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:

$a\ast b=a^2+b^2$

Answer 1

$\text{Checking commutative.}$

Given: $a\ast b=a^2+b^2$

$\forall a,b\in Q,a^2+b^2\in Q.$ So, it is binary

$\ast$ is commutative if,

$a\ast b=b\ast a$

Now $a\ast b=a^2+b^2$

And, $b\ast a=b^2+a^2$

$=a^2+b^2$

Since $a\ast b=b\ast a\forall a,b\in Q,$

$\ast$ is commutative.

$\text{Checking associative.}$

$\ast$ is associative if,

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

Now, $(a\ast b)\ast c=(a^2+b^2)\ast c$

$=(a^2+b^2{)}^2+c^2$

And, $a\ast (b\ast c)=a(b^2+c^2)$

$=a^2+(b^2+c^2{)}^2$

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

$\ast$ is not an associative binary operation.