Given a non-empty set X, consider the binary operation ∗: P(X) × P(X) → P(X) given by A ∗ B = A ∩ B ∀ A, B in P(X), where P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation ∗.

It is given that ∗: P(X) × P(X) → P(X) given by

A ∗ B = A ∩ B ∀ A, B ϵ P(X).

As we know that,

⇒ A * X = A = x * A A ϵ P(X).

Thus, X is the identity element for the given binary operation *.

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

A * B = X = B * A (As X is the identity element)

A ∩ B = X = B ∩ A

This can be possible only when A = X = B.

Therefore, X is the only invertible element in P(X) w.r.t. given operation *.

Hence Proved.