Question

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a^2+b^2$
Find which of the binary operation are commutative and which are associative?

Answer 1

On Q, the operation $\ast$ is defined as a$\ast b=a^2+b^2.$
For $a,b\in Q$,we have
$a\ast b=a^2+b^2=b^2+a^2=b\ast a$
Therefore, $a\ast b=b\ast a$
Thus, the operation $\ast$ is commutative.
It can be observed that
$(1\ast 2)\ast 3=(1^2+2^2)\ast 3=(1+4)\ast 4=5\ast 4=5^2+4^2=411\ast (2\ast 3)=1\ast (2^2+3^2)=1\ast (4+9)=1\ast 13=1^2+{13}^2=170$
$(1\ast 2)\ast 3\ne 1\ast (2\ast 3)$ where $1,2,3,\in Q$
Thus, the operation $\ast$ is not associative.