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Question
Let A, B and C be sets. Then show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Solution
We first show that A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)
Let x ∈ A ∪ (B ∩ C).
Then x ∈ A or x ∈ B ∩ C
⇒ x ∈ A or (x ∈ B and x ∈ C)
⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)
⇒ (x ∈ A ∪ B) and (x ∈ A ∪ C)
⇒ x ∈ (A ∪ B) ∩ (A ∪ C)
Thus, A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C) ......(1)
Now we will show that (A ∪ B) ∩ (A ∪ C) ⊂ (A ∪ C)
Let x ∈ (A ∪ B) ∩ (A ∪ C)
⇒ x ∈ A ∪ B and x ∈ A ∪ C
⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)
⇒ x ∈ A or (x ∈ B and x ∈ C)
⇒ x ∈ A or (x ∈ B ∩ C)
⇒ x ∈ A ∪ (B ∩ C)
Thus, (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C) ......(2)
So, from (1) and (2)
We have A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)
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