Question

Show that for any sets A and B ,

(i) A = (A intersection B) intersection (A-B)

(ii) A U (B-A) = A U B​

Answer 1

Answer:

Step-by-step explanation:

(A ∩ B) ∪ (A - B)

Use relation, A – B = A ∩ B’

⇒ (A ∩ B) ∪ (A ∩ B’)

⇒A ∩ (B ∪ B’)

Use relation B ∪ B’ = U

⇒A ∩ U

⇒ A

LHS

Hence A = (A ∩ B) ∪ (A - B)

(2)A ∪ (B - A) = (A ∪ B)

LHS

A ∪ (B - A)

Use relation, B – A = B ∩ A’

⇒ A ∪ (B ∩ A’)

⇒ (A ∪ B) ∩ (A ∪ A’)

Use relation A ∪ A’ = U.

⇒ (A ∪ B) ∩ U

⇒ (A ∪ B)

RHS

Hence, A ∪ (B - A) = (A ∪ B)

Answer 2

Answer:

(1) (A ∩ B) ∪ (A - B)

Use relation, A – B = A ∩ B’

⇒ (A ∩ B) ∪ (A ∩ B’)

⇒A ∩ (B ∪ B’)

Use relation B ∪ B’ = U

⇒A ∩ U

⇒ A

LHS

Hence A = (A ∩ B) ∪ (A - B)