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Let A, B and C be sets. Then show that

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

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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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