Question

Let * be a binary operation on N defined by a * b = a + b + 10 for all a, b ∈ N. The identity element for * in N is
(a) −10
(b) 0
(c) 10
(d) non-existent

Answer 1

(d) non-existent

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

$$\begin{gathered} a*e=a=e*a,\forall a\in N \\ a*e=a\text{and}e*a=a,\forall a\in N \\ \mathrm{Then}, \\ a+e+10=a\text{and}e+a+10=a,\forall a\in N \\ e=-10\notin N \end{gathered}$$

So, the identity element with respect to * does not exist in N.