Question

Given 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(X), where P(X)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(X) with respect to the operation.

Answer 1

Given that
$\ast :P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ is defined as $A\ast B=A\cap B\forall A,B\in P\left(X\right)$
We know that $A\ast X=A\cap X=A=X\cap A=X\ast A\forall A\in P\left(X\right)$
Therefore, the operation o is not commutative.
Let $a,b,c\in R$. Then, we have
(a o b)o c = a o c=a, a o (b o c)=a o b $=a\Rightarrow$ (a o b)o c =a o (b o c)
Therefore, the operation o is associative.
Now, let $a,b,c\in R$ then we have
$a\ast \left(boc\right)=a\ast b=|a-b|(a\ast b)o(a\ast c)=(|a-b|)o(|a-c|)=|a-b|$
Hence, $a\ast \left(boc\right)=1o(|2-3|)=1o1=1$
$\left(1o2\right)\ast \left(1o3\right)=1\ast 1=|1-1|=0$
Therefore, $1o(2\ast 3)\ne \left(1o2\right)\ast \left(1o3\right)$ where $1,2,3,\in R$
Therefore, the operation o does not distribute over $\ast .$