Question

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

Answer 1

$a\ast b=(a-b{)}^2$ is binarry operation on $Q$

(1)Associative: $(a\ast b)\ast c=(a-b{)}^2\ast c=((a-b{)}^2-c{)}^2$

$a\ast (b\ast c)=a\ast (b-c{)}^2=(a-(b-c{)}^2{)}^2\ne (a\ast b)\ast c$

(2) Commutative: $a\ast b=(a-b{)}^2=a^2+b^2-2ab$

$b\ast a=(b-a{)}^2=a^2+b^2-2ab=a\ast b.\Rightarrow$ Commutative.