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Thread: Binary operation

  1. #1
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    Binary operation

    Hi everybody,

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

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

    I find that $\displaystyle _*$ is commutative and associative.

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

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

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

    Then for the inverses, solve the equation$\displaystyle x * x^{-1} = e$
    so $\displaystyle x* x^{-1}=xx^{-1}-2(x+x^{-1})+6= e$
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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 $\displaystyle \mathbb{R}$ as follows:

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

    I find that $\displaystyle _*$ is commutative and associative.

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

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

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

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

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

    Tonio
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