3, 6},<\/p>\n<p>To find: the ordered pair (a, b) such that a + b = 5<\/p>\n<p>Then the ordered pair (a, b) such that a + b = 5 are as follows<\/p>\n<p>(a, b) \u2208\u00a0{(- 1, 6), (2, 3), (5, 0)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-if-a-%e2%88%88-2-4-6-9-and-b-%e2%88%88-4-6-18-27-then-form-the-set-of-all-ordered-pairs-a-b-such-that-a-divides-b-and-a\"><\/span>4. If a\u00a0\u2208\u00a0{2, 4, 6, 9} and b\u00a0\u2208 {4, 6, 18, 27}, then form the set of all ordered pairs (a, b) such that a divides b and a&lt;b.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>a\u00a0\u2208\u00a0{2, 4, 6, 9} and b\u00a0\u2208{4, 6, 18, 27}<\/p>\n<p>Here,<\/p>\n<p>2 divides 4, 6, and 18 and is also less than all of them<\/p>\n<p>4 divides 4 and is also less than none of them<\/p>\n<p>6 divides 6, 18 and is less than 18 only<\/p>\n<p>9 divides 18, 27 and is less than all of them<\/p>\n<p>\u2234 Ordered pairs (a, b) are (2, 4), (2, 6), (2, 18), (6, 18), (9, 18), and (9, 27)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-if-a-1-2-and-b-1-3-find-a-x-b-and-b-x-a\"><\/span>5. If A = {1, 2} and B = {1, 3}, find A x B and B x A.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2} and B = {1, 3}<\/p>\n<p>A \u00d7 B = {1, 2} \u00d7 {1, 3}<\/p>\n<p>= {(1, 1), (1, 3), (2, 1), (2, 3)}<\/p>\n<p>B \u00d7 A = {1, 3} \u00d7 {1, 2}<\/p>\n<p>= {(1, 1), (1, 2), (3, 1), (3, 2)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-let-a-1-2-3-and-b-3-4-find-a-x-b-and-show-it-graphically\"><\/span>6. Let A = {1, 2, 3} and B = {3, 4}. Find A x B and show it graphically<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2, 3} and B = {3, 4}<\/p>\n<p>A x B = {1, 2, 3} \u00d7 {3, 4}<\/p>\n<p>= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}<\/p>\n<p>Steps to follow to represent A \u00d7 B graphically,<\/p>\n<p>Step 1: One horizontal and one vertical axis should be drawn<\/p>\n<p>Step 2: Element of set A should be represented in the horizontal axis and on the vertical axis elements of set B should be represented<\/p>\n<p>Step 3: Draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B<\/p>\n<p>Step 4: The point of intersection of these perpendicular represents A \u00d7 B<\/p>\n<h3><img class=\"alignnone size-full wp-image-117698\" src=\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/01\/rd.png\" alt=\"\" width=\"750\" height=\"350\"><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"7-if-a-1-2-3-and-b-2-4-what-are-a-x-b-b-x-a-a-x-a-b-x-b-and-a-x-b-%e2%88%a9-b-x-a\"><\/span>7. If A = {1, 2, 3} and B = {2, 4}, what are A x B, B x A, A x A, B x B, and (A x B)\u00a0\u2229\u00a0(B x A)?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2, 3} and B = {2, 4}<\/p>\n<p>Now let us find: A \u00d7 B, B \u00d7 A, A \u00d7 A, (A \u00d7 B)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0A)<\/p>\n<p>A \u00d7 B = {1, 2, 3} \u00d7 {2, 4}<\/p>\n<p>= {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}<\/p>\n<p>B \u00d7 A = {2, 4} \u00d7 {1, 2, 3}<\/p>\n<p>= {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}<\/p>\n<p>A \u00d7 A = {1, 2, 3} \u00d7 {1, 2, 3}<\/p>\n<p>= {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}<\/p>\n<p>B \u00d7 B = {2, 4} \u00d7 {2, 4}<\/p>\n<p>= {(2, 2), (2, 4), (4, 2), (4, 4)}<\/p>\n<p>The intersection of two sets represents common elements of both the sets<\/p>\n<p>So,<\/p>\n<p>(A \u00d7 B)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0A) = {(2, 2)}<\/p>\n<p>EXERCISE 2.2 PAGE NO: 2.12<\/p>\n<h3><span class=\"ez-toc-section\" id=\"1-given-a-1-2-3-b-3-4-c-4-5-6-find-a-x-b-%e2%88%a9-b-x-c\"><\/span>1. Given A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, find (A x B)\u00a0\u2229\u00a0(B x C).<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}<\/p>\n<p>Let us find: (A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p>(A \u00d7 B) = {1, 2, 3} \u00d7 {3, 4}<\/p>\n<p>= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}<\/p>\n<p>(B \u00d7 C) = {3, 4} \u00d7 {4, 5, 6}<\/p>\n<p>= {(3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}<\/p>\n<p>\u2234\u00a0(A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C) = {(3, 4)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-if-a-2-3-b-4-5-c-5-6-find-a-x-b-%e2%88%aa-c-a-x-b-%e2%88%aa-a-x-c\"><\/span>2. If A = {2, 3}, B = {4, 5}, C = {5, 6} find A x (B\u00a0\u222a\u00a0C), (A x B)\u00a0\u222a\u00a0(A x C).