Class 12 Maths Chapter 1: Relations and Functions

Relations and Functions is Chapter 1 of NCERT Class 12 Maths. It builds on what you learnt in Class 11 and introduces the different types of relations (reflexive, symmetric, transitive and equivalence) and the different types of functions (one-one, onto and bijective). These notes give you clear definitions, simple tests and solved examples so you can revise the chapter quickly before the CBSE board exam.

What you will learn

The meaning of empty, universal and identity relations
How to test a relation for reflexive, symmetric and transitive properties
What an equivalence relation and an equivalence class are
How to check whether a function is one-one (injective) or onto (surjective)
When a function is bijective and how to count relations and functions on finite sets

Key Concepts and Formulas

Relations on a Set

A relation R on a set A is any subset of A × A. If (a, b) ∈ R, we say a is related to b and write a R b. If A has n elements, then A × A has n² elements, so a relation can be chosen in many ways.

Relation:           R ⊆ A × A
Empty relation:     R = φ
Universal relation: R = A × A
Identity relation:  R = {(a, a) : a ∈ A}
Relations on a set of n elements = 2^(n²)

In the empty relation no element is related to any element; in the universal relation every element is related to every element.

Reflexive, Symmetric and Transitive Relations

These three tests decide the nature of a relation. Always check each property separately and give a counter-example when a property fails.

Reflexive:  (a, a) ∈ R for every a ∈ A
Symmetric:  (a, b) ∈ R ⇒ (b, a) ∈ R
Transitive: (a, b) ∈ R and (b, c) ∈ R
            ⇒ (a, c) ∈ R

For reflexive, the condition must hold for every element of A.
One missing pair is enough to prove that a property fails.

Equivalence Relations and Classes

A relation that is reflexive, symmetric and transitive at the same time is called an equivalence relation. It splits the set into separate groups that do not overlap.

Equivalence class: [a] = {x ∈ A : x R a}
Either [a] = [b]  or  [a] ∩ [b] = φ

Two equivalence classes are either identical or disjoint.
The union of all equivalence classes is the whole set A.
Example of an equivalence relation: "a − b is divisible by m" on the set of integers.

Types of Functions

A function f : A → B assigns to each element of A exactly one element of B. A is the domain and B is the co-domain. The set of actual outputs is the range.

One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂
Onto:    range of f = co-domain B
Bijective = one-one and onto
If n(A) = m and n(B) = n:
Number of functions A → B = n^m
One-one function possible only if m ≤ n
Onto function possible only if m ≥ n

Arrow diagrams comparing one-one, onto and bijective functions for Class 12 Maths Chapter 1 Relations and Functions — One-one, onto and bijective functions as arrow diagrams

One-one (injective): different inputs always give different outputs.
Many-one: two different inputs give the same output.
Onto (surjective): every element of B has at least one pre-image.

Solved Examples

Example 1: Let A = {1, 2, 3} and R = {(1, 1), (2, 2), (3, 3), (1, 3)}. Check whether R is reflexive, symmetric and transitive.

(1,1), (2,2), (3,3) ∈ R  → reflexive
(1,3) ∈ R but (3,1) ∉ R  → not symmetric
(1,1), (1,3) ∈ R ⇒ (1,3) ∈ R
(1,3), (3,3) ∈ R ⇒ (1,3) ∈ R  → transitive
∴ R is reflexive and transitive, not symmetric

Example 2: On the set of integers Z, a R b if a − b is divisible by 4. Show that R is an equivalence relation and find the equivalence class of 1.

a − a = 0 = 4 × 0             → reflexive
a − b = 4k ⇒ b − a = 4(−k)    → symmetric
a − b = 4k and b − c = 4m
⇒ a − c = 4(k + m)            → transitive
[1] = {x ∈ Z : x − 1 = 4k}
∴ [1] = {..., −7, −3, 1, 5, 9, ...}

Four equivalence classes of integers under the relation a minus b divisible by 4, Class 12 Relations — The four equivalence classes modulo 4

Example 3: Show that f : R → R, f(x) = 5x − 2 is bijective.

f(a) = f(b) ⇒ 5a − 2 = 5b − 2
⇒ a = b                      → one-one
For any y ∈ R, take x = (y + 2)/5
f(x) = 5 × (y + 2)/5 − 2 = y → onto
∴ f is bijective

Example 4: Check whether f : R → R, f(x) = x² + 3 is one-one or onto.

f(2) = 4 + 3 = 7, f(−2) = 4 + 3 = 7
2 ≠ −2 but f(2) = f(−2)      → not one-one
x² ≥ 0 ⇒ f(x) ≥ 3
y = 1 has no pre-image       → not onto
∴ f is neither one-one nor onto

Example 5: Check whether f : N → N, f(n) = 2n + 1 is one-one or onto.

2a + 1 = 2b + 1 ⇒ a = b      → one-one
2n + 1 = 2 ⇒ n = ½ ∉ N       → not onto
∴ f is one-one but not onto

Example 6: If A = {p, q, r} and B = {5, 6}, find the number of onto functions and one-one functions from A to B.

Total functions = 2³ = 8
Constant functions (not onto) = 2
Onto functions = 8 − 2 = 6
One-one: n(A) = 3 > n(B) = 2, not possible
∴ Onto functions = 6, one-one functions = 0

Example 7: How many relations are possible on a set with 3 elements?

n(A × A) = 3² = 9
Number of relations = 2⁹
∴ Number of relations = 512

Common Mistakes to Avoid

Do not call a relation reflexive just because a few (a, a) pairs are present; every element must be related to itself.
When a property fails, write a specific counter-example pair; examiners give marks for it.
To prove onto, always solve for x in terms of y and check that x actually lies in the domain (for example, a natural number or a real number).
Remember that the answer to "one-one or onto" depends on the domain and co-domain; x² is not one-one on R but is one-one on [0, ∞).

Quick Revision

Relation on A: R ⊆ A × A
Relations on n elements = 2^(n²)
One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂
Onto: range = co-domain
Bijective = one-one + onto
Functions from m elements to n elements = n^m