Class 11 Maths NCERT Solutions Chapter 2 Ex 2.2 – Relations

3x − y = 0 ⟹ y = 3x, and y must also lie in A, i.e. y ≤ 14.

So 3x ≤ 14 ⟹ x ≤ 4.67, giving x ∈ {1, 2, 3, 4}.

x = 1 → y = 3; x = 2 → y = 6; x = 3 → y = 9; x = 4 → y = 12

x is a natural number less than 4, so x ∈ {1, 2, 3}.

x = 1 → y = 6; x = 2 → y = 7; x = 3 → y = 8

Check |x − y| for every pair with x ∈ {1,2,3,5}, y ∈ {4,6,9}, keeping only odd differences.

x=1: |1−4|=3 (odd) ✓, |1−6|=5 (odd) ✓, |1−9|=8 (even) ✗

x=2: |2−4|=2 (even) ✗, |2−6|=4 (even) ✗, |2−9|=7 (odd) ✓

x=3: |3−4|=1 (odd) ✓, |3−6|=3 (odd) ✓, |3−9|=6 (even) ✗

x=5: |5−4|=1 (odd) ✓, |5−6|=1 (odd) ✓, |5−9|=4 (even) ✗

Each arrow subtracts 2: 5 → 3, 6 → 4, 7 → 5, so the rule is y = x − 2.

For each a ∈ A, find every b ∈ A that a divides exactly.

a=1: divides 1,2,3,4,6 → (1,1),(1,2),(1,3),(1,4),(1,6)

a=2: divides 2,4,6 → (2,2),(2,4),(2,6)

a=3: divides 3,6 → (3,3),(3,6)

a=4: divides 4 → (4,4)

a=6: divides 6 → (6,6)

Substituting each value of x: 0→5, 1→6, 2→7, 3→8, 4→9, 5→10

Prime numbers less than 10: 2, 3, 5, 7.

2³ = 8, 3³ = 27, 5³ = 125, 7³ = 343

n(A) = 3, n(B) = 2, so n(A × B) = 3 × 2 = 6.

The number of relations from A to B equals the number of subsets of A × B, which is 2⁶.

Since a and b are both integers, a − b is always an integer, no matter what values a and b take.

So every possible pair (a, b) with a, b ∈ Z satisfies the condition — R is the universal relation on Z.