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Question
Mon June 25, 2012

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

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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