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Math Help - Binary operation

  1. #1
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    Binary operation

    Hi everybody,

    f is a binary operation defined in \mathbb{R} as follows:

    (\forall (x,y) \in \mathbb{R}^2)   x_* y=xy-2(x+y)+6

    I find that _* is commutative and associative.

    I must show that _* takes a neutral element; and i must find if each element of \mathbb{R} has an inverse element?

    How to do it??
    Thanks.
    Last edited by lehder; March 24th 2010 at 06:41 AM.
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  2. #2
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    For the identity (neutral) element.

    You must solve x * e = x
    so  x * e=xe-2(x+e)+6 = x

    Then for the inverses, solve the equation  x * x^{-1} = e
    so x* x^{-1}=xx^{-1}-2(x+x^{-1})+6= e
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  3. #3
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    Quote Originally Posted by lehder View Post
    Hi everybody,

    f is a binary operation defined in \mathbb{R} as follows:

    (\forall (x,y) \in \mathbb{R}^2) x_* y=xy-2(x+y)+6

    I find that _* is commutative and associative.

    I must show that _* takes a neutral element; and i must find if each element of \mathbb{R} has an inverse element?

    How to do it??
    Thanks.
    Denote by z the neutral element, in case it exists. Then it must fulfill x=x_*z=xz-2(x+z)+6\,,\,\,\forall x\in\mathbb{R} .

    In particular, this must be true for x=0: 0=0_*z=0\cdot z-2(0+z)+6=-2z+6\Longrightarrow z=3 , and now we check with a general element x\in\mathbb{R} :

    3_*x=3x-2(3+x)+6=3x-6-2x+6=x ...check!

    Well, now find \forall x\in\mathbb{R} an element x'\,\,\,s.t.\,\,\,x_*x'=3 .

    Tonio
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