Relations and Functions
Learn Relations and Functions with notes, examples, and practice questions.
Chapter Overview
A relation is any subset of A × B. A function is a relation with no input repeated with different outputs.
Relations describe how elements of one set connect to another. Functions are special relations where every input has exactly one output.
Class 12 extends Class 11 ideas: types of relations, types of functions, composition, and invertibility — all tested heavily in board exams and JEE.
Types of Relations on a Set A
- Reflexive: (a,a) ∈ R for every a ∈ A
- Symmetric: (a,b) ∈ R ⇒ (b,a) ∈ R
- Transitive: (a,b) and (b,c) ∈ R ⇒ (a,c) ∈ R
- Equivalence relation: reflexive + symmetric + transitive
- Empty relation is reflexive; universal relation A×A is equivalence
Types of Functions
One-one (injective)
f(a) = f(b) ⇒ a = b
Onto (surjective)
Every element of codomain has a preimage
Bijective
One-one and onto — invertible
- Many-one: two inputs, same output
- Into: some codomain elements unused
- Constant function: one-one only if |A| = 1
Composition and Inverse
Composition
(f ∘ g)(x) = f(g(x))
Inverse property
f⁻¹(f(x)) = x and f(f⁻¹(y)) = y
(f ∘ g)(x) = f(g(x)) — apply g first, then f. Domain of f ∘ g: all x in domain of g where g(x) is in domain of f.
f is invertible iff bijective. Then f⁻¹ exists and (f⁻¹ ∘ f)(x) = x.
Binary Operations
- Commutative: a * b = b * a
- Associative: (a * b) * c = a * (b * c)
- Identity e: a * e = a = e * a
- Inverse of a: element b such that a * b = e
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Example 1: Check Reflexivity
Is R = {(1,1), (2,2), (1,2)} on A = {1,2} reflexive?
- 1Need (2,2) — present ✓
- 2Need (1,1) — present ✓
- 3All pairs (a,a) exist for a ∈ A.
Answer
Yes, R is reflexive on A
Example 2: Composition
If f(x) = 2x + 1 and g(x) = x², find (f ∘ g)(3).
- 1g(3) = 9
- 2f(9) = 2(9) + 1 = 19
Answer
19
Example 3: Inverse Function
Find f⁻¹ if f(x) = (3x − 2)/5.
- 1Let y = (3x−2)/5
- 25y = 3x − 2
- 3x = (5y + 2)/3
- 4f⁻¹(y) = (5y + 2)/3
Answer
f⁻¹(x) = (5x + 2)/3
Key Points to Remember
- ✓ Equivalence relations partition a set into equivalence classes
- ✓ Bijective functions have inverses
- ✓ Composition order matters: f ∘ g ≠ g ∘ f in general
- ✓ Graph test: vertical line test for functions
Exam Tips
- • Prove equivalence by checking all three properties separately
- • For onto: solve y = f(x) for x in terms of y
- • Composition questions: work inside-out
Formula Cheat Sheet
Quick reference — NCERT board exam formulas
Sets & Functions
- n(A × B) = n(A) × n(B)
- One-one:f(x₁) = f(x₂) ⇒ x₁ = x₂
- Onto:Range(f) = Codomain
- Composition:(f ∘ g)(x) = f(g(x))
Practice MCQs — Relations and Functions
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
A function must assign to each input: