Question

On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.

Answer 1

Let e be the identity element in Z with respect to * such that

\[a * e = a = e * a, \forall a \in Z\] 
\[a * e = a \text{ and }e * a = a, \forall a \in Z\] 
\[a + e + 2 = a \text{ and }e + a + 2 = a, \forall a \in Z\] 
\[e = - 2 , \forall a \in Z\]

Thus, -2 is the identity element in Z with respect to *.