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
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)
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