If an operation \ast  on the set of integers Z is defined by a\ast b= 2a^{2}+b,  then find (i) whether it is a binary or not, and (ii) if a binary,then is it commutative or not.

 

 

 

 
 
 
 
 

Answers (1)

(i) As a^{2}\epsilon Z\; \; \; \therefore 2a^{2}+b\epsilon Z\; \; for \: all\; \; a,b\epsilon Z
That is , a\ast b\, \: \epsilon Z\; \; so \: \ast \: binary
(ii) Note that  1\ast 2= 2\times 1^{2}+2= 4
             and  2\ast 1= 2\times 2^{2}+1= 9
       ie 1\ast 2\neq 2\ast 1
\therefore  It isn't commutative

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