Question
Mon June 25, 2012

# Q1.For a*b= a+b-4 for a,b belongs to Z show that * is both commutative & associative also find identity element in Z. Q2.For a*b= 3ab/5 for a,b belongs to Q . check for commutativity & associativity.

Tue June 26, 2012
Answer 1 . Given : .For a*b= a+b-4 for a,b belongs to Z
to prove : * is both commutative & associative
To find : identity element in Z

for commutativity : b*a = b+a - 4  for a,b belongs to Z

therefore a*b is equal to b*a

hence it is commutative.

for associativity :

(a*b) *c=  (a+b -4)*c

=(a+b -4) +c - 4

= a+b+c - 8........(1)

a*(b*c)= a*(b + c -4)

=a+ b+c -4 - 4

=a+b+c- 8....................(2)

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

now to find the identity element in a*b=a+b+1 in Z

Let e be the identity element in Z for the binary operator * on Z .

then ,

a * e = a  =e * a  for all a belongs to Z

a*e=a and e*a =a  for all a belongs to Z

a+e -4 =a  and e+a - 4 =a

hence e=  4

e= 4 is the identity element in Z

Answer 2 . Given : .For a*b= (3ab) /5 for a,b belongs to Q

to check : * is both commutative & associative

for commutativity : b*a = (3ba ) /5  for a,b belongs to Q

as multipliaction is communicative operation

therefore a*b is equal to b*a

hence it is commutative.

for associativity :

(a*b) *c=  ( (3ab) /5 )*c

= ( 3 ((3ab)/5 ) c) /5

=(9abc) /25.....................(1)

a*(b*c)= a* ((3bc) / 5)

= ( 3 ( a ) ( ( 3bc ) / 5 ) ) / 5

=(9abc) / 25....................(2)

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

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