Let \ast :N\times N\rightarrow N  be an operation defined as a\ast b= a+ab,\, \forall \, a,b\, \epsilon \, N. check if \ast . is a binary operation. If yes, find if it is associative too.

 

 

 

 
 
 
 
 

Answers (1)

For all a,b\, \epsilon \, N  we have that a+b\, \epsilon \, N
\therefore a\ast b\,\epsilon \, N
Hence \ast is a binary operation.
Now \left ( a\ast b \right )\ast c= \left ( a+ab \right )\ast c= a+ab+ac+abc
a\ast \left ( b\ast c \right )= a\ast \left ( b+bc \right )= a+ab+abc
That is a\ast \left ( b\ast c \right )\neq \left ( a\ast b \right )\ast c   which implies  \ast is not associative.

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