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

[ktheo]

Homework Statement

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

1. ≽ is not symmetric $\Rightarrow$ ≽ is not asymmetric
2. ≽ is not symmetric $\Rightarrow$ ≽ is not antisymmetric
3. ≽ is not antisymmetric $\Rightarrow$ ≽ is not asymmetric

Homework Equations

Symmetric:
For any x,y$\in$X, x≽y $\Rightarrow$ y≽x

Antisymmetric:
For any x,y$\in$X, x≽y and y≽x and x=y

Asymmetric:
For any x,y$\in$X, x≽y$\neq$y≽x

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 $\neq$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?

 

[verty]

Hint: When you have "not" on both sides of the implication, use the contrapositive instead.

 

[ktheo]

Hint: When you have "not" on both sides of the implication, use the contrapositive instead.

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?