- #1
ktheo
- 51
- 0
Homework Statement
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
Homework Equations
Symmetric:
For any x,y[itex]\in[/itex]X, x≽y [itex]\Rightarrow[/itex] y≽x
Antisymmetric:
For any x,y[itex]\in[/itex]X, x≽y and y≽x and x=y
Asymmetric:
For any x,y[itex]\in[/itex]X, x≽y[itex]\neq[/itex]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 [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?