- #1
gottfried
- 119
- 0
According to wikipedia a total order ≤ on a set X is one such that
If a ≤ b and b ≤ a then a = b (antisymmetry);
If a ≤ b and b ≤ c then a ≤ c (transitivity);
a ≤ b or b ≤ a (totality).
I'm wondering why antisymmetry is a condition since, as far as I can see, totality discounts antisymmetry. So suppose I'm trying to prove that R is a total order would it be sufficient to prove only transitivity and totality?
If a ≤ b and b ≤ a then a = b (antisymmetry);
If a ≤ b and b ≤ c then a ≤ c (transitivity);
a ≤ b or b ≤ a (totality).
I'm wondering why antisymmetry is a condition since, as far as I can see, totality discounts antisymmetry. So suppose I'm trying to prove that R is a total order would it be sufficient to prove only transitivity and totality?