Question

Let 'o' be a binary operation on the set Q₀ of all non-zero rational numbers defined by   \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].

Show that 'o' is both commutative and associate ?

Answer 1

Commutativity :

\[\text{ Let }a, b \in Q_0 . \text{Then}, \] 
\[a o b = \frac{ab}{2}\] 
      \[ = \frac{ba}{2}\] 
      \[ = b o a\] 
\[\text{Therefore},\] 
\[a o b = b o a, \forall a, b \in Q_0\]

Thus, o is commutative on Q_o.

Associativity:

\[\text{ Let }a, b, c \in Q_0 . \text{ Then }, \] 
\[a o \left( b o c \right) = a o \left( \frac{bc}{2} \right)\] 
               \[ = \frac{a\left( \frac{bc}{2} \right)}{2}\] 
               \[ = \frac{abc}{4}\] 
\[\left( a o b \right) o c = \left( \frac{ab}{2} \right) o c\] 
               \[ = \frac{\left( \frac{ab}{2} \right)c}{2}\] 
                \[ = \frac{abc}{4}\] 
\[\text{Therefore},\] 
\[a o \left( b o c \right) = \left( a o b \right) o c, \forall a, b, c \in Q_0 \] 

Thus, o is associative on Q_o.