Question

Let * be a binary operation on the set $Q$ of rational numbers as follows:
$a\ast b=a+ab$
Find which of the binary operations are commutative and which are associative.

Answer 1

$a\ast b=a+ab$ is binary operation on $Q$

Checking Associative property $\Rightarrow (a\ast b)\ast =(a+ab)+(a+ab)c$

$=a+ab+ac+abc$

and $a\ast (b\ast c)=a\ast (b+bc)=a+a(b+bc)$

$=a+ab+abc$

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

Not Associative:

$a\ast b=a+ab$

Now , $b\ast a=b+ab\ne a\ast b$ Not commutative.