<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given: A = {2, 3}, B = {4, 5} and C = {5, 6}<\/p>\n<p>Let us find: A x (B\u00a0\u222a\u00a0C) and (A x B)\u00a0\u222a\u00a0(A x C)<\/p>\n<p>(B\u00a0\u222a\u00a0C) = {4, 5, 6}<\/p>\n<p>A \u00d7 (B\u00a0\u222a\u00a0C) = {2, 3} \u00d7 {4, 5, 6}<\/p>\n<p>= {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<p>(A\u00a0\u00d7\u00a0B) = {2, 3} \u00d7 {4, 5}<\/p>\n<p>= {(2, 4), (2, 5), (3, 4), (3, 5)}<\/p>\n<p>(A\u00a0\u00d7\u00a0C) = {2, 3} \u00d7 {5, 6}<\/p>\n<p>= {(2, 5), (2, 6), (3, 5), (3, 6)}<\/p>\n<p>\u2234\u00a0(A\u00a0\u00d7\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u00d7\u00a0C) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<p>A \u00d7 (B\u00a0\u222a\u00a0C) = {(2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-if-a-1-2-3-b-4-c-5-then-verify-that-i-a-x-b-%e2%88%aa-c-a-x-b-%e2%88%aa-a-x-c-ii-a-x-b-%e2%88%a9-c-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%80%93-c-a-x-b-%e2%80%93-a-x-c\"><\/span>3.\u00a0If A = {1, 2, 3}, B = {4}, C = {5}, then verify that:<br \/>(i) A x (B\u00a0\u222a\u00a0C) = (A x B)\u00a0\u222a\u00a0(A x C)<br \/>(ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C)<br \/>(iii) A x (B \u2013 C) = (A x B) \u2013 (A x C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2, 3}, B = {4} and C = {5}<\/p>\n<p><strong>(i)<\/strong>\u00a0A \u00d7 (B\u00a0\u222a\u00a0C) = (A\u00a0\u00d7\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<p>Let us consider LHS: (B\u00a0\u222a\u00a0C)<\/p>\n<p>(B\u00a0\u222a\u00a0C) = {4, 5}<\/p>\n<p>A \u00d7 (B\u00a0\u222a\u00a0C) = {1, 2, 3} \u00d7 {4, 5}<\/p>\n<p>= {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u00d7 B) = {1, 2, 3} \u00d7 {4}<\/p>\n<p>= {(1, 4), (2, 4), (3, 4)}<\/p>\n<p>(A \u00d7 C) = {1, 2, 3} \u00d7 {5}<\/p>\n<p>= {(1, 5), (2, 5), (3, 5)}<\/p>\n<p>(A \u00d7 B)\u00a0\u222a\u00a0(A\u00a0\u00d7\u00a0C) = {(1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5)}<\/p>\n<p>\u2234\u00a0LHS = RHS<\/p>\n<p>\u00a0<\/p>\n<p><strong>(ii)<\/strong>\u00a0A \u00d7 (B\u00a0\u2229\u00a0C) = (A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<p>Let us consider LHS: (B\u00a0\u2229\u00a0C)<\/p>\n<p>(B\u00a0\u2229\u00a0C) =\u00a0\u2205\u00a0(No common element)<\/p>\n<p>A \u00d7 (B\u00a0\u2229\u00a0C) = {1, 2, 3} \u00d7 \u2205<\/p>\n<p>=\u00a0\u2205<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u00d7 B) = {1, 2, 3} \u00d7 {4}<\/p>\n<p>= {(1, 4), (2, 4), (3, 4)}<\/p>\n<p>(A \u00d7 C) = {1, 2, 3} \u00d7 {5}<\/p>\n<p>= {(1, 5), (2, 5), (3, 5)}<\/p>\n<p>(A \u00d7 B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C) =\u00a0\u2205<\/p>\n<p>\u2234\u00a0LHS = RHS<\/p>\n<p>\u00a0<\/p>\n<p><strong>(iii)<\/strong>\u00a0A \u00d7 (B \u2212 C) = (A \u00d7 B) \u2212 (A \u00d7 C)<\/p>\n<p>Let us consider LHS: (B \u2212 C)<\/p>\n<p>(B \u2212 C) =\u00a0\u2205<\/p>\n<p>A \u00d7 (B \u2212 C) = {1, 2, 3} \u00d7 \u2205<\/p>\n<p>=\u00a0\u2205<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u00d7 B) = {1, 2, 3} \u00d7 {4}<\/p>\n<p>= {(1, 4), (2, 4), (3, 4)}<\/p>\n<p>(A \u00d7 C) = {1, 2, 3} \u00d7 {5}<\/p>\n<p>= {(1, 5), (2, 5), (3, 5)}<\/p>\n<p>(A \u00d7 B) \u2212 (A \u00d7 C) =\u00a0\u2205<\/p>\n<p>\u2234\u00a0LHS = RHS<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-let-a-1-2-b-1-2-3-4-c-5-6-and-d-5-6-7-8-verify-that-i-a-x-c-%e2%8a%82-b-x-d-ii-a-x-b-%e2%88%a9-c-a-x-b-%e2%88%a9-a-x-c\"><\/span>4. Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that:<br \/>(i) A x C \u2282\u00a0B x D<br \/>(ii) A x (B\u00a0\u2229\u00a0C) = (A x B)\u00a0\u2229\u00a0(A x C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}<\/p>\n<p><strong>(i)\u00a0<\/strong>A x C \u2282\u00a0B x D<\/p>\n<p>Let us consider LHS A x C<\/p>\n<p>A \u00d7 C = {1, 2} \u00d7 {5, 6}<\/p>\n<p>= {(1, 5), (1, 6), (2, 5), (2, 6)}<\/p>\n<p>Now, RHS<\/p>\n<p>B \u00d7 D = {1, 2, 3, 4} \u00d7 {5, 6, 7, 8}<\/p>\n<p>= {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}<\/p>\n<p>Since, all elements of A \u00d7 C is in B \u00d7 D.<\/p>\n<p>\u2234We can say A\u00a0\u00d7\u00a0C\u00a0\u2282\u00a0B\u00a0\u00d7\u00a0D<\/p>\n<p><strong>(ii)<\/strong>\u00a0A \u00d7 (B\u00a0\u2229\u00a0C) = (A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<p>Let us consider LHS A \u00d7 (B\u00a0\u2229\u00a0C)<\/p>\n<p>(B\u00a0\u2229\u00a0C) =\u00a0\u2205<\/p>\n<p>A \u00d7 (B\u00a0\u2229\u00a0C) = {1, 2} \u00d7 \u2205<\/p>\n<p>= \u2205<\/p>\n<p>Now, RHS<\/p>\n<p>(A \u00d7 B) = {1, 2} \u00d7 {1, 2, 3, 4}<\/p>\n<p>= {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}<\/p>\n<p>(A \u00d7 C) = {1, 2} \u00d7 {5, 6}<\/p>\n<p>= {(1, 5), (1, 6), (2, 5), (2, 6)}<\/p>\n<p>Since, there is no common element between A \u00d7 B and A \u00d7 C<\/p>\n<p>(A \u00d7 B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C) =\u00a0\u2205<\/p>\n<p>\u2234 A \u00d7 (B\u00a0\u2229\u00a0C) = (A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-if-a-1-2-3-b-3-4-and-c-4-5-6-find-i-a-x-b-%e2%88%a9-c-ii-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%88%aa-c-iv-a-x-b-%e2%88%aa-a-x-c5-if-a-1-2-3-b-3-4-and-c-4-5-6-find-i-a-x-b-%e2%88%a9-c-ii-a-x-b-%e2%88%a9-a-x-c-iii-a-x-b-%e2%88%aa-c-iv-a-x-b-%e2%88%aa-a-x-c\"><\/span>5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find<br \/>(i) A x (B\u00a0\u2229\u00a0C)<br \/>(ii) (A x B)\u00a0\u2229\u00a0(A x C)<br \/>(iii) A x (B\u00a0\u222a\u00a0C)<br \/>(iv) (A x B)\u00a0\u222a\u00a0(A x C)5. If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find<br \/>(i) A x (B\u00a0\u2229\u00a0C)<br \/>(ii) (A x B)\u00a0\u2229\u00a0(A x C)<br \/>(iii) A x (B\u00a0\u222a\u00a0C)<br \/>(iv) (A x B)\u00a0\u222a\u00a0(A x C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}<\/p>\n<p><strong>(i)<\/strong>\u00a0A \u00d7 (B\u00a0\u2229\u00a0C)<\/p>\n<p>(B\u00a0\u2229\u00a0C) = {4}<\/p>\n<p>A\u00a0\u00d7\u00a0(B\u00a0\u2229\u00a0C) = {1, 2, 3} \u00d7 {4}<\/p>\n<p>= {(1, 4), (2, 4), (3, 4)}<\/p>\n<p><strong>(ii)<\/strong>\u00a0(A \u00d7 B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<p>(A\u00a0\u00d7\u00a0B) = {1, 2, 3} \u00d7 {3, 4}<\/p>\n<p>= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}<\/p>\n<p>(A\u00a0\u00d7\u00a0C) = {1, 2, 3} \u00d7 {4, 5, 6}<\/p>\n<p>= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<p>(A\u00a0\u00d7\u00a0B)\u00a0\u2229\u00a0(A\u00a0\u00d7\u00a0C) = {(1, 4), (2, 4), (3, 4)}<\/p>\n<p><strong>(iii)<\/strong>\u00a0A \u00d7 (B\u00a0\u222a\u00a0C)<\/p>\n<p>(B\u00a0\u222a\u00a0C) = {3, 4, 5, 6}<\/p>\n<p>A \u00d7 (B\u00a0\u222a\u00a0C) = {1, 2, 3} \u00d7 {3, 4, 5, 6}<\/p>\n<p>= {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}<\/p>\n<p><strong>(iv)<\/strong>\u00a0(A \u00d7 B)\u00a0\u222a\u00a0(A\u00a0\u00d7\u00a0C)<\/p>\n<p>(A \u00d7 B) = {1, 2, 3} \u00d7 {3, 4}<\/p>\n<p>= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}<\/p>\n<p>(A \u00d7 C) = {1, 2, 3} \u00d7 {4, 5, 6}<\/p>\n<p>= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<p>(A\u00a0\u00d7\u00a0B)\u00a0\u222a\u00a0(A\u00a0\u00d7\u00a0C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-prove-that-i-a-%e2%88%aa-b-x-c-a-x-c-a-x-c-%e2%88%aa-b-x-c\"><\/span>6. Prove that:<br \/>(i) (A\u00a0\u222a\u00a0B) x C = (A x C) = (A x C)\u00a0\u222a\u00a0(B x C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-%e2%88%a9-b-x-c-a-x-c-%e2%88%a9-b-x-c\"><\/span>(ii) (A\u00a0\u2229\u00a0B) x C = (A x C)\u00a0\u2229\u00a0(B x C)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>(A\u00a0\u222a\u00a0B) x C = (A x C) = (A x C)\u00a0\u222a\u00a0(B x C)<\/p>\n<p>Let (x, y) be an arbitrary element of (A\u00a0\u222a\u00a0B)\u00a0\u00d7\u00a0C<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B) C<\/p>\n<p>Since, (x, y) are elements of Cartesian product of (A\u00a0\u222a\u00a0B)\u00a0\u00d7\u00a0C<\/p>\n<p>x\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x\u00a0\u2208\u00a0A or x \u2208 B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x\u00a0\u2208\u00a0A and y\u00a0\u2208\u00a0C)\u00a0or\u00a0(x\u00a0\u2208\u00a0Band y\u00a0\u2208\u00a0C)<\/p>\n<p>(x, y)\u00a0\u2208\u00a0A\u00a0\u00d7\u00a0C\u00a0or\u00a0(x, y)\u00a0\u2208\u00a0B \u00d7 C<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C)\u00a0\u222a\u00a0(B\u00a0\u00d7\u00a0C) \u2026 (1)<\/p>\n<p>Let (x, y) be an arbitrary element of (A\u00a0\u00d7\u00a0C)\u00a0\u222a\u00a0(B\u00a0\u00d7\u00a0C).<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C)\u00a0\u222a\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C) or (x, y)\u00a0\u2208\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p>(x\u00a0\u2208\u00a0A and y\u00a0\u2208\u00a0C)\u00a0or\u00a0(x\u00a0\u2208\u00a0B and y\u00a0\u2208\u00a0C)<\/p>\n<p>(x\u00a0\u2208\u00a0A or x\u00a0\u2208\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>x\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u222a\u00a0B)\u00a0\u00d7\u00a0C \u2026 (2)<\/p>\n<p>From 1 and 2, we get: (A\u00a0\u222a\u00a0B)\u00a0\u00d7\u00a0C = (A\u00a0\u00d7\u00a0C)\u00a0\u222a\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p><strong>(ii)<\/strong>\u00a0(A\u00a0\u2229\u00a0B) x C = (A x C)\u00a0\u2229\u00a0(B x C)<\/p>\n<p>Let (x, y) be an arbitrary element of (A\u00a0\u2229\u00a0B)\u00a0\u00d7\u00a0C.<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)\u00a0\u00d7\u00a0C<\/p>\n<p>Since, (x, y) are elements of Cartesian product of (A\u00a0\u2229\u00a0B) \u00d7\u00a0C<\/p>\n<p>x\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x\u00a0\u2208\u00a0A and x\u00a0\u2208 B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x\u00a0\u2208\u00a0A and y\u00a0\u2208\u00a0C)\u00a0and\u00a0(x\u00a0\u2208\u00a0Band y\u00a0\u2208\u00a0C)<\/p>\n<p>(x, y)\u00a0\u2208\u00a0A\u00a0\u00d7\u00a0C\u00a0and\u00a0(x, y)\u00a0\u2208\u00a0B\u00a0\u00d7\u00a0C<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C) \u2026 (1)<\/p>\n<p>Let (x, y) be an arbitrary element of (A\u00a0\u00d7\u00a0C)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C).