Question

Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?

Answer 1

 Commutativity :

\[\text{Let }a, b \in Z . \text{Then}, \]

\[a * b = a - b\]

\[b * a = b - a\]

\[\text{Therefore},\]

\[a * b \neq b * a\]

Thus, * not is commutative on Z.

Associativity:

\[\text{Let }a, b, c \in Z . \text{Then}, \]

\[a * \left( b * c \right) = a * \left( b - c \right)\]

\[ = a - \left( b - c \right)\]

\[ = a - b + c\]

\[\left( a * b \right) * c = \left( a - b \right) - c\]

\[ = a - b - c\]

\[\text{Therefore},\]

\[a * \left( b * c \right) \neq \left( a * b \right) * c\]

Thus, * is not associative on Z.