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1. |
For any two sets A and B, prove that A ∪ (B – A) = A ∪ B |
Answer» Let x ϵ A ∪ (B –A) ⇒ x ϵ A or x ϵ (B – A) ⇒ x ϵ A or x ϵ B and x ∉ A ⇒ x ϵ B ⇒ x ϵ (A ∪ B) [∵ B ⊂ (A ∪ B)] This is true for all x ϵ A ∪ (B–A) ∴ A∪(B–A)⊂(A∪B)……(1) Conversely, Let x ϵ (A ∪ B) ⇒ x ϵ A or x ϵ B ⇒ x ϵ A or x ϵ (B–A) [∵ B ⊂ (A ∪ B)] ⇒ x ϵ A ∪ (B–A) ∴ (A∪B)⊂ A∪(B–A)……(2) From 1 and 2 we get… A ∪ (B – A) = A ∪ B |
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