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Given a non -empty set X, let **: P(X) x...

Given a non -empty set X, let `**: P(X) xx P(X) ->P(X)` be defined as `A **B = (A - B) uu(B - A), AAA , B in P(X)`. Show that the empty set `phi` is the identity for the operation `**` and all the elements A of P(A) are invertible with `A^-1=A`.

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It is given that `**: P(X) xx P(X) ->P(X)` be defined as `A **B = (A - B) uu(B - A), AAA , B in P(X)`.
`A ** phi = (A - phi) uu (phi-A) = A uu phi = A`
`phi ** A = (phi-A) uu (A-phi) = phi uu A = A`
`therefore A ** phi = A = phi ** A \ \ AA A in P(X)`.
`phi` is the identity for the operation `**`.
Element `A in P(X)` will be invertible if there exists `B in P(X)` such that `A**B = phi = B ** A`.
`A **A = (A - A) uu (A - A) = phi uu phi = phi \ \ AA A in P(X)`.
All the elements `A` of P(X) are invertible with `A^(-1) = A`.
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