<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u00d7\u00a0C) and (x, y)\u00a0\u2208\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<p>(x\u00a0\u2208A and y\u00a0\u2208\u00a0C)\u00a0and\u00a0(x\u00a0\u2208\u00a0Band y\u00a0\u2208\u00a0C)<\/p>\n<p>(x\u00a0\u2208A and x\u00a0\u2208\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>x\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)\u00a0and\u00a0y\u00a0\u2208\u00a0C<\/p>\n<p>(x, y)\u00a0\u2208\u00a0(A\u00a0\u2229\u00a0B)\u00a0\u00d7\u00a0C \u2026 (2)<\/p>\n<p>From 1 and 2, we get: (A\u00a0\u2229\u00a0B)\u00a0\u00d7\u00a0C = (A\u00a0\u00d7\u00a0C)\u00a0\u2229\u00a0(B\u00a0\u00d7\u00a0C)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-if-a-x-b-%e2%8a%86-c-x-d-and-a-%e2%88%a9-b-%e2%88%88-%e2%88%85-prove-that-a-%e2%8a%86-c-and-b-%e2%8a%86-d\"><\/span>7. If A x B\u00a0\u2286\u00a0C x D and A\u00a0\u2229\u00a0B\u00a0\u2208\u00a0\u2205, Prove that A\u00a0\u2286\u00a0C and B\u00a0\u2286\u00a0D.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given:<\/p>\n<p>A \u00d7 B\u00a0<strong>\u2286<\/strong>\u00a0C x D and A\u00a0\u2229\u00a0B\u00a0\u2208\u00a0\u2205<\/p>\n<p>A \u00d7 B\u00a0<strong>\u2286<\/strong>\u00a0C x D denotes A \u00d7 B is subset of C \u00d7 D that is every element A \u00d7 B is in C \u00d7 D.<\/p>\n<p>And A\u00a0\u2229\u00a0B\u00a0\u2208\u00a0\u2205\u00a0denotes A and B does not have any common element between them.<\/p>\n<p>A \u00d7 B = {(a, b): a\u00a0\u2208\u00a0A and b\u00a0\u2208\u00a0B}<\/p>\n<p>\u2234We can say (a, b)\u00a0<strong>\u2286<\/strong>\u00a0C \u00d7 D [Since, A \u00d7 B\u00a0<strong>\u2286<\/strong>\u00a0C x D is given]<\/p>\n<p>a\u00a0\u2208\u00a0C and b\u00a0\u2208\u00a0D<\/p>\n<p>a\u00a0\u2208\u00a0A = a\u00a0\u2208\u00a0C<\/p>\n<p>A\u00a0<strong>\u2286<\/strong>\u00a0C<\/p>\n<p>And<\/p>\n<p>b\u00a0\u2208\u00a0B = b\u00a0\u2208\u00a0D<\/p>\n<p>B\u00a0<strong>\u2286<\/strong>\u00a0D<\/p>\n<p>Hence proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"exercise-23-page-no-220\"><\/span>EXERCISE 2.3 PAGE NO: 2.20<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"1-if-a-1-2-3-b-4-5-6-which-of-the-following-are-relations-from-a-to-b-give-reasons-in-support-of-your-answer\"><\/span><strong>1. If A = {1, 2, 3}, B = {4, 5, 6}, which of the following are relations from A to B?<br \/>Give reasons in support of your answer.<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"i-1-6-3-4-5-2\"><\/span>(i) {(1, 6), (3, 4), (5, 2)}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-1-5-2-6-3-4-3-6\"><\/span>(ii) {(1, 5), (2, 6), (3, 4), (3, 6)}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-4-2-4-3-5-1\"><\/span>(iii) {(4, 2), (4, 3), (5, 1)}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-%c3%97-b\"><\/span>(iv) A \u00d7 B<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p>A = {1, 2, 3}, B = {4, 5, 6}<\/p>\n<p>A relation from A to B can be defined as:<\/p>\n<p>A \u00d7 B = {1, 2, 3} \u00d7 {4, 5, 6}<\/p>\n<p>= {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6)}<\/p>\n<p><strong>(i)<\/strong>\u00a0{(1, 6), (3, 4), (5, 2)}<\/p>\n<p>No, it is not a relation from A to B. The given set is not a subset of A \u00d7 B as (5, 2) is not a part of the relation from A to B.<\/p>\n<p><strong>(ii)<\/strong>\u00a0{(1, 5), (2, 6), (3, 4), (3, 6)}<\/p>\n<p>Yes, it is a relation from A to B. The given set is a subset of A \u00d7 B.<\/p>\n<p><strong>(iii)<\/strong>\u00a0{(4, 2), (4, 3), (5, 1)}<\/p>\n<p>No, it is not a relation from A to B. The given set is not a subset of A \u00d7 B.<\/p>\n<p><strong>(iv)<\/strong>\u00a0A \u00d7 B<\/p>\n<p>A \u00d7 B is a relation from A to B and can be defined as:<\/p>\n<p>{(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6),(3, 4),(3, 5),(3, 6)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"2-a-relation-r-is-defined-from-a-set-a-2-3-4-5-to-a-set-b-3-6-7-10-as-follows-x-y-r-x-is-relatively-prime-to-y-express-r-as-a-set-of-ordered-pairs-and-determine-its-domain-and-range\"><\/span>2. A relation R is defined from a set A = {2, 3, 4, 5} to a set B = {3, 6, 7, 10} as follows: (x, y)\u00a0R\u00a0x is relatively prime to y. Express R as a set of ordered pairs and determine its domain and range.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Relatively prime numbers are also known as co-prime numbers. If there is no integer greater than one that divides both (that is, their greatest common divisor is one).<\/p>\n<p>Given: (x, y) \u2208\u00a0R\u00a0= x is relatively prime to y<\/p>\n<p>Here,<\/p>\n<p>2 is co-prime to 3 and 7.<\/p>\n<p>3 is co-prime to 7 and 10.<\/p>\n<p>4 is co-prime to 3 and 7.<\/p>\n<p>5 is co-prime to 3, 6, and 7.