Q:

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

Accepted Solution

A:
Answer: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......