Question

For the binary operation * on Z defined by a * b = a + b + 1 the identity element is
(a) 0
(b) −1
(c) 1
(d) 2

Answer 1

(b) −1

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

$$\begin{gathered} a*e=a=e*a,\forall a\in Z \\ a*e=a\text{and}e*a=a,\forall a\in Z \\ \mathrm{Then},a+e+1=a\text{and}e+a+1=a,\forall a\in Z \\ e=-1\in Z,\forall a\in Z \end{gathered}$$

Thus, −1 is the identity element in Z with respect to *.