A description/proof of that inverse of partial ordering is partial ordering
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of partial ordering.
- The reader knows a definition of inverse of ordering.
Target Context
- The reader will have a description and a proof of the proposition that the inverse of any partial ordering is a partial ordering.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
For any set, \(S\), and any partial ordering, \(R \subseteq S \times S\), the inverse of \(R\), \(R^{-1}\), is a partial ordering.
2: Proof
\(\langle s_1, s_1 \rangle \notin R^{-1}\), because \(\langle s_1, s_1 \rangle \notin R\).
If \(\langle s_1, s_2 \rangle \in R^{-1}\) and \(\langle s_2, s_3 \rangle \in R^{-1}\), \(\langle s_1, s_3 \rangle \in R^{-1}\), because if \(\langle s_2, s_1 \rangle \in R\) and \(\langle s_3, s_2 \rangle \in R\), \(\langle s_3, s_1 \rangle \in R\).
