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Math Help - Question on how to tell if its reflective, symmetric, antisymmetric, & transitive?

  1. #1
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    Question on how to tell if its reflective, symmetric, antisymmetric, & transitive?

    If I have t his:

    x + y = 0

    How do I tell if its: Reflective, Symmetric, Antisymmetric, and Transitive?

    For Reflective, it says: check f(x,x) = 0

    For symmetry, it says: check f(x,y) = f(y,x)

    For Transitive, it says: check f(x,y)=0 and f(y,z)=0, then f(x,z)=0

    What is Antisymmetric?

    This is where I am getting this at: Is the x^2 - xy +2x -2y= 0 reflective/symmetric/transitive??? - Yahoo! Answers
    (Look at User JCS)
    __________________________________________________ _____________

    Am I doing this correct?

    f(x, x) = x + y
    = x + x
    Reflective

    f(y, x ) = x + y
    = y + x
    Symmetry

    I am not sure on transitive and i do not know antisymmetric?
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Paris
    Posts
    354
    Hi

    When you define a relation, a good thing to do is to define its domain

    As you write x,y for the elements, maybe is it \mathbb{R} ?

    If so,

    x\mathcal{R}y \Leftrightarrow x+y=0<br />

    The relation \mathcal{R} is reflexive if \forall x\in\mathbb{R},\ x\mathcal{R}x i.e. x+x=0. Is this true for any real x ? ( A "no" means \mathcal{R} isn't reflexive)

    Using the same notation, \mathcal{R} is:

    symmetric if \forall x\forall y(x\mathcal{R}y\Rightarrow y\mathcal{R}x)

    antisymmetric if \forall x\forall y(x\mathcal{R}y\ \text{and}\ y\mathcal{R}x)\Rightarrow x=y

    transitive if \forall x\forall y\forall z(x\mathcal{R}y\ \text{and}\ y\mathcal{R}z)\Rightarrow x\mathcal{R}z
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