Question

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) 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*.

Answer 1

It is given the binary operation *:

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

Now, an element A ∈ P(X) is invertible if there exists B ∈ P(X) such that
A * B = X = B * A [As X is the identity element]
or
A∩ B = X = B ∩ A
This case is possible only when A = X = B.

Thus, X is the only invertible element in P(X) with respect to the given operation*.
Hence, the given result is proved.