28 lines
660 B
TeX
28 lines
660 B
TeX
|
|
||
|
|
In order for this relation to be a partial order, it has to be reflexive, antisymmetric and transitive.
|
||
|
|
|
||
|
|
\textbf{Reflexive:}
|
||
|
|
|
||
|
|
All elements are related to themself
|
||
|
|
|
||
|
|
\[ AA, BB, CC, DD \]
|
||
|
|
|
||
|
|
\textbf{Antisymmetric:}
|
||
|
|
|
||
|
|
No relation have a symmetric counterpart \\
|
||
|
|
|
||
|
|
(Listing the ones that don't have a symmetric counterpart would just be listing the whole set) \\
|
||
|
|
|
||
|
|
\textbf{Transitive:}
|
||
|
|
|
||
|
|
All pair of relations where $xRy$ and $yRz$ has its transitive counterpart
|
||
|
|
|
||
|
|
\begin{gather*}
|
||
|
|
AB\text{ and }BD\text{ with }AD \\
|
||
|
|
AB\text{ and }BC\text{ with }AC \\
|
||
|
|
BC\text{ and }CD\text{ with }BD \\
|
||
|
|
AC\text{ and }CD\text{ with }AD
|
||
|
|
\end{gather*}
|
||
|
|
|
||
|
|
Hence the relation is a partial order
|