Question

Find the identity element in the set I⁺ of all positive integers defined by a * b = a + b for all a, b ∈ I⁺.

Answer 1

Let e be the identity element in I⁺ with respect to * such that
$$\begin{gathered} a*e=a=e*a,\forall a\in I^{+} \\ a*e=a\text{and}e*a=a,\forall a\in I^{+} \\ a+e=a\text{and}e+a=a,\forall a\in I^{+} \\ e=0,\forall a\in I^{+} \end{gathered}$$

Thus, 0 is the identity element in I⁺ with respect to *.