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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 theempty set `varphi`is the identity for the |
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Answer» 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` |
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