Complement laws: (i) A ∪ A ′ = U (ii) A ∩ A ′ = φ . De Morgan’s law: (i) (A ∪ B)´ = A ′ ∩ B ′ (ii) (A ∩ B ) ′ = A ′ ∪ B ′ . Law of double complementation : (A ′ ) ′ = A . Laws of empty set and universal set φ′ = U and U ′ = φ .
These laws can be verified by using Venn diagrams. EXERCISE . . Let U = { , , , , , , , , }, A = { , , , }, B = { , , , } and C = { , , , }.
Find (i) A ′ (ii) B ′ (iii) (A ∪ C) ′ (iv) (A ∪ B) ′ (v) (A ′ ) ′ (vi) (B – C) ′ . If U = { a, b, c, d, e, f, g, h }, find the complements of the following sets : (i) A = { a, b, c } (ii) B = { d, e, f, g } (iii) C = { a, c, e, g } (iv) D = { f, g, h, a } . Taking the set of natural numbers as the universal set, write down the complements of the following sets: { x : x is an even natural number} (ii) { x : x is an odd natural number } (iii) { x : x is a positive multiple of } (iv) { x : x is a prime number } (v) { x : x is a natural number divisible by and } (vi) { x : x is a perfect square } (vii) { x : x is a perfect cube} (viii) { x : x + = } (ix) { x : x + = } (x) { x : x ≥ } (xi) { x : x ∈ N and x + > } . If U = { , , , , ,