Relations and Functions — Mathematics Class 12 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Relations and Functions (Class 12) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Relations and Functions (CBSE & HBSE)
Relations and Functions forms the theoretical backbone of Class 12 Maths. CBSE typically allocates 8 marks and HBSE 10 marks. Expect questions on identifying relation types (reflexive/symmetric/transitive/equivalence), one-one/onto functions, and composition.
Types of Relations
Types of Relations
A relation R on a set A is a subset of A × A. We write aRb or (a,b) ∈ R.
| Relation Type | Definition | Example on A={1,2,3} |
|---|---|---|
| Empty | R = ∅ | No pair related |
| Universal | R = A×A | Every pair related |
| Reflexive | (a,a) ∈ R ∀ a ∈ A | {(1,1),(2,2),(3,3),…} |
| Symmetric | (a,b) ∈ R ⇒ (b,a) ∈ R | {(1,2),(2,1)} |
| Transitive | (a,b),(b,c) ∈ R ⇒ (a,c) ∈ R | {(1,2),(2,3),(1,3)} |
| Equivalence | Reflexive + Symmetric + Transitive | See below |
Checking Relations Step-by-Step
Step 1 — Reflexive: Check if (a,a) is in R for EVERY element a of A. Step 2 — Symmetric: For every (a,b) in R, verify (b,a) is also in R. Step 3 — Transitive: For every (a,b) and (b,c) in R, verify (a,c) is in R.
Equivalence Relations
R is an equivalence relation iff it is reflexive, symmetric AND transitive simultaneously.
Equivalence Class of a ∈ A: [a] = {x ∈ A : xRa}
- Equivalence classes partition the set A into disjoint subsets.
- Every element belongs to exactly one equivalence class.
Classic Example: R on ℤ defined by aRb iff (a – b) is divisible by n is an equivalence relation. The equivalence classes are {0,n,2n,…}, {1,n+1,2n+1,…}, …, {n-1,2n-1,…}.
Important Tips
- Empty relation is symmetric and transitive but NOT reflexive (if A ≠ ∅).
- Universal relation is always an equivalence relation.
- Identity relation I = {(a,a) : a ∈ A} is always an equivalence relation.
CBSE Tip: CBSE frequently asks: "Is the relation R = {(a,b): a ≤ b} on ℝ an equivalence relation?" — Answer: Reflexive ✓, NOT symmetric (2 ≤ 3 but 3 ≰ 2), transitive ✓. So NOT equivalence.
Types of Functions
Types of Functions
A function f: A → B assigns each element of A to exactly one element of B.
| Type | Definition | Test |
|---|---|---|
| One-One (Injective) | f(a₁)=f(a₂) ⇒ a₁=a₂ | Horizontal line meets graph at most once |
| Onto (Surjective) | Range of f = B (every b ∈ B has a pre-image) | No element in co-domain is "missed" |
| Bijective | One-one AND onto | Horizontal line meets graph exactly once |
Methods to Check One-One
Algebraic: Assume f(x₁) = f(x₂), then show x₁ = x₂. Derivative method (for continuous functions): If f'(x) > 0 or f'(x) < 0 throughout domain, then f is strictly monotone ⇒ one-one.
Methods to Check Onto
Find the range of f. If Range = Co-domain B, then onto.
Example: f: ℝ → ℝ, f(x) = x² is neither one-one (f(2) = f(−2) = 4) nor onto (range = [0,∞) ≠ ℝ).
Example: f: ℝ → ℝ, f(x) = 2x+3 is one-one and onto (bijective).
Counting Injective and Surjective Functions
| Type | A has m elements, B has n elements |
|---|---|
| Total functions | n^m |
| Injective (one-one) | n × (n-1) × … × (n-m+1) = ⁿPₘ (requires n ≥ m) |
| Surjective (onto) | Inclusion-exclusion formula (complex) |
| Bijective | n! (requires n = m) |
Inverse of a Function
A function f: A → B has an inverse f⁻¹: B → A iff f is bijective. f⁻¹(y) = x ⟺ f(x) = y
CBSE Tip: To prove f is invertible, always first prove it is bijective (both one-one and onto). The inverse is then found by solving y = f(x) for x.
Composition of Functions and Binary Operations
Composition of Functions
If f: A → B and g: B → C, then gof: A → C is defined as: (gof)(x) = g(f(x))
Key Results:
- In general, gof ≠ fog (not commutative)
- (hog)of = ho(gof) (associative)
- If f and g are one-one, then gof is one-one
- If f and g are onto, then gof is onto
- If gof is one-one, then f is one-one
- If gof is onto, then g is onto
Invertible Functions
f: A → B is invertible iff f is bijective. If fog = I_B and gof = I_A, then f and g are inverses of each other.
Finding f⁻¹: Set y = f(x), solve for x, then write f⁻¹(y) = x.
Binary Operations
A binary operation ∗ on set A is a function ∗: A×A → A.
| Property | Definition |
|---|---|
| Commutative | a∗b = b∗a ∀ a,b ∈ A |
| Associative | (a∗b)∗c = a∗(b∗c) ∀ a,b,c ∈ A |
| Identity element e | a∗e = e∗a = a ∀ a ∈ A |
| Inverse of a | a∗b = b∗a = e (b is inverse of a) |
Examples of binary operations on ℝ:
- a∗b = a+b (commutative, associative, identity=0)
- a∗b = a−b (neither commutative nor associative)
- a∗b = ab (commutative, associative, identity=1)
- a∗b = max(a,b) (commutative, associative)
CBSE Tip: CBSE often asks to find identity element and inverse for a given binary operation. Set a∗e = a to find e, then a∗b = e to find b = a⁻¹.
Frequently asked questions
Are these Relations and Functions notes free?
Yes — the Relations and Functions notes for Mathematics (Class 12) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Relations and Functions notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Relations and Functions chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Relations and Functions.