1.

Let A, B and C be sets. Then show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Answer»

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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