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
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)