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:

(i) $a\ast b=a-b$

Answer 1

$\text{Checking commutative.}$

Given $a\ast b=a-b$

$\forall a,b\in Q,a-b\in Q.$ So, it is binary

$\ast$ is commutative if

$a\ast b=b\ast a$

Now, $a\ast b=a-b$

And, $b\ast a=b-a$

Since

$a\ast b\ne b\ast a$

$\ast$ is not commutative.

$\text{Checking associative.}$

$\ast$ is associative if,

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

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

$=(a-b)-c$

$=a-b-c$

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

$=a-(b-c)$

$=a-b+c$

Since

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

$\ast$ is not an associative binary operations.