| Line 8: | Line 8: | ||
<p> | <p> | ||
| − | <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><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> | ||
Revision as of 07:58, 1 June 2013
Equivalences of Well-ordered Relation
Definitions
$ \langle A, R \rangle $ is an totally ordered class iff
- $ 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 $
