Question

Let * be a binary operation on the set $Q$ of rational numbers as follows:
$a\ast b=a^2+b^2$
Find which of the binary operations are commutative and which are associative.

Answer 1

Consider given the given binary operation,

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

Commutative-

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

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

$a\ast b=b\ast a$

Hence, given operation is commutative.

Associative –

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

$=a^4+{(b^2+c^2)}^2$

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

$={(a^2+b^2)}^2+c^4$

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

Hence, given operation is not binary.