<\/p>\n<p>\u2234\u00a0R = {(2, 3), (2, 7), (3, 7), (3, 10), (4, 3), (4, 7), (5, 3), (5, 6), (5, 7)}<\/p>\n<p>The domain of relation R = {2, 3, 4, 5}<\/p>\n<p>Range of relation R = {3, 6, 7, 10}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"3-let-a-be-the-set-of-the-first-five-natural-and-let-r-be-a-relation-on-a-defined-as-follows-x-y-r-x-%e2%89%a4-y-express-r-and-r-1-as-sets-of-ordered-pairs-determine-also-i-the-domain-of-r%e2%80%911-ii-the-range-of-r\"><\/span>3. Let A be the set of the first five natural and let R be a relation on A defined as follows: (x, y) R x \u2264 y<br \/>Express R and R<sup>-1<\/sup>\u00a0as sets of ordered pairs. Determine also<br \/>(i) The domain of R<sup>\u20111<\/sup><br \/>(ii) The Range of R.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>A is a set of the first five natural numbers.<\/p>\n<p>So, A= {1, 2, 3, 4, 5}<\/p>\n<p>Given: (x, y)\u00a0R\u00a0x\u00a0\u2264\u00a0y<\/p>\n<p>1 is less than 2, 3, 4, and 5.<\/p>\n<p>2 is less than 3, 4, and 5.<\/p>\n<p>3 is less than 4 and 5.<\/p>\n<p>4 is less than 5.<\/p>\n<p>5 is not less than any number\u00a0A<\/p>\n<p>\u2234\u00a0R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 4), (2, 5), (3, 3), (3, 4), (3, 5), (4, 4), (4, 5), (5, 5)}<\/p>\n<p>\u201cAn inverse relation is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original relation. If the graph of a function contains a point (a, b), then the graph of the inverse relation of this function contains the point (b, a)\u201d.<\/p>\n<p>\u2234\u00a0R<sup>-1<\/sup>\u00a0= {(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (2, 2), (3, 2), (4, 2), (5, 2), (3, 3), (4, 3), (5, 3), (4, 4), (5, 4) (5, 5)}<\/p>\n<p>(i) Domain of R<sup>\u20111<\/sup>\u00a0= {1, 2, 3, 4, 5}<\/p>\n<p>(ii) Range of R = {1, 2, 3, 4, 5}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"4-find-the-inverse-relation-r-1-in-each-of-the-following-cases-i-r-1-2-1-3-2-3-3-2-5-6-ii-r-x-y-x-y-%e2%88%88-n-x-2y-8-iii-r-is-a-relation-from-11-12-13-to-8-10-12-defined-by-y-x-%e2%80%93-3\"><\/span>4. Find the inverse relation R<sup>-1<\/sup>\u00a0in each of the following cases:<br \/>(i) R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}<br \/>(ii) R= {(x, y) : x, y\u00a0\u2208\u00a0N; x + 2y = 8}<br \/>(iii) R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x \u2013 3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)<\/strong>\u00a0Given:<\/p>\n<p>R= {(1, 2), (1, 3), (2, 3), (3, 2), (5, 6)}<\/p>\n<p>So, R<sup>\u20111<\/sup>\u00a0= {(2, 1), (3, 1), (3, 2), (2, 3), (6, 5)}<\/p>\n<p><strong>(ii)<\/strong>\u00a0Given,<\/p>\n<p>R= {(x, y): x, y \u2208\u00a0N; x + 2y = 8}<\/p>\n<p>Here, x + 2y = 8<\/p>\n<p>x = 8 \u2013 2y<\/p>\n<p>As y \u2208\u00a0N, Put the values of y = 1, 2, 3,\u2026\u2026 till x \u2208\u00a0N<\/p>\n<p>When, y = 1, x = 8 \u2013 2(1) = 8 \u2013 2 = 6<\/p>\n<p>When, y = 2, x = 8 \u2013 2(2) = 8 \u2013 4 = 4<\/p>\n<p>When, y = 3, x = 8 \u2013 2(3) = 8 \u2013 6 = 2<\/p>\n<p>When, y = 4, x = 8 \u2013 2(4) = 8 \u2013 8 = 0<\/p>\n<p>Now, y cannot hold the value 4 because x = 0 for y = 4 which is not a natural number.<\/p>\n<p>\u2234\u00a0R = {(2, 3), (4, 2), (6, 1)}<\/p>\n<p>R<sup>\u20111<\/sup>\u00a0= {(3, 2), (2, 4), (1, 6)}<\/p>\n<p><strong>(iii)<\/strong>\u00a0Given,<\/p>\n<p>R is a relation from {11, 12, 13} to (8, 10, 12} defined by y = x \u2013 3<\/p>\n<p>Here,<\/p>\n<p>x =\u00a0{11, 12, 13} and y =\u00a0(8, 10, 12}<\/p>\n<p>y = x \u2013 3<\/p>\n<p>When, x = 11, y = 11 \u2013 3 = 8 \u2208\u00a0(8, 10, 12}<\/p>\n<p>When, x = 12, y = 12 \u2013 3 = 9\u00a0\u2209\u00a0(8, 10, 12}<\/p>\n<p>When, x = 13, y = 13 \u2013 3 = 10 \u2208\u00a0(8, 10, 12}<\/p>\n<p>\u2234\u00a0R = {(11, 8), (13, 10)}<\/p>\n<p>R<sup>\u20111<\/sup>\u00a0= {(8, 11), (10, 13)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"5-write-the-following-relations-as-sets-of-ordered-pairs-i-a-relation-r-from-the-set-2-3-4-5-6-to-the-set-1-2-3-defined-by-x-2y\"><\/span>5. Write the following relations as sets of ordered pairs:<br \/>(i) A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-a-relation-r-on-the-set-1-2-3-4-5-6-7-defined-by-x-y-%e2%88%88-r-%e2%87%94-x-is-relatively-prime-to-y\"><\/span>(ii) A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y)\u00a0\u2208\u00a0R \u21d4 x is relatively prime to y.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iii-a-relation-r-on-the-set-0-1-2%e2%80%a610-defined-by-2x-3y-12\"><\/span>(iii) A relation R on the set {0, 1, 2,\u2026,10} defined by 2x + 3y = 12.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"iv-a-relation-r-form-a-set-a-5-6-7-8-to-the-set-b-10-12-15-16-18-defined-by-x-y-r-x-divides-y\"><\/span>(iv) A relation R form a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by (x, y)\u00a0R\u00a0x divides y.