Question

Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.

Answer 1

 Commutativity:

\[\text{Let }a, b \in Q - \left\{ - 1 \right\} . \text{Then}, \] 
  \[a * b = a + b + ab\] 
           \[ = b + a + ba\] 
           \[ = b * a\] 
\[\text{Therefore},\] 

\[a * b = b * a, \forall a, b \in Q - \left\{ - 1 \right\}\]

Thus, * is commutative on Q - {-1}

Associativity:

\[\text{Let }a, b, c \in Q - \left\{ - 1 \right\} . \text{ Then }, \] 
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\] 
                  \[ = a + b + c + bc + a \left( b + c + bc \right)\] 
                  \[ = a + b + c + bc + ab + ac + abc\] 
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\] 
                   \[ = a + b + ab + c + \left( a + b + ab \right)c\] 
                    \[ = a + b + c + ab + ac + bc + abc\] 
\[\text{Therefore},\] 
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q - \left\{ - 1 \right\} . \] 

Thus, * is associative on Q -{-1]