# Using properties of sets show that (i) A ∪ (A ∩ B) = A (ii) A ∩ (A ∪ B) = A.

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Using properties of sets show that

(i) A ∪ (A ∩ B) = A

(ii) A ∩ (A ∪ B) = A.

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(i) To show: A ∪ (A ∩ B) = A

We know that

A ⊂ A A ∩ B ⊂ A

∴ A ∪ (A ∩ B) ⊂ A … (1)

Also,

A ⊂ A ∪ (A ∩ B) … (2)

∴ From (1) and (2),

A ∪ (A ∩ B) = A

(ii)To show:

A ∩ (A ∪ B) = A

A ∩ (A ∪ B) = (A ∩ A) ∪ (A ∩ B)

= A ∪ (A ∩ B)  9 = A

{from (1)}