Question

Verify whether the operation $\ast$ defined on $\varrho$ by $a\ast b=\frac{ab}{2}$ is associative or not.

Answer 1

Operation $\ast$ is defined by $a\ast b=\frac{ab}{2}$

Now we have to check whether the operation is associative or not.

Let us consider another variable $c$, the operation is said to be associative if $a\ast (b\ast c)=(a\ast b)\ast c$

Consider $a\ast (b\ast c)=a\ast \left(\frac{bc}{2}\right)$

$=\frac{abc}{4}$

Also, $(a\ast b)\ast c=\left(\frac{ab}{2}\right)\ast c=\frac{abc}{4}$

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

Hence, $\ast$ is associative.