Question

Given a non-empty set X, let * : P(X) x P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A) for all A, B ∈ P(X). Show that emptry set Φ is the identity for the operation * and all the elements of P(X) are invertible with A-1 = A.

Answer 1

Φ is the empty set.

The operation * is defined as

A * B = (A – B) ∪ (B-A)

Putting B = Φ, we get

A * Φ = (A – Φ) ∪ (Φ – A) = A ∪ Φ = A

Φ * A = (Φ – A) ∪ (A – Φ) = Φ ∪ A = A

⇒ A * Φ = Φ * A

Also, A * A = (A – A) ∪ (A – A) = Φ ∪ Φ = Φ

⇒ is an identity element.

Also, A * A = Φ ⇒ A-1 = A.