Question

Check the commutativity and associativity of the following binary operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?

Answer 1

Commutativity:

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

\[\text{a o b }= \frac{ab}{2}\]

\[ = \frac{ba}{2}\]

\[ = \text{b o a} \]

\[\text{Therefore},\]

\[ \text{a  o  b }= \text{b  o  a}, \forall a, b \in Q\]

Thus, o is commutative on Q.

Associativity: 

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

\[a o \left( b o c \right) = a o \left( \frac{bc}{2} \right)\]

\[ = \frac{a \left( \frac{bc}{2} \right)}{2}\]

\[ = \frac{abc}{4}\]

\[\left( a o b \right) o c = \left( \frac{ab}{2} \right) o c\]

\[ = \frac{\left( \frac{ab}{2} \right)c}{2}\]

\[ = \frac{abc}{4}\]

\[\text{Therefore},\]

\[a o \left( b o c \right) = \left( a o b \right) o c, \forall a, b, c \in Q\]

Thus, o is  associative on Q.