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

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

Answer
(i) A ∪ (A ∩ B) = A
LHS
⇒ A∪ ( A ∩ B)
Use distribution property
⇒ (A ∪ A) ∩ ( A ∪ B)
Use relation A ∪ A = A
⇒ (A) ∩ ( A ∪ B)
⇒ A
RHS
Hence, A ∪ (A ∩ B) = A
(ii) A ∩ (A ∪ B) = A
LHS
⇒A ∩ (A ∪ B)
Use distribution property we get
⇒ (A ∩ A) ∪ (A ∩ B)
Use relation A ∩ A = A
⇒ (A ∩ A) ∪ (A ∩ B)
⇒ A ∪ (A ∩ B)
⇒ A
RHS
Hence, A ∩ (A ∪ B) = A