Question

Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows. Find the binary operation given below is associative or commutative:

(v) $a\ast b=\frac{ab}{4}$

Answer 1

$\text{Checking commutative.}$

Given: $a\ast b=\frac{ab}{4}$

$\forall a,b\in Q,\frac{ab}{4}\in Q.$ So, it is binary

$\ast$ commutative if,

$a\ast b=b\ast a$

Now, $a\ast b=\frac{ab}{4}$

And, $b\ast a=\frac{ab}{4}$

Since $a\ast b=b\ast a\forall a,b\in Q$

$\ast$ is commutative.

$\text{Checking associative..}$

$\ast$ is associative if,

$(a\ast b)\ast c=a\ast (b\ast c)$

Now, $(a\ast b)\ast c=\left(\frac{ab}{4}\right)\ast c$

$=\frac{\frac{ab}{4}\times c}{4}=\frac{abc}{16}$

And, $a\ast (b\ast c)=a\ast \left(\frac{bc}{4}\right)$

$=\frac{a\times \frac{bc}{4}}{4}=\frac{abc}{16}$

Since $(a\ast b)\ast c\ne a\ast (b\ast c)\forall a,b,c\in Q$

$\ast$ is an associative binary operation.