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>A relation R from the set {2, 3, 4, 5, 6} to the set {1, 2, 3} defined by x = 2y.<\/p>\n<p>Let A = {2, 3, 4, 5, 6} and B = {1, 2, 3}<\/p>\n<p>Given, x = 2y where y\u00a0= {1, 2, 3}<\/p>\n<p>When, y = 1, x = 2(1) = 2<\/p>\n<p>When, y = 2, x = 2(2) = 4<\/p>\n<p>When, y = 3, x = 2(3) = 6<\/p>\n<p>\u2234\u00a0R = {(2, 1), (4, 2), (6, 3)}<\/p>\n<p><strong>(ii)<\/strong>\u00a0A relation R on the set {1, 2, 3, 4, 5, 6, 7} defined by (x, y) \u2208\u00a0R \u21d4 x is relatively prime to y.<\/p>\n<p>Given:<\/p>\n<p>(x, y)\u00a0R\u00a0x is relatively prime to y<\/p>\n<p>Here,<\/p>\n<p>2 is co-prime to 3, 5, and 7.<\/p>\n<p>3 is co-prime to 2, 4, 5, and 7.<\/p>\n<p>4 is co-prime to 3, 5, and 7.<\/p>\n<p>5 is co-prime to 2, 3, 4, 6, and 7.<\/p>\n<p>6\u00a0is co-prime to 5 and 7.<\/p>\n<p>7 is co-prime to 2, 3, 4, 5, and 6.<\/p>\n<p>\u2234\u00a0R ={(2, 3), (2, 5), (2, 7), (3, 2), (3, 4), (3, 5), (3, 7), (4, 3), (4, 5), (4, 7), (5, 2), (5, 3), (5, 4), (5, 6), (5, 7), (6, 5), (6, 7), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6), (7, 7)}<\/p>\n<p><strong>(iii)\u00a0<\/strong>A relation R on the set {0, 1, 2,\u2026, 10} defined by 2x + 3y = 12.<\/p>\n<p>Given,<\/p>\n<p>(x, y)\u00a0R\u00a02x + 3y = 12<\/p>\n<p>Where x and y = {0, 1, 2,\u2026, 10}<\/p>\n<p>2x + 3y = 12<\/p>\n<p>2x = 12 \u2013 3y<\/p>\n<p>x = (12-3y)\/2<\/p>\n<p>When, y = 0, x = (12-3(0))\/2 = 12\/2 = 6<\/p>\n<p>When, y = 2, x = (12-3(2))\/2 = (12-6)\/2 = 6\/2 = 3<\/p>\n<p>When, y = 4, x = (12-3(4))\/2 = (12-12)\/2 = 0\/2 = 0<\/p>\n<p>\u2234\u00a0R = {(0, 4), (3, 2), (6, 0)}<\/p>\n<p><strong>(iv)\u00a0<\/strong>A relation R form a set A = {5, 6, 7, 8} to the set B = {10, 12, 15, 16, 18} defined by (x, y) \u2208\u00a0R \u21d4\u00a0x divides y.<\/p>\n<p>Given,<\/p>\n<p>(x, y)\u00a0R\u00a0x divides y<\/p>\n<p>Where, x\u00a0= {5, 6, 7, 8} and y =\u00a0{10, 12, 15, 16, 18}<\/p>\n<p>Here,<\/p>\n<p>5 divides 10 and 15.<\/p>\n<p>6 divides 12 and 18.<\/p>\n<p>7 divides none of the values of set B.<\/p>\n<p>8 divides 16.<\/p>\n<p>\u2234\u00a0R = {(5, 10), (5, 15), (6, 12), (6, 18), (8, 16)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"6-let-r-be-a-relation-in-n-defined-by-x-y-%e2%88%88-r-%e2%87%94-x-2y-8-express-r-and-r-1-as-sets-of-ordered-pairs\"><\/span>6. Let R be a relation in N defined by (x, y)\u00a0\u2208\u00a0R\u00a0\u21d4\u00a0x + 2y = 8. Express R and R<sup>-1<\/sup>\u00a0as sets of ordered pairs.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p>(x, y)\u00a0R\u00a0x + 2y = 8 where x \u2208\u00a0N and y \u2208\u00a0N<\/p>\n<p>x + 2y= 8<\/p>\n<p>x = 8 \u2013 2y<\/p>\n<p>Putting the values y = 1, 2, 3,\u2026\u2026 till x \u2208\u00a0N<\/p>\n<p>When, y = 1, x = 8 \u2013 2(1) = 8 \u2013 2 = 6<\/p>\n<p>When, y = 2, x = 8 \u2013 2(2) = 8 \u2013 4 = 4<\/p>\n<p>When, y = 3, x = 8 \u2013 2(3) = 8 \u2013 6 = 2<\/p>\n<p>When, y = 4, x = 8 \u2013 2(4) = 8 \u2013 8 = 0<\/p>\n<p>Now, y cannot hold the value 4 because x = 0 for y = 4 which is not a natural number.<\/p>\n<p>\u2234\u00a0R = {(2, 3), (4, 2), (6, 1)}<\/p>\n<p>R<sup>\u20111<\/sup>\u00a0= {(3, 2), (2, 4), (1, 6)}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"7-let-a-3-5-and-b-7-11-let-r-a-b-a-%e2%88%88-a-b-%e2%88%88-b-a-b-is-odd-show-that-r-is-an-empty-relation-from-a-to-b\"><\/span>7. Let A = {3, 5} and B = {7, 11}. Let R = {(a, b): a \u2208 A, b \u2208 B, a-b is odd}. Show that R is an empty relation from A to B.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p>A = {3, 5} and B = {7, 11}<\/p>\n<p>R = {(a, b): a \u2208\u00a0A, b \u2208\u00a0B, a-b is odd}<\/p>\n<p>On putting a = 3 and b = 7,<\/p>\n<p>a \u2013 b = 3 \u2013 7 = -4 which is not odd<\/p>\n<p>On putting a = 3 and b = 11,<\/p>\n<p>a \u2013 b = 3 \u2013 11 = -8 which is not odd<\/p>\n<p>On putting a = 5 and b = 7:<\/p>\n<p>a \u2013 b = 5 \u2013 7 = -2 which is not odd<\/p>\n<p>On putting a = 5 and b = 11:<\/p>\n<p>a \u2013 b = 5 \u2013 11 = -6 which is not odd<\/p>\n<p>\u2234\u00a0R = { } =\u00a0\u03a6<\/p>\n<p>R is an empty relation from A to B.<\/p>\n<p>Hence proved.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"8-let-a-1-2-and-b-3-4-find-the-total-number-of-relations-from-a-to-b\"><\/span>8. Let A = {1, 2} and B = {3, 4}. Find the total number of relations from A to B.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>Given,<\/p>\n<p>A= {1, 2}, B= {3, 4}<\/p>\n<p>n (A) = 2 (Number of elements in set A).<\/p>\n<p>n (B) = 2 (Number of elements in set B).<\/p>\n<p>We know,<\/p>\n<p>n (A \u00d7 B) = n (A) \u00d7 n (B)<\/p>\n<p>= 2 \u00d7 2<\/p>\n<p>= 4 [since, n(x) = a, n(y) = b. total number of relations = 2<sup>ab<\/sup>]<\/p>\n<p>\u2234 A number of relations from A to B are 2<sup>4<\/sup>\u00a0= 16.