Well-Order Relation

Definition

A Total Order Relation in which every non-empty subset has a smallest element.

• 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: $\forall a,b\in X,(a\le b\text{or}b\le a)$
• Well-ordering: $\forall A\subset W\text{s.t.}A\ne 0,\exists a\in A\text{s.t.}\forall x\in A,a\le x$