# Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

Define $*$ on $\mathrm{Q}-\{-1\}$ by $\mathrm{a} * \mathrm{~b}=\mathrm{a}+\mathrm{b}+\mathrm{ab}$. Then, $*$ on $\mathrm{Q}-\{-1\}$ 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,

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

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 same and will always be true.

Therefore, $*$ is commutative...........(1)

Check for associative

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

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

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

And,$a^{*}\left(b^{*} c\right)=a^{*}(b+c+b c)$

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

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

Both the equation are 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)