Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Let $Z$ be the set of all integers. Then, the operation * on $Z$ defined by $a * b=a+b-a b$ is

A. commutative but not associative

B. associative but not commutative

C. neither commutative nor associative

D. both commutative and associative

 

Solution:

According to the question,

$\mathrm{Q}=\{$ All integers $\}$

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

Formula

* is commutative if $\mathrm{a}^{*} \mathrm{~b}=\mathrm{b}^{*} \mathrm{a}$

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

Check for commutative

Consider, $a^{*} b=a+b-a b$

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

Both equations are the same and will always be true .

Therefore , * is commutative ……. (1)

Check for associative

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

$=a+b-a b+c-(a+b-a b) c$

$=a+b-a b+c-a c-b c+a b c$

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

$=a+(b+c-b c)-a(b+c-b c)$

$=a+b+c-b c-a b-a c+a b c$

Both the equation are the same and therefore will always be true.

Therefore, $*$ is associative $\ldots \ldots$.. (2)

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

Correct option will be (D)

 

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