Let G be a set of rational numbers other than 1. Let * be an operation on G defined by a*b=a + b + ab for all a, b € G. Prove that {G, *} is a group. -

Mumbai University > Computer Engineering > Sem 3 > Discrete Structures

Marks: 8 Marks

Year: May 2015

1. CLOSOURE

   Let A and B be elements of real numbers R. Then

   a * b=a + b + ab is also real. So it is in R.

2. ASSOCIATIVE

   a * (b * c)=a * {b+c+bc)=a+b+c+bc+ab+ac+abc

   (a * b) * c=(a+b+ab) * c=a+b+ab+c+ac+bc+abc=a * (b * c)

3. IDENTITY ELEMENT

   LET a * e=a

   a+e+ae=a

   e+ae=0

   e=0

   LET US CHECK FOR

   e * a=e+a+ea=0+a+0a

4. INVERSE

   Let a * i=e=0

   a+i+ai=0

   i(1+a)=-a

   i=-a/(1+a)...THIS EXISTS SINCE A IS NOT EQUAL TO -1 AS GIVEN.R IS Reals(R){-1}

   Let us check

   i * a=e=0

   -a/(1+a) * a=-a/(1+a)+a+(-a.a)/(1+a)=(1/(1+a)){-a+a(1+a)-a.a}

   =(1/(1+a)){-a+a+a.a-a.a}=0...PROVED

5. ABELIAN TEST

   a * b=a+b+ab

   b * a=b+a+ba=a * b...Hence it is Group.