NCERT Chapter Summary: Relations and Functions (Class 12)

Class 12 Maths

In this chapter, you study different types of relations and equivalence relation, composition of functions, invertible functions and binary operations.

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

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

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

Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

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

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

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.

A function f : X → Y is one-one (or injective) if f(x₁) = f(x₂) ⇒ x₁ = x₂ ∀ x₁, x₂ ∈ X.

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

A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.

The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof(x) = g(f(x)) ∀ x ∈ A.

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

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

Given a finite set X, a function f : X → X 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.

A binary operation ∗ on a set A is a function ∗ from A × A to A.

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

An 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.

An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.

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