Given a non-empty set X, consider the binary operation

Given a non-empty set X, consider the binary operation $\ast :P\left ( X \right )\times P\left ( X \right )\rightarrow P\left ( X \right )$ 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 $\ast$ is commutative and associative and 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 $\ast$ .

Answers (1)

(i) for any $A,B\in P\left ( X \right ),A\ast B= A\cap B$ and $B\ast A= B\cap A$
as $A\cap B= B\cap A\, \therefore A\ast B= B\ast A$
$\Rightarrow \ast is \, commutative$
(ii) for any $A,B,C\in P\left ( X \right )$
$\left ( A\ast B \right )\ast C= \left ( A\cap B \right )\ast C= \left ( A\cap B \right )\cap C$
and $A\ast \left ( B\ast C \right )= A\ast \left ( B\cap C \right )= A\cap \left ( B\cap C \right )$
$\therefore \left ( A\cap B \right )\cap C= A\cap \left ( B\cap C \right )\Rightarrow \ast$ is associative
(iii) for any $A\in P\left ( X \right ),A\ast X= A\cap X= A$
$X\ast A= X\cap A= A$
$\Rightarrow$ X is the identity elementry
(iv) $X\ast X= X\cap X= X\Rightarrow X$ is the only innvertible element
$\because$ It is true only for X