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
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)