Question

The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.

Answer 1

\[\text{Let }a, b, c \in Q . \text{Then}, \] 
\[a * \left( b * c \right) = a * \left( \frac{bc}{7} \right)\] 
           \[ = \frac{a\left( \frac{bc}{7} \right)}{7}\] 
           \[ = \frac{abc}{49}\] 
\[\left( a * b \right) * c = \left( \frac{ab}{7} \right) * c\] 
           \[ = \frac{\left( \frac{ab}{7} \right)c}{7}\] 
           \[ = \frac{abc}{49}\] 
\[\text{Therefore},\] 
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q\] 

Thus, * is associative on Q.