Question

Let $\ast$ be a binary operation on the Q of rational number as follows:
$\left(v\right)a\ast b=\frac{ab}{4}$
Find which of the binary operation are commutative and which are associative?

Answer 1

On Q, the operation $\ast$ is defined as $a\ast b=\frac{ab}{4}.$
For $a,b\in Q$, we have a $\ast b=\frac{ab}{4}=\frac{ba}{4}=b\ast a$
Therefore, $a\ast b=b\ast a$
Thus, the operation $\ast$ is commutative.
For $a,b,c\in Q$, we have $(a\ast b)\ast c=\frac{ab}{4}\ast c=\frac{\frac{ab}{4}.c}{4}=\frac{abc}{16}$
$a\ast (b\ast c)=a\ast \frac{bc}{4}=\frac{a.\frac{bc}{4}}{4}=\frac{abc}{16}$
Therefore, $(a\ast b)\ast c=a\ast (b\ast c)$. Thus, the operation $\ast$ is associative.