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"9-determine-the-domain-and-range-of-the-relation-r-defined-by-i-r-x-x5-x-%e2%88%88-0-1-2-3-4-5\"><\/span>9. Determine the domain and range of the relation R defined by<br \/>(i) R = {(x, x+5): x\u00a0\u2208\u00a0{0, 1, 2, 3, 4, 5}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-r-x-x3-x-is-a-prime-number-less-than-10\"><\/span>(ii) R= {(x, x<sup>3<\/sup>): x is a prime number less than 10}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>R = {(x, x+5): x \u2208 {0, 1, 2, 3, 4, 5}<\/p>\n<p>Given,<\/p>\n<p>R = {(x, x+5): x \u2208\u00a0{0, 1, 2, 3, 4, 5}<\/p>\n<p>\u2234\u00a0R = {(0, 0+5), (1, 1+5), (2, 2+5), (3, 3+5), (4, 4+5), (5, 5+5)}<\/p>\n<p>R = {(0, 5), (1, 6), (2, 7), (3, 8), (4, 9), (5, 10)}<\/p>\n<p>So,<\/p>\n<p>The domain of relation R = {0, 1, 2, 3, 4, 5}<\/p>\n<p>Range of relation R = {5, 6, 7, 8, 9, 10}<\/p>\n<p><strong>(ii)\u00a0<\/strong>R= {(x, x<sup>3<\/sup>): x is a prime number less than 10}<\/p>\n<p>Given,<\/p>\n<p>R = {(x, x<sup>3<\/sup>): x is a prime number less than 10}<\/p>\n<p>Prime numbers less than 10 are 2, 3, 5 and 7<\/p>\n<p>\u2234\u00a0R = {(2, 2<sup>3<\/sup>), (3, 3<sup>3<\/sup>), (5, 5<sup>3<\/sup>), (7, 7<sup>3<\/sup>)}<\/p>\n<p>R = {(2, 8), (3, 27), (5, 125), (7, 343)}<\/p>\n<p>So,<\/p>\n<p>Domain of relation R = {2, 3, 5, 7}<\/p>\n<p>Range of relation R = {8, 27, 125, 343}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"10-determine-the-domain-and-range-of-the-following-relations-i-r-a-b-a-%e2%88%88-n-a-%3c-5-b-4\"><\/span>10. Determine the domain and range of the following relations:<br \/>(i) R= {a, b): a\u00a0\u2208\u00a0N, a &lt; 5, b = 4}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<h3><span class=\"ez-toc-section\" id=\"ii-s-a-b-b-a-1-a-%e2%88%88-z-and-a-%e2%89%a4-3\"><\/span>(ii) S= {a, b): b = |a-1|, a \u2208\u00a0Z and |a|\u00a0\u2264\u00a03}<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p><strong>(i)\u00a0<\/strong>R= {a, b): a\u00a0\u2208 N, a &lt; 5, b = 4}<\/p>\n<p>Given,<\/p>\n<p>R= {a, b): a \u2208\u00a0N, a &lt; 5, b = 4}<\/p>\n<p>Natural numbers less than 5 are 1, 2, 3 and 4<\/p>\n<p>a\u00a0= {1, 2, 3, 4} and b\u00a0= {4}<\/p>\n<p>R = {(1, 4), (2, 4), (3, 4), (4, 4)}<\/p>\n<p>So,<\/p>\n<p>The domain of relation R = {1, 2, 3, 4}<\/p>\n<p>Range of relation R = {4}<\/p>\n<p><strong>(ii)\u00a0<\/strong>S= {a, b): b = |a-1|, a \u2208\u00a0Z and |a|\u00a0\u2264\u00a03}<\/p>\n<p>Given,<\/p>\n<p>S= {a, b): b = |a-1|, a \u2208\u00a0Z and |a|\u00a0\u2264\u00a03}<\/p>\n<p>Z denotes integer which can be positive as well as negative<\/p>\n<p>Now, |a|\u00a0\u2264\u00a03 and b = |a-1|<\/p>\n<p>\u2234\u00a0a\u00a0= {-3, -2, -1, 0, 1, 2, 3}<\/p>\n<p>For, a = -3, -2, -1, 0, 1, 2, 3 we get,<\/p>\n<p>S = {(-3, |-3 \u2013 1|), (-2, |-2 \u2013 1|), (-1, |-1 \u2013 1|), (0, |0 \u2013 1|), (1, |1 \u2013 1|), (2, |2 \u2013 1|), (3, |3 \u2013 1|)}<\/p>\n<p>S = {(-3, |-4|), (-2, |-3|), (-1, |-2|), (0, |-1|), (1, |0|), (2, |1|), (3, |2|)}<\/p>\n<p>S = {(-3, 4), (-2, 3), (-1, 2), (0, 1), (1, 0), (2, 1), (3, 2)}<\/p>\n<p>b = 4, 3, 2, 1, 0, 1, 2<\/p>\n<p>So,<\/p>\n<p>Domain of relation S = {0, -1, -2, -3, 1, 2, 3}<\/p>\n<p>Range of relation S = {0, 1, 2, 3, 4}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"11-let-a-a-b-list-all-relations-on-a-and-find-their-number\"><\/span>11. Let A = {a, b}. List all relations on A and find their number.<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Solution:<\/strong><\/p>\n<p>The total number of relations that can be defined from a set A to a set B is the number of possible subsets of A \u00d7 B. If n (A) = p and n (B) = q, then n (A \u00d7 B) = pq.<\/p>\n<p>So, the total number of relations is 2<sup>pq<\/sup>.<\/p>\n<p>Now,<\/p>\n<p>A \u00d7 A = {(a, a), (a, b), (b, a), (b, b)}<\/p>\n<p>Total number of relations are all possible subsets of A \u00d7 A:<\/p>\n<p>[{(a, a), (a, b), (b, a), (b, b)}, {(a, a), (a, b)}, {(a, a), (b, a)},{(a, a), (b, b)}, {(a, b), (b, a)}, {(a, b), (b, b)}, {(b, a), (b, b)}, {(a, a), (a, b), (b, a)}, {(a, b), (b, a), (b, b)}, {(a, a), (b, a), (b, b)}, {(a, a), (a, b), (b, b)}, {(a, a), (a, b), (b, a), (b, b)}]<\/p>\n<p>n (A) = 2\u00a0\u21d2\u00a0n (A \u00d7 A) = 2 \u00d7 2 = 4<\/p>\n<p>\u2234\u00a0Total number of relations = 2<sup>4<\/sup>\u00a0= 16<\/p>\n<h2><span class=\"ez-toc-section\" id=\"important-topics-from-rd-sharma-solutions-class-11-maths-chapter-2\"><\/span>Important Topics From RD Sharma Solutions Class 11 Maths Chapter 2<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">The chapter has only a few topics important being listed below:<\/span><\/p>\n<ul>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Relation<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Domain<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Domain Range<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Using results and theorems<\/span><\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"key-take-away-from-rd-sharma-solutions-class-11-maths-chapter-2\"><\/span>Key Take Away From RD Sharma Solutions Class 11 Maths Chapter 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Understanding is the key.