Question

Given a 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) 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: $\ast :P\left(X\right)\times P\left(X\right)\to P\left(X\right)$ given by

$A\ast B=A\cap B,\forall A,B$ in P(X) is the power set of X
e is the identity of * if
a * e = e * a = a

Every set is a subset of its superset. So,

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

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

So, $A\ast X=A=X\ast A,forallA\in P\left(X\right)$
Thus, X is the only identity element in P(X).

Solve for Invertible.
An element a in set is invertible if, there is an element in set such that,
a * b = e = b * a
Here, e = X
So, A * B = X = B * A

$i.e.,A\cap B=X$
This is only possible if A = B = X

So, A * X = A = X * A, for all $A\in P\left(X\right)$
Thus, X is the onlyl invertible element in P(X).