Q5. U = {1, 2, …, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8}. Find B′ and (A′)′ (NCERT Exercise 1.5, Q1 (ii), (v)).
B′ = {1, 3, 5, 7, 9}; (A′)′ = A = {1, 2, 3, 4}.
Q6. With U = N, find the complements of (i) the perfect squares; (ii) {x : 2x + 5 = 9}; (iii) {x : x ∈ N and 2x + 1 > 10} (NCERT Exercise 1.5, Q3 (vi), (ix), (xi)).
(i) The natural numbers that are not perfect squares: {2, 3, 5, 6, 7, 8, 10, …}. (ii) The set is {2}, so its complement is {1, 3, 4, 5, …}. (iii) 2x + 1 > 10 means x ≥ 5, so the complement is {1, 2, 3, 4}.
Q7. U = {a, b, c, d, e, f, g, h}. Find the complements of B = {d, e, f, g} and D = {f, g, h, a} (NCERT Exercise 1.5, Q2 (ii), (iv)).
B′ = {a, b, c, h}; D′ = {b, c, d, e}.
Q8. Show that A ∩ B and A ∩ B′ are disjoint and that their union is A.
An element of both would be in B and in B′ at once, which is impossible, so they are disjoint. Every element of A is either in B or not, so it lies in A ∩ B or in A ∩ B′; hence their union is A. (By the distributive law, (A ∩ B) ∪ (A ∩ B′) = A ∩ (B ∪ B′) = A ∩ U = A.)
Q9. With U = N, find the complements of (i) the odd natural numbers; (ii) {x : x is a natural number divisible by 3 and 5}; (iii) the perfect cubes (NCERT Exercise 1.5, Q3 (ii), (v), (vii)).
(i) The even natural numbers. (ii) The natural numbers not divisible by 15, that is, not divisible by 3 or not divisible by 5 (De Morgan’s law in words): {1, 2, …, 14, 16, …}. (iii) The natural numbers that are not perfect cubes: {2, 3, 4, 5, 6, 7, 9, …}.