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Prove that {G, *} is a group.

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

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  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.

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