Question

If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.

Answer 1

Let e be the identity element in Z with respect to * such that
$$\begin{gathered} a*e=a=e*a,\forall a\in Z \\ a*e=a\text{and}e*a=a,\forall a\in Z \\ a+e-5=a\text{and}e+a-5=a,\forall a\in Z \\ e=5,\forall a\in Z \end{gathered}$$

Thus, 5 is the identity element in Z with respect to *.