Question

Show that (A | B) U (B \ A) = (AUB) \(B n A).

Answer 1

We have to prove,

(A \ B) ∪ ( B \ A ) = (A U B) \ (B ∩ A).

Suppose,

x ∈ (A \ B) ∪ ( B \ A ), where x is an arbitrary,

⇒ x ∈ A \ B or x ∈ B \ A

⇒ x ∈ A and x ∉ B or x ∈ B and x ∉ A

⇒ x ∈ A or x ∈ B and x ∉ B and x ∉ A

⇒ x ∈ A ∪ B and x ∉ B ∩ A

⇒ x ∈ ( A ∪ B ) \ ( B ∩ A )

Conversely,

Suppose,

y ∈ ( A ∪ B ) \ ( B ∩ A ), where, y is an arbitrary.

⇒ y ∈ A ∪ B and x ∉ B ∩ A

⇒ y ∈ A or y ∈ B and y ∉ B or y ∉ A

⇒ y ∈ A and y ∉ B or y ∈ B and y ∉ A

⇒ y ∈ A \ B or y ∈ B \ A

⇒ y ∈ ( A \ B ) ∪ ( B \ A )

Hence, proved......