Here operation ‘*‘ is defined as
* : P(X) × P(X) → P(X) such that
A * B = (A – B) ∪ (B – A) ∀ A, B ∈ P(X)
Existence of identity :
Let E ∈ P(X) be identity for ‘*‘ in set P(X)
⇒ A * E = A = E * A
⇒ (A – E) ∪ (E – A) = A = (E – A) ∪ (A – E)
It is possible only when E = Φ, Because
(A – Φ) ∪ (Φ – A) = A ∪ Φ = A and
(Φ – A) ∪ (A – Φ) = Φ ∪ A = A
Hence, Φ is identity element.
Existence of inverse :
Let A–1 be the inverse of A for ‘*‘ on set P(X).
∴ A * A–1 = Φ = A–1 * A
⇒ (A – A–1 ) ∪ (A–1 – A) = Φ
⇒ A – A–1 = Φ = A–1 – A = Φ
⇒ A ⊂ A–1 and A–1 ⊂ A
⇒ A = A–1
Hence, each element of P(X) is inverse of itself.