Let × be a binary operation on set of rational number Q defined as follows:...

FizeauV

FizeauV

Answered

2020-12-16

Let × be a binary operation on set of rational number Q defined as follows: ab=a+b+2ab, where a,bQ
a) Prove that × is commutative, associate algebraic operation on Q

Answer & Explanation

Mitchel Aguirre

Mitchel Aguirre

Expert

2020-12-17Added 94 answers

a) For commutative
Prove ab=ba
Now ab=a+b+2ab
and ba=b+a+2ba
ba=a+b=a+b+2ab

a+b=b+aandab=ba
Hence ab=ba
For associatvie
Prove(ab)c=a(bc)
Now(ab)c=(a+b+2ab)c
=a+b+2ab+c+2(a+b+2ab)c
=a+b+2ab+2ac+abc+4abc
and a(bc)=a(b+c+2bc)
=a+b+c+2bc+2a(b+c+2bc)
=a+b+c+2bc+2ab+2ac+4abc
=a+b+c+2ab+2ac+2bc+4abc
Hence (ab)c=a(bc)

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