Question

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

(i) $a\ast b=1\forall a,b\in R$

Answer 1

$\text{Checking commutative.}$

Given : $a\ast b=1\forall a,b\in R$

$\ast$ is commutative if

$a\ast b=b\ast a$

Now,

$a\ast b=1$

And,

$b\ast a=1$

Since

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

$\ast$ is commutative.

$\text{Checking associative}$

$\ast$ is associative if

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

Now,

$(a\ast b)\ast c=1\ast c$

$=1$

$a\ast (b\ast c)=a\ast 1$

$=1$

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

$\ast$ is an associative binary operation.