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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 ?
Solution
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.
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