# Relations and Functions

This page contains the 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
Relations and Functions – Summary
The following is the summary of the main features of this chapter.

Empty relation is the relation R in X given by R = ϕX × X
Universal Relation is the relation in R in X given by R = X × X
Reflexive relation R in X is a relation with {(a, a)}RaX
Symmetric relation R in X is a relation satisfying (a, b)R and (b, c)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 aX 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(x_1) = f(x_2) ⇒ x_1 = x_2} x_1, x_2X
A function f : X → Y is onto (or surjective) if given any yY, xX such that f(x) = y.
A function f : X → Y and g : B → C is the function gof : A → C given by {gof(x) = g(f(x))} xA.
A function f : X → Y is invertible if g : Y → X such that gof = Ix and fog = IY.
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 → 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.
A binary operation * on a set A is a function * from A × A to A
An element eX is the identity element for binary operation * : X × X → X, if {a * e = a = e * a} aX.
A element aX is invertible for binary operation * : X × X → X, if there exists bX 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}.
A 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