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