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Using the properties of sets and their complements prove thatA – (B ∪ C) = (A - B) ∩ (A - C) |
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Answer» Let, x ∈ {A − (B ∪ C)} x ∈ A and x ∉ (B ∪ C). (x ∈ A and x ∉ B) and (x ∈ A and x ∉ C) x ∈ (A − B) x ∈ (A − C) x ∈ {(A − B) ∩ (A − C)} A− (A − B) ⊆ (A − B) ∩ (A − C) …(i) Again, let y ∈ (A − B) ∩ (A − C) y ∈ (A − B) and y ∈ (A − C) (y ∈ A and y ∉ B) and (y ∈ A and y ∉ C y ∈ A and y ∉ B ∪ C) y ∈ {A − (B − C)} (A − B) ∩ (A − C) ⊆ A − (B ∪ C) …(ii) From eqs. (i) & (ii) A – (B ∪ C) = (A - B) ∩ (A - C) |
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