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Question
Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Solution
A, B and C are three given sets
To prove: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Let x ∈ A ∩ (B ∪ C)
⇒ x ∈ A and x ∈ (B ∪ C)
⇒ x ∈ A and (x ∈ B or x ∈ C)
⇒ (x ∈ A or x ∈ B) or (x ∈ A or x ∈ C)
⇒ x ∈ A ∩ B or x ∈ A ∩ C
⇒ x ∈ (A ∩ B) ∪ (A ∩ C)
⇒ A ∩ (B ∩ C) ⊂ (A ∩ B) ∪(A ∩ C) ......(i)
Let y ∈ (A ∩ B) ∪ (A ∩ C)
⇒ y ∈ A ∩ B or x ∈ A ∩ C
⇒ (y ∈ A or y ∈ B) or (y ∈ A or y ∈ C)
⇒ y ∈ A and (y ∈ B or y ∈ C)
⇒ y ∈ A and y ∈ (B ∪ C)
⇒ y ∈ A ∩ (B ∩ C)
⇒ (A ∩ B) ∪ (A ∩ C) ⊂ A ∩ (B ∪ C) ......(ii)
We know that:
P ⊂ Q and Q ⊂ P
⇒ P = Q
From (i) and (ii)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Hence Proved
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