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

$$\begin{gathered} a*e=a=e*a,\forall a\in Z \\ a*e=a\text{and}e*a=a,\forall a\in Z \\ a+e+2=a\text{and}e+a+2=a,\forall a\in Z \\ e=-2,\forall a\in Z \end{gathered}$$

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