Partial Order Relation

Definition

Partial order relation is a binary Relation $\le$ on set $X$ that satisfies the following conditions:

• Reflectivity: $\forall a\in X,a\le a$
• Antisymmetry: $\forall a,b\in X,(a\le b\wedge b\le a\Rightarrow a=b)$
• Transitivity: $\forall a,b,c\in X,(a\le b\wedge b\le c\Rightarrow a\le c)$

Strict Partial Order Relation

Strict partial 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: $\forall a,b\in X,a<b\Rightarrow \neg (b<a)$