Mark the tick against the correct answer in the following:
Let $Q$ be the set of all rational numbers, and $*$ be the binary operation, defined by $a * b=a+2 b$, then
A. ${ }^{*}$ is commutative but not associative
B. $*$ is associative but not commutative
C. $*$ is neither commutative nor associative
D. $*$ is both commutative and associative
According to the question,
$\mathrm{Q}$ is set of all rarional numbers
$R=\{(a, b): a * b=a+2 b\}$
Formula
$*$ is commutative if $a * b=b * a$
$*$ is associative if $\left(a^{*} b\right) * c=a^{*}\left(b^{*} c\right)$
Check for commutative
Consider, $a * b=a+2 b$
Both equations will not always be true .
Therefore , * is not commutative ……. (1)
Check for associative
Consider, $(a * b) * c=(a+2 b) * c=a+2 b+2 c$
And,$a^{*}\left(b^{*} c\right)=a^{*}(b+2 c)=a+2(b+2 c)=a+2 b+4 c$
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)