Question

Determine the given below binary operation on the set $R$ is associative or commutative.

$a\ast b=\frac{(a+b)}{2}\forall a,b\in R$

Answer 1

$\text{Checking commutative.}$

Given : $a\ast b=\frac{(a+b)}{2}\forall a,b\in R$

$\ast$ is commutative if

$a\ast b=b\ast a$

$a\ast b=\frac{a+b}{2}$

$b\ast a=\frac{b+a}{2}$

$b\ast a=\frac{a+b}{2}$

$a\ast b=b\ast a\forall a,b\in R$

$\ast$ is commutative.

$\text{Checking associative}$

$\ast$ is associative if

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

Now,

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

$=\frac{\frac{a+b}{2}+c}{2}$

$=\frac{\left(\frac{a+b+2c}{2}\right)}{2}$

$=\frac{a+b+2c}{2}$

And,

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

$=\frac{a+\frac{b+c}{2}}{2}$

$=\frac{\frac{2a+b+c}{2}}{2}$

$=\frac{2a+b+c}{4}$

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

$\ast$ is not an associative binary operation.