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**NCERT mathematics class 12 chapter Relations and Functions summary.**. You can find the key concepts summary for the**chapter 1**of**NCERT class 12 mathematics**in this page. So is the case if you are looking for**NCERT class 12 Maths**related topic**Relations and Functions**. This page contains the summary of the chapter. If you’re looking for solutions to exercise problems, you can find them at●

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Relations and Functions – Summary

The following is the summary of the main features of this chapter.

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Empty relation is the relation R in X given by R = ϕ ⊂ X × X

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Universal Relation is the relation in R in X given by R = X × X

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Reflexive relation R in X is a relation with {(a, a)} ∈ R ∀ a ∈ X

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Symmetric relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R

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Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R

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Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.

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Equivalence class {[a]} containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.

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A function f : X → Y is one-one (or injective) if

{f(x_1) = f(x_2) ⇒ x_1 = x_2} ∀ x_1, x_2 ∈ X

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A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

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A function f : X → Y and g : B → C is the function gof : A → C given by {gof(x) = g(f(x))} ∀ x ∈ A.

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A function f : X → Y is invertible if ∃ g : Y → X such that gof = Ix and fog = IY.

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A function f : X → Y is invertible if and only if f is one-one and onto.

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Given a finite set X, a function f : X → Y is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set.

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A binary operation * on a set A is a function * from A × A to A

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An element e ∈ X is the identity element for binary operation * : X × X → X, if {a * e = a = e * a} ∀ a ∈ X.

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A element a ∈ X is invertible for binary operation * : X × X → X, if there exists b ∈ X such that {a * b = e = b * a} where, e is the identity for the binary operation *. The element b is called inverse of a and is denoted by a^{-1}.

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A operation * on X is commutative if {a * b = b * a} ∀ a, b in X

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An operation * on X is associative if (a * b) * c = a * (b * c) ∀ a, b, c in X