Answer :

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.


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