Given $\ast$ defined : $P(X) \times p(X) \to P(X)$ and given by $A*B =A \cap B, A,B \in P(X)$

An element X is identify element for a binary operation * if $ A*X=A=X*A$

$\Rightarrow A \cap X =A=X \cap A \qquad for A \in p(X)$

Therefore we know that $A*X=A=X*A$

Hence X is the identity element .

An element A is invertible if then exists B such that $ A*B=X=B*A$

$A \cap B =X \;and\;B \cap A=X \rightarrow$ This is possible only if $A=B=X$

ie $A *B=X=B*A$ only possible element satisfying the relation is the element X.