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

\[a * e = a = e * a, \forall a \in I^+ \] 
\[a * e = a\text{ and }e * a = a, \forall a \in I^+ \] 
\[a + e = a \text{ and }e + a = a, \forall a \in I^+ \] 
\[e = 0 , \forall a \in I^+\]

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