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

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 ...