# Let $\ast$ be an operation defined as $\ast : R\times R\rightarrow R$ such that $a\ast b= 2a+b,a,b\epsilon R.$  Check if $\ast$ is a binary operation. If yes, find if it is associative too.

As $a,b\epsilon R\; \; \therefore 2a+b\: \epsilon R$  that is  $a\ast b\epsilon R$
Hence $\ast$ is a binary operation.
Assocoativily  Now $\left ( a\ast b\ast c \right )= \left ( 2a+b \right )\ast c, a,b,c\: \epsilon R$
$\Rightarrow \left ( a\ast b \right )\ast c= 4a+2b+c$
Also, $a\ast \left ( b\ast c \right )= a\ast \left ( 2b+c \right )= 2a+2b+c$
$\neq \left ( a\ast b \right )\ast c$.
Hence $\ast$ isn't associative.

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