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:

(iii) $a\ast b=a+ab$

Answer 1

$\text{Checking commutative.}$

Given: $a\ast b=a+ab$

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

$\ast$ commutative if,

$a\ast b=b\ast a$

Now, $a\ast b=a+ab$

And, $b\ast a=b+ba$

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

$\ast$ is not commutative.

$\text{Checking associative..}$

$\ast$ is associative if,

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

Now, $(a\ast b)\ast c=(a+ab)\ast c$

$=(a+ab)+(a+ab)c$

And, $a\ast (b\ast c)=a\ast (b+bc)$

$=a+a(b+bc)$

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

$\ast$ is not an associative binary operation.