Q7. Find the union of each pair: (i) A = {x : x is a natural number and a multiple of 3}, B = {x : x is a natural number less than 6}; (ii) A = {x : x is a natural number and 1 < x ≤ 6}, B = {x : x is a natural number and 6 < x < 10}; (iii) A = {1, 2, 3}, B = ϕ (NCERT Exercise 1.4, Q1 (iii)–(v)).
(i) {1, 2, 3, 4, 5, 6, 9, 12, 15, …}: the natural numbers below 6 together with all multiples of 3. (ii) {2, 3, 4, 5, 6, 7, 8, 9}. (iii) {1, 2, 3}.
Q8. For A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}, find A ∩ D, A ∩ (B ∪ D), (A ∩ B) ∩ (B ∪ C) and (A ∪ D) ∩ (B ∪ C) (NCERT Exercise 1.4, Q6 (vii)–(x)).
A ∩ D = ϕ; A ∩ (B ∪ D) = {7, 9, 11}; (A ∩ B) ∩ (B ∪ C) = {7, 9, 11}; (A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}.
Q9. State whether true or false: (i) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint; (ii) {2, 6, 10} and {3, 7, 11} are disjoint (NCERT Exercise 1.4, Q12 (iii), (iv)).
Both true: neither pair has a common element (the first sets are even, the second odd).
Q10. Verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) for A = {1, 2}, B = {2, 3}, C = {3, 4}.
B ∩ C = {3}, so the left side is {1, 2, 3}. A ∪ B = {1, 2, 3} and A ∪ C = {1, 2, 3, 4}, whose intersection is {1, 2, 3}. Both sides agree.
Q11. Show that A ∪ B = A ∩ B implies A = B (NCERT Example 25, page 21).
If a ∈ A, then a ∈ A ∪ B = A ∩ B, so a ∈ B: A ⊂ B. In the same way, if b ∈ B, then b ∈ A ∪ B = A ∩ B, so b ∈ A: B ⊂ A. Hence A = B.