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

\[a * e = a = e * a, \forall a \in Z\] 
\[a * e = a \text{ and }e * a = a, \forall a \in Z\] 
\[a + e - 5 = a \text{ and } e + a - 5 = a, \forall a \in Z\] 
\[e = 5, \forall a \in Z\]

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