We observe the following:

a * b = b * a = a a * a = a

c * a = a * c = a b * b = b

a * d = d * a = a c * c = d

b * c = c * b = c d * d = c

b * d = d * b = d

c * d = d * c = b

There ‘ * ’ is commutative.

Also,

a * (b * c) = a * (c) = a (From above)

(a * b) * c = (a) * c = a (Also from above)

Hence, ‘ * ’ is associative too.

Therefore, to find the identity element, e for e belong to S, we need:

a * e = e * a = a, a belong to S.

Therefore, a * e = a

We find that there is no unique element e which satisfies the condition.

e = a or b or c or d for a.

Since the identity is not unique, the inverse will also be not unique.