# Q 13 ) Given a non-empty set X, let ∗ : P(X) × P(X) → P(X) be defined as A * B = (A – B) ∪ (B – A), ∀ A, B ∈ P(X). Show that the empty set φ is the identity for the operation ∗ and all the elements A of P(X) are invertible with A–1 = A. (Hint : (A – φ) ∪ (φ – A) = A and (A – A) ∪ (A – A) = A ∗ A = φ).

It is given that

be defined as

Now, let  .
Then,

And

Therefore,

Therefore, we can say that    is the identity element for the given operation *.

Now, an element A  P(X) will be invertible if there exists B  P(X) such that

Now, We can see that

such that

Therefore, by this we can say that all the element A of P(X) are invertible with

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