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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 $a * b=a+a b$ for all $a, b \in Q$. Then,

A. * is not a binary composition

B. $*$ is not commutative

C. $*$ is commutative but not associative

D. $*$ is both commutative and associative

Solution:

According to the question ,

$\mathrm{Q}=\{\mathrm{a}, \mathrm{b}\}$

$\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a} * \mathrm{~b}=\mathrm{a}+\mathrm{ab}\}$

Formula

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

$*$ is associative if $(\mathrm{a} * \mathrm{~b}) * \mathrm{c}=\mathrm{a} *(\mathrm{~b} * \mathrm{c})$

Check for commutative

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

And, $b * a=b+b a$

Both equations will not always be true .

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

Check for associative

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

And, $a *(b * c)=a *(b+b c)=a+a(b+b c)=a+a b+a b c$

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 (B)

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