Total Order Relation

Definition

Total order relation is a Partial Order Relation in which any two elements are comparable

• Reflectivity: $\forall a\in X,a\le a$
• Antisymmetry: $\forall a,b\in X,(a\le b\text{and}b\le a\Rightarrow a=b)$
• Transitivity: $\forall a,b,c\in X,(a\le b\text{and}b\le c\Rightarrow a\le c)$
• Comparability: $\forall a,b\in X,(a\le b\text{or}b\le a)$

Strict Total Order Relation

Strict total order relation is a binary Relation $<$ on set $X$ that satisfies the following conditions:

• Transitivity: $\forall a,b,c\in X,(a<b\text{and}b\le c\Rightarrow a<c)$
• Irreflexivity: $\forall a\in X,\neg (a<a)$
• Asymmetry: $\forall a,b\in X,a<b\Rightarrow \neg (b<a)$
• Comparability: $\forall a,b\in X\text{s.t.}a\ne b,a<b\text{or}b<a$