Question

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?

Answer 1

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

\[a * e = a = e * a, \forall a \in Z\] 
\[a * e = a \text { and }e * a = a, \forall a \in Z\] 
\[a + e - 4 = a \text{ and }e + a - 4 = a, \forall a \in Z\] 
\[e = 4 , \forall a \in Z\]

Thus, 4 is the identity element in Z with respect to *.