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Homework Help: True Or False: Symmetry, anti-symmetric, asymmetry.

  1. May 19, 2013 #1
    1. The problem statement, all variables and given/known data

    State whether the following are true or false. If false, give a counter-example:

    1. ≽ is not symmetric [itex]\Rightarrow[/itex] ≽ is not asymmetric
    2. ≽ is not symmetric [itex]\Rightarrow[/itex] ≽ is not antisymmetric
    3. ≽ is not antisymmetric [itex]\Rightarrow[/itex] ≽ is not asymmetric

    2. Relevant equations

    For any x,y[itex]\in[/itex]X, x≽y [itex]\Rightarrow[/itex] y≽x

    For any x,y[itex]\in[/itex]X, x≽y and y≽x and x=y

    For any x,y[itex]\in[/itex]X, x≽y[itex]\neq[/itex]y≽x

    3. The attempt at a solution

    1. False. Lack of symmetry does not mean you can't be asymmetrical. Lack of symmetry in which x≽y [itex]\neq[/itex]y≽x is the very definition of anti-symmetry.

    2. False. Lacking symmetry does not mean you lack anti-symmetry. I don't know how to explain this one.

    3. True. A relation is asymmetric if and only if it is anti-symmetric. I can however, be anti-symmetric and not be asymmetric.

    Could you guys look this over and give me some guidance on number 2?
  2. jcsd
  3. May 19, 2013 #2


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    Homework Helper

    Hint: When you have "not" on both sides of the implication, use the contrapositive instead.
  4. May 19, 2013 #3
    Okay. So by that I assume you mean just prove that when I am anti-symmetric, I can be symmetric.

    So could I say that given the set X: {(1,1)} in ℝ Is both anti-symmetric and symmetric?
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