Relations and Functions — Class 12 Maths Solution

It is given that, * be the binary operation defined on $Q$.

(i) $a*b = a - b,\forall a,b \in Q$ and $b*a = b - a$

So, $a*b \ne b*a$

Hence we can say that, * is not commutative.
(ii) $a*b = {a^2} + {b^2}$

$b*a = {b^2} + {a^2}$

So, * is commutative. [since, '+' is on rational is commutative]

(iii) $a*b = a + ab$

$b*a = b + ab$

Clearly, $a + ab \ne b + ab$

So, * is not commutative.
(iv) $a*b = {(a - b)^2},\forall a,b \in Q$

$b*a = {(b - a)^2}$

${(a - b)^2} = {(b - a)^2}$

Hence we can say that * is commutative.