Question

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a+ab$
Find which of the binary operation are commutative and which are associative?

Answer 1

On Q, the operation $\ast$ is defined as $a\ast b=a+ab$
It can be observed that
$1\ast 2=1+21\times 2=1+2=3,2\ast 1=2+2\times 1=2+2=4$
$\therefore 1\ast 2\ne 2\ast 1$; where $1,2\in Q$
Thus, the operation $\ast$ is not commutative.
It can also be observed that
$(1\ast 2)\ast 3=(1+1\times 2)\ast 3=3\ast 3=3+3\times 3=3+9=121\ast (2\ast 3)=1\ast (2+2\times 3)=1\ast 8=1+1\times 8=9$
$\therefore (1\ast 2)\ast 3\ne 1\ast (2\ast 3)$ where $1,2,3\in Q$
Thus, the operation $\ast$ is not associative.