Question
Mon June 25, 2012 By: Arvindh Jambunathan
 

Q3.For a*b=a+b-ab on R-{1}. prove that * is commutative & associative. find identity element if any .

Expert Reply
Tue June 26, 2012
Given :  a*b=a+b-ab on R-{1} 

for commutativity : b*a = b+a - ba  for R-{1} 

 as addition is communicative operation   

  therefore a*b is equal to b*a 

hence it is commutative.

 

for associativity :

(a*b) *c=  (a+b -ab)*c 

            =(a+b -ab) +c -( c (a+b -ab))

         = a+b+c - ab-ac -bc+abc........(1)

 

a*(b*c)= a*(b + c -bc) 

             =a+ b+c -bc - (a(b+c-bc))

            =a+b+c-bc-ab -ac +abc....................(2)

 

since (1) is equal to (2) , therefore it  is  associative.

now to find out the identity element for a*b=a+b - ab on R -{1}

 Let e be the identity element in R - {1} for the binary operator * on R -{1} . 

 

then , 

 

a*e=a =e*a for all a belongs to R -{1}

 

a*e=a and e*a =a  for all a belongs to R - {1}

 a+e -ae =a  and e+a -ea =a

 e= 0

     

e= 0 is the identity element in R - {1} 

 

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