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 and let $a * b=a-b+a b$. Then, $*$ 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 $a * b=b * a$

$*$ is associative if $(a * b) * c=a *(b * c)$

Check for commutative

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

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

Both equations are not the same and will not always be true .

Therefore , * is not 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 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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