Question

Let $X$ be a non- empty set and $\ast$ be a binary operation on $P\left(x\right)$ defined by $A\ast B=A\cap B$ for all $A,B\in P\left(x\right)$. Find the identity and invertible elements

Answer 1

Let $X=\{1,2,3\}$

$P\left(X\right)=\{\varphi ,\left\{1\right\},\left\{2\right\},\left\{3\right\},\{1,2\},\{2,3\},\{1,3\},\{1,2,3\}\}$

$A\ast x=A\cap X=A$

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

$\left(i\right)e$ is the identity of $\ast$

$a\ast X=A\cap X=A$ for $a\in A$

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

$\therefore A\ast X=X\ast A=A$ for $A\in P\left(x\right)$

Hence $X$ is an identity element.

$\left(ii\right)a\ast b=e=b\ast a$

We have $e=X$

$\Rightarrow A\ast B=e=B\ast A$

$\Rightarrow A\cap B=X$

This is possible if $A=B=X$

$A\ast X=A=X\ast A$ for all $A\in P\left(X\right)$

Hence $X$ is the only invertible element in $P\left(X\right)$