<\/span><\/h3>\nDefinition\/Theorems<\/em><\/strong><\/p>\n\n- Empty relation holds a specific relation R in X as: R = \u03c6 \u2282 X \u00d7 X<\/strong>.<\/li>\n
- A Symmetric relation R in X satisfies a certain relation as: (a, b) \u2208 R<\/strong> implies (b, a) \u2208 R<\/strong>.<\/li>\n
- A Reflexive relation R in X can be given as: (a, a) \u2208 R; for all \u2200 a \u2208 X<\/strong>.<\/li>\n
- A Transitive relation R in X can be given as: (a, b) \u2208 R and (b, c) \u2208 R, thereby, implying (a, c) \u2208 R<\/strong>.<\/li>\n
- A Universal relation is the relation R in X can be given by R = X \u00d7 X.<\/li>\n
- Equivalence relation R in X is a relation that shows all the reflexive, symmetric, and transitive relations.<\/li>\n<\/ol>\n
Properties & CBSE Class 12 Maths Formulas<\/em><\/strong><\/p>\n\n- A function f: X \u2192 Y is one-one\/injective; if f(x1<\/sub>) = f(x2<\/sub>) \u21d2 x1<\/sub> = x2<\/sub> \u2200 x1<\/sub> , x2<\/sub> \u2208 X.<\/li>\n
- A function f: X \u2192 Y is onto\/surjective; if given any y \u2208 Y, \u2203 x \u2208 X such that f(x) = y.<\/li>\n
- A function f: X \u2192 Y is one-one and onto or bijective; if f follows both the one-one and onto properties.<\/li>\n
- A function f: X \u2192 Y is invertible if \u2203 g: Y \u2192 X such that gof = IX<\/sub> and fog = IY<\/sub>. This can happen only if f is one-one and onto.<\/li>\n
- A binary operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> performed on a set A is a function ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> from A \u00d7 A to A.<\/li>\n
- An element e \u2208 X possess the identity element for binary operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> : X \u00d7 X \u2192 X, if a ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> e = a = e ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> a; \u2200 a \u2208 X.<\/li>\n
- An element a \u2208 X shows the invertible property for binary operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> : X \u00d7 X \u2192 X, if there exists b \u2208 X such that a ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> b = e = b ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> a where e is said to be the identity for the binary operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span>. The element b is called the inverse of a and is denoted by a\u20131<\/sup>.<\/li>\n
- An operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> on X is said to be commutative if a ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> b = b ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> a; \u2200 a, b in X.<\/li>\n
- An operation ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> on X is said to associative if (a ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> b) ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> c = a ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> (b ∗<\/mo><\/math>“>\u2217<\/span><\/span><\/span>\u2217<\/span><\/span> c); \u2200 a, b, c in X.<\/li>\n<\/ol>\n
<\/span>CBSE Class 12 Maths Formulas: Inverse Trigonometric Functions<\/strong><\/span><\/h3>\nInverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.<\/p>\n
Properties\/Theorems<\/em><\/strong><\/p>\nThe domain and range of inverse trigonometric functions are given below:<\/p>\n
\n\n\nFunctions<\/strong><\/td>\nDomain<\/strong><\/td>\nRange<\/strong><\/td>\n<\/tr>\n\ny = sin-1<\/sup> x<\/td>\n| [\u20131, 1]<\/td>\n | [<\/mo>−<\/mo>π<\/mi><\/mrow>2<\/mn><\/mfrac>,<\/mo>π<\/mi>2<\/mn><\/mfrac>]<\/mo><\/mrow><\/math>“>[<\/span>\u2212<\/span>\u03c0<\/span><\/span>2<\/span><\/span>,<\/span>\u03c0<\/span>2<\/span><\/span>]<\/span><\/span><\/span><\/span>[\u2212\u03c02,\u03c02]<\/span><\/span><\/td>\n<\/tr>\n\n| y = cos | | | | | |
![\"CBSE](\"https:\/\/www.kopykitab.com\/blog\/wp-content\/uploads\/2021\/04\/CBSE-Class-12-Maths-Formulas-1.jpg\")
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