Let $X$ be a nonempty set and $*$ be a binary operation on $P(X)$, the power set of $X$, defined by $A * B=$ $A \cap B$ for all $A, B \in P(X)$.
(i) Find the identity element in P(X).
(ii) Show that X is the only invertible element in P(X).
$\mathrm{e}$ is the identity of $*$ if $\mathrm{e}^{*} \mathrm{a}=\mathrm{a}$
From the above Venn diagram,
$A^{*} X=A \cap X=A$
$X^{*} A=X \cap A=A$
$\Rightarrow X$ is the identity element for binary operation $*$
Let B be the invertible element
$\Rightarrow \mathrm{A} * \mathrm{~B}=\mathrm{X}$
$\Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{X}$
This is only possible if $A=B=X$
Thus $X$ is the only invertible element in $P(X)$
Hence proved.
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