Relations and Functions

part 1

Types of Relations

• Recall of relations and functions from Class 11
• Types of relations — reflexive, symmetric, transitive
• Equivalence relations
• Equivalence classes

part 2

Types of Functions

• One-one (injective) functions
• Onto (surjective) functions
• Bijective functions
• Composition of functions

part 3

Inverse Functions and Binary Operations

• Invertible functions
• Inverse of a function
• Binary operations — definition
• Properties of binary operations

Equivalence relation

Reflexive + Symmetric + Transitive

★ board favourite

Reflexive relation

$aRa \;\forall a \in A$

★ frequently asked

Symmetric relation

$aRb \Rightarrow bRa \;\forall a,b \in A$

★ frequently asked

Transitive relation

$aRb \;\&\; bRc \Rightarrow aRc$

★ frequently asked

One-one (injective)

$f(x_1) = f(x_2) \Rightarrow x_1 = x_2$

★ frequently asked

Onto (surjective)

Range of f = Codomain

★ frequently asked

Show that the relation R on the set Z of integers, given by R = {(a, b) : 5 divides (a − b)}, is an equivalence relation. Also write the equivalence class [0].

Check whether the relation R on the set A = {1, 2, 3} defined by R = {(1,1),(2,2),(3,3),(1,2),(2,3)} is reflexive, symmetric and transitive.

Show that the relation R on the set A = {1, 2, 3, 4, 5} defined by R = {(a, b) : |a − b| is even} is an equivalence relation.

If f(x) = 2x + 1 and g(x) = x² − 2, find (f ∘ g)(x) and (g ∘ f)(x).

Let f : R − {2/3} → R be defined by f(x) = (4x + 3)/(6x − 4). Show that f is one-one and onto. Hence find f⁻¹.

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a − b| is divisible by 2} is reflexive and symmetric.

If a set A has 3 elements, how many reflexive relations can be defined on A?

Show that the function f : N → N given by f(x) = 2x is one-one but not onto.

Show that the function f : R → R defined by f(x) = x/(x² + 1) is neither one-one nor onto.

How many marks is Relations and Functions worth in the Class 12 board exam?

Around 5 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Bijective functions and composition — usually 2-mark MCQs.

Is Relations and Functions easy or hard?

It's rated medium — a moderate-difficulty chapter in Class 12 Maths. Most students need about 3 hours to cover it well.

What are the important topics in Relations and Functions?

The chapter covers 9 NCERT topics in 3 areas: Types of Relations; Types of Functions; Inverse Functions and Binary Operations.

What questions come from Relations and Functions in board exams?

Between 2019–2024, CBSE board papers asked questions from this chapter on Equivalence Relations, Types of Relations, Composition of Functions, Bijective Functions — 1- to 5-mark questions.