Given a non -empty set X, let `*:" "P(X)" "xx" "P(X) ->P(X)` be defined as A * B = (A B) `uu` (B A), `AA` A, B ` in ` P(X)`A" "*" "B" "=" "(A" "-" "B) uu(B" "-" "A), AA""""A ," "B in P(X)` . Show that the empty set `varphi` is the identity for the

Explanation

Operation on set P(X) is
`A ** B =(A-B) cup (B-A) AA A ,B in p(X)`
Let `A in P(X)`
`therefore A ** phi =(A - phi ) cup (phi -A)`
and `phi ** A =(phi-A)cup (A- phi)`
`therefore A ** =A = phi ** A`
`rArr phi` is the identity element in the binary peration .operation Again `A in P(X)` will be invertible if and only if exists an element B in P(X) such that `A ** B =Phi = B**A `
Now `Lambda ** Lambda=(A-A)cup (A-A)`
`= phi cup phi = phi `
`therefore` All elements A of P(X) are invertible
and `A^(-1)=A`