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Let $X$ be a nonempty set and $*$ be a binary operation on $P(X)$, the power set of $X$, defined by $A *$ E $A \cap B$ for all $A, B \in P(X)$ (i) Find the identity element in $\mathrm{P}(\mathrm{X})$. (ii) Show that $X$ is the only invertible element in $P(X)$.
Let $X$ be a nonempty set and $*$ be a binary operation on $P(X)$, the power set of $X$, defined by $A *$ E $A \cap B$ for all $A, B \in P(X)$ (i) Find the identity element in $\mathrm{P}(\mathrm{X})$. (ii) Show that $X$ is the only invertible element in $P(X)$.
Binary Operations | CBSE | Class 12 | Exercise 3A | Maths | RS Aggarwal
Let $X$ be a nonempty set and $*$ be a binary operation on $P(X)$, the power set of $X$, defined by $A *$ E $A \cap B$ for all $A, B \in P(X)$ (i) Find the identity element in $\mathrm{P}(\mathrm{X})$. (ii) Show that $X$ is the only invertible element in $P(X)$.

e is the identity of $*$ if $e^{*} a=a$

$\mathrm{A}$

From the Venn diagram,

$\begin{array}{l}
A * X=A \cap X=A \\
X * A=X \cap A=A
\end{array}$

$\Rightarrow \mathrm{X}$ is the identity element for binary operation *

Let $B$ be the invertible element

$\begin{array}{l}
\Rightarrow \mathrm{A}^{*} \mathrm{~B}=\mathrm{X} \\
\Rightarrow \mathrm{A} \cap \mathrm{B}=\mathrm{X}
\end{array}$

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

Thus $X$ is the only invertible element in $P(X)$

i

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