Let * be the binary operation defined on Q. Find which of the fol… - Vidyadip

Question

Let $*$ be the binary operation defined on $Q$. Find which of the following binary operations are commutative:

$a * b = a – b \forall a, b \in Q$
$a * b = a^2 + b^2 \forall a, b \in Q$
$a * b = a + ab \forall a, b \in Q$
$a * b = (a – b)^2 \forall a, b \in Q$

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Answer

Given that $*$ be the binary operation defined on $Q.$
$i. a * b = a – b \forall a, b \in Q$
$= -b + a$
$= -(b - a)$
$= -b * a$
$\therefore\ \text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Hence, $*$ is not commutative.
$ii.a * b = a^2 + b^2$
$= b^2 + a^2$
$= b * a$
Hence $, *$ is commutative.
$iii$. We have $a^* b=a+a b$ and $b * a=b+a b$
Clearly, $a+a b \neq b+a b$
So $,*$ is not communicative.
$iv.$ We have $a ^* b=( a - b )^2 \forall a , b \in Q$
$= (-b + a)^2$
$= {-(b - a)}^2$
$= (b - a)^2$
$= b * a$
Hence $, *$ is communicative.

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