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Question
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
Solution
Commutativity:
\[\text{Let }a, b \in Q . \text{Then}, \]
\[a * b = \frac{ab}{4}\]
\[ = \frac{ba}{4}\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in Q\]
Thus, * is commutative on Q.
Associativity :
\[\text{Let}a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( \frac{bc}{4} \right)\]
\[ = \frac{a\left( \frac{bc}{4} \right)}{4}\]
\[ = \frac{abc}{16}\]
\[\left( a * b \right) * c = \left( \frac{ab}{4} \right) * c\]
\[ = \frac{\left( \frac{ab}{4} \right)c}{4}\]
\[ = \frac{abc}{16}\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Q\]
Thus, * is associative on Q.
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