Relations and Functions – Class 12 Mathematics notes, important questions and MCQs

Chapter overview

Relations connect elements of sets; special types (reflexive, symmetric, transitive) make equivalence relations. Functions can be one-one, onto or both (bijective).

Relations

• Reflexive: (a, a) ∈ R for all a; symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R; transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R.
• Equivalence relation = reflexive + symmetric + transitive; it partitions a set into equivalence classes.

Functions

• One-one (injective): f(x₁) = f(x₂) ⇒ x₁ = x₂; onto (surjective): range = codomain.
• Bijective functions have inverses; composition (g∘f)(x) = g(f(x)).
• Number of functions from A (m elements) to B (n elements) = nᵐ; one-one functions (m ≤ n) = ⁿPₘ.

Key terms

Equivalence relation

relation that is reflexive, symmetric and transitive

Bijection

function that is both one-one and onto

Invertible function

a function f for which there is g with g∘f = identity and f∘g = identity; exists exactly when f is bijective

Formulas

• Functions A → B: nᵐ
• (g∘f)(x) = g(f(x))

Important questions with answers

Quick revision

• Equivalence = R + S + T.
• Only bijections are invertible.
• Onto: every element of codomain has a pre-image.

Exam tips

• f(x) = x² on ℝ is not one-one (f(2) = f(−2)); on ℕ it is.
• To show onto, take any y in the codomain and solve f(x) = y for x inside the domain.

Practice MCQs

1. A relation that is reflexive, symmetric and transitive is

   1. identity
   2. universal only
   3. empty
   4. equivalence
   Show answer

   Answer: (d) equivalence. Equivalence relation.

2. f: ℝ → ℝ, f(x) = 2x + 3 is

   1. one-one only
   2. onto only
   3. bijective
   4. neither
   Show answer

   Answer: (c) bijective. Linear with non-zero slope.

3. f(x) = x² from ℝ to ℝ is

   1. one-one
   2. onto
   3. neither one-one nor onto
   4. bijective
   Show answer

   Answer: (c) neither one-one nor onto. f(1) = f(−1) and negatives have no pre-image.

4. If f(x) = x + 1 and g(x) = 2x, then (g∘f)(3) is

   1. 7
   2. 6
   3. 8
   4. 10
   Show answer

   Answer: (c) 8. g(4) = 8.

5. Number of reflexive relations on a set with 3 elements is

   1. 2⁶
   2. 2⁹
   3. 2³
   4. 3²
   Show answer

   Answer: (a) 2⁶. Diagonal fixed; 6 other pairs free.

6. Number of functions from a set of 3 elements to a set of 2 elements is

   1. 6
   2. 9
   3. 8
   4. 5
   Show answer

   Answer: (c) 8. 2³.

7. Relation R on {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2)} is

   1. reflexive and transitive but not symmetric
   2. equivalence
   3. symmetric only
   4. not reflexive
   Show answer

   Answer: (a) reflexive and transitive but not symmetric. (2, 1) missing.

8. If f(x) = (3x + 2)/(x − 1), x ≠ 1, then f⁻¹(x) is

   1. (x + 3)/(x − 2)
   2. (x − 2)/(x + 3)
   3. (x + 2)/(x − 3)
   4. (3x − 2)/(x + 1)
   Show answer

   Answer: (c) (x + 2)/(x − 3). Solve y = (3x + 2)/(x − 1) for x.