Question

Let * be a binary operation on the set $Q$ of rational numbers as follows:
$a\ast b=\frac{ab}{4}$
Find which of the binary operations are commutative and which are associative.

Answer 1

$a\ast b=\frac{ab}{4}$ is a binnary operation on set $Q$.

(1) Associative : $(a\ast b)\ast c=\frac{ab}{4}\ast c=\frac{ab}{4}\times \frac{c}{4}=\frac{abc}{16}$

$a\ast (b\ast c)=a\ast \frac{bc}{4}=\frac{a}{4}\times \frac{bc}{4}=\frac{abc}{16}=(a\ast b)\ast c$$\Rightarrow$ Associative.

(2) Commutative : $a\ast b=\frac{ab}{4}$

$b\ast a=\frac{ab}{4}=a\ast b\Rightarrow$ Commutative.