Question

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

Answer 1

Let a ∈ Z and b ∈ Z be the inverse of a. Then,
\[a * b = e = b * a\] 
\[a * b = e \text{ and }b * a = e\] 
\[a + b - 4 = 4 \text{ and } b + a - 4 = 4\] 
\[b = 8 - a \in Z\] 
\[\text{Thus},8 - \text{a is the inverse of a} \in Z . \]