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

MCQ

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

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

✓

Answer

Given that $*$ be the binary operation defined on $Q$.

$a * b = a – b \forall a, b \in Q$

$= -b + a$
$= -(b - a)$
$= -b * a$
$\therefore\ \text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Hence, $*$ is not commutative.

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

$= b^2 + a^2$
$= b * a$
Hence $, *$ is commutative.

We have $a * b = a + ab$ and $b * a = b + ab$

Clearly, $\text{a}+\text{ab}\neq\text{b}+\text{ab}$
So, $*$ is not communicative.

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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