Question

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

Answer 1

 Commutativity :

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

\[a * b = a + b - ab\]

       \[ = b + a - ba\]

       \[ = b * a \]

\[\text{Therefore},\]

\[a * b = b * a, \forall a, b \in Z\]

Thus, * is commutative on Z.

Associativity:

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

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

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

\[ = a + b + c - bc - ab - ac + abc\]

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

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

\[ = a + b + c - ab - ac - bc + abc\]

\[\text{Therefore}\]

\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Z\]

Thus, * is associative on Z.