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.

 

 

 

 
 
 
 
 

Answers (1)

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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