Question

Given a non-empty set $X$, consider the binary operation $\ast :P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ given by $A\ast B=A\cap B\forall A,B$ in $P\left(X\right)$, where $P\left(X\right)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P\left(X\right)$ with respect to the operation $\ast$

Answer 1

Given defined :$P\left(X\right)\times P\left(X\right)\to P\left(X\right)$

and $A\ast B=A\cap B,A,B\in P\left(X\right)$

An element X is identify element for a binary operation if

$A\ast X=A=X\ast A$

$⟹A\cap X=A=X\cap A$

for $A\in P\left(X\right)$

Weknow that

$A\ast X=A=X\ast A$

Hence X is the identity element

An element A is invertible if then exists B such that $A\ast B=X=B\ast A,A\cap B=X$

and $B\cap A=X$

This is possible only if $A=B=X$

that is $A\ast B=X=B\ast A$ only possible element satisfying the relation is the element $X$.Hence $X$ is the identity element is proved.