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Do not learn answers or the questions for the ease of solving them.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Analyze every problem you did wrong.<\/span><\/li>\n<li><span style=\"font-weight: 400;\"> \u00a0 \u00a0 \u00a0 <\/span><span style=\"font-weight: 400;\">Do not leave any concept in between. Go through again because everything you leave will create a void for further levels.<\/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\u00a0<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 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><span style=\"font-weight: 400;\">Doing this much is sufficient for Relations. Do let us know if you have any queries. We would like to answer your queries.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">All the best for your <a href=\"https:\/\/cbse.nic.in\/\" target=\"_blank\" rel=\"noopener noreferrer\">CBSE<\/a>\u00a0 class 11 examinations.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"faqs-on-rd-sharma-solutions-class-11-maths-chapter-2\"><\/span>FAQs on RD Sharma Solutions Class 11 Maths Chapter 2<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-1629718975403\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"are-the-solutions-of-rd-sharma-solutions-class-11-maths-chapter-2-reliable\"><\/span>Are the solutions of RD Sharma Solutions Class 11 Maths Chapter 2 reliable?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>The solutions of RD Sharma Solutions Class 11 Maths Chapter 2 are prepared by Subject matter experts and hence very reliable. <\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629719148088\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"can-i-access-the-rd-sharma-solutions-class-11-maths-chapter-2-pdf-offline\"><\/span>Can I access the RD Sharma Solutions Class 11 Maths Chapter 2 PDF offline?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>Yes, once you have downloaded the PDF online you can access it offline as well. <\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629719453563\" class=\"rank-math-list-item\">\n<h3 class=\"rank-math-question \"><span class=\"ez-toc-section\" id=\"from-where-can-i-download-the-rd-sharma-solutions-class-11-maths-chapter-2-pdf\"><\/span>From where can I download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download the PDF from the above link. <\/p>\n\n<\/div>\n<\/div>\n<div id=\"faq-question-1629719755077\" 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-rd-sharma-solutions-class-11-maths-chapter-2-pdf\"><\/span>How much does it cost to download the RD Sharma Solutions Class 11 Maths Chapter 2 PDF?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<div class=\"rank-math-answer \">\n\n<p>You can download the PDF of RD Sharma Solutions Class 11 Maths Chapter 2 for free. <\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>RD Sharma Solutions Class 11 Maths Chapter 2:\u00a0In this blog, we are going to talk about RD Sharma Solutions Class 11 Maths Chapter 2- Relations. For proper exam preparation, refer to RD Sharma Solutions Class 11 Maths Chapter 2. Get the exam ready with RD Sharma Solutions Class 11 Maths Chapter 2.\u00a0 Download RD Sharma &#8230; <a title=\"RD Sharma Solutions Class 11 Math&#8217;s Chapter 2 &#8211; Relations (Updated For 2024)\" class=\"read-more\" href=\"https:\/\/www.kopykitab.com\/blog\/rd-sharma-solutions-class-11-maths-chapter-2-relations\/\" aria-label=\"More on RD Sharma Solutions Class 11 Math&#8217;s Chapter 2 &#8211; Relations (Updated For 2024)\">Read more<\/a><\/p>\n","protected":false},"author":243,"featured_media":119177,"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\/62807"}],"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=62807"}],"version-history":[{"count":5,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/62807\/revisions"}],"predecessor-version":[{"id":500429,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/posts\/62807\/revisions\/500429"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media\/119177"}],"wp:attachment":[{"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/media?parent=62807"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/categories?post=62807"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.kopykitab.com\/blog\/wp-json\/wp\/v2\/tags?post=62807"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}