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:

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

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

\[ = b + a + ba\]

\[ = b * a \]

\[\text{Therefore},\]

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

Associativity :

\[\text{Let } a, b, c \in Z . \text{Then}, \] 
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\] 
\[ = a + \left( b + c + bc \right) + a\left( b + c + bc \right)\] 
\[ = a + b + c + bc + ab + ac + abc\] 
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\] 
\[ = a + b + ab + c + \left( a + b + ab \right)c\] 
\[ = a + b + ab + c + ac + bc + abc\] 
\[\text{Therefore},\] 
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Z\]

Thus, * is associative on Z.