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| − | <math>\langle A, R \rangle</math> is an ordered class iff | + | <math>\langle A, R \rangle</math> is an totally ordered class iff |
<ol> | <ol> | ||
| − | <li><math>R\subseteq A\times A</math></li> | + | <li>(R is a relation on A) <math>R\subseteq A\times A</math></li> |
| − | <li>(irreflexivity) <math>\forall x \in A \langle x,x \rangle \notin R</math></li> | + | <li>(irreflexivity) <math>\forall x \in A\, \langle x,x \rangle \notin R</math></li> |
| − | <li>(transitivity) <math>\forall x,y,z \in A \langle x,y \rangle \in R \wedge \langle y,z \rangle \in R \rightarrow \langle x,z \rangle \in R</math></li> | + | <li>(transitivity) <math>\forall x,y,z \in A\, \langle x,y \rangle \in R \wedge \langle y,z \rangle \in R \rightarrow \langle x,z \rangle \in R</math></li> |
| − | <li>(trichotomy) <math>\forall x,y \in A \langle x,y \rangle \in R \ | + | <li>(trichotomy) <math>\forall x,y \in A\, \langle x,y \rangle \in R \vee \langle y,x \rangle \in R \vee x=y</math></li> |
</ol> | </ol> | ||
</p> | </p> | ||
Latest revision as of 07:59, 1 June 2013
Equivalences of Well-ordered Relation
Definitions
$ \langle A, R \rangle $ is an totally ordered class iff
- (R is a relation on A) $ R\subseteq A\times A $
- (irreflexivity) $ \forall x \in A\, \langle x,x \rangle \notin R $
- (transitivity) $ \forall x,y,z \in A\, \langle x,y \rangle \in R \wedge \langle y,z \rangle \in R \rightarrow \langle x,z \rangle \in R $
- (trichotomy) $ \forall x,y \in A\, \langle x,y \rangle \in R \vee \langle y,x \rangle \in R \vee x=y $
