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Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $Q$ be the set of all rational numbers, and $*$ be the binary operation, defined by $a * b=a+2 b$, then

A. ${ }^{*}$ is commutative but not associative

B. $*$ is associative but not commutative

C. $*$ is neither commutative nor associative

D. $*$ is both commutative and associative

 

Solution:

According to the question,

$\mathrm{Q}$ is set of all rarional numbers

$R=\{(a, b): a * b=a+2 b\}$

Formula

$*$ is commutative if $a * b=b * a$

$*$ is associative if $\left(a^{*} b\right) * c=a^{*}\left(b^{*} c\right)$

Check for commutative

Consider, $a * b=a+2 b$

Both equations will not always be true .

Therefore , * is not commutative ……. (1)

Check for associative

Consider, $(a * b) * c=(a+2 b) * c=a+2 b+2 c$

And,$a^{*}\left(b^{*} c\right)=a^{*}(b+2 c)=a+2(b+2 c)=a+2 b+4 c$

Both the equation are not the same and therefore will not always be true.

Therefore , * is not associative ……. (2)

Now , according to the equations (1) , (2)

Correct option will be (C)

 

 

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