Consider the binary operation * and `o` defined by the following tables on set `S={a ,\ b ,\ c ,\ d}` . (FIGURE) Show that both the binary operations are commutative and associative. Write down the identities and list the inverse of elements.

Explanation

From the given composition table it is clear that for all `a, b in S, a circ b in S` and `a circ b=b circ a`.


This gives that `circ` is commutative on `S`.


The identity element of `S` with respect to the `circ` is `b` since `a circ b=b circ a=` a, `forall a, b in S`.


But the inverse of a doesn't exist as `notin` any element `x` in `S` such that `{a} circ x=x circ a=` b.


But the inverses of `b, c, d` are `b, d, c` since `b circ b=b, c circ d=d circ c=b`.


So `b^{-1}=b, c^{-1}=d, d^{-1}=c`.