Given a non-empty set X, consider the binary operation `*: P(X)xx P(X) ->P(X)`given by `A * B = AnnB AAA , 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 i

Explanation

A binary operation * on set P(X) is lefined as follows : `AA A, B in P (X)`
`A ** B =A cup B`
Now , `A ** X = A cap X =A =X cap A = X ** A `
`therefore` X is the identity element for binary operation `A ** B=A cap B in P(X)`
Again `A in P(X)` will be inbertible if `B in P(X)` is such that
A * B=X=B * A
`rArr A cap B =X= B cap A `
It is possible when A=B+X Therefore, only one element X is invertible with respect to the binary operation `A ** B= A cup B "in" P (x)`