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# Given a non-empty set $X$, consider the binary operation $\ast : P(X) × P(X) \to P(X)$ given by $A \ast B=A \cap B\; \forall A,$ $B \;in\; P(X),$ 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 $\ast$.

Toolbox:
• An element X is identify element for a binary operation * if $A*X=A=X*A$
• An element A is invertible if then exists B such that $A*B=X=B*A$
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.
edited Mar 20, 2013