A description/proof of that well-ordered subset with inclusion ordering is chain in base set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of well-ordered set.
- The reader knows a definition of chain in set.
Target Context
- The reader will have a description and a proof of the proposition that for any set, any well-ordered subset with the inclusion ordering is a chain in the base set.
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, \(S\), any subset, \(S_1 \subseteq S\), that is well-ordered by the inclusion ordering is a chain in \(S\).
2: Proof
For any elements, \(p_1, p_2 \in S_1\), as \(\{p_1, p_2\}\) is a subset of \(S_1\), there is the smallest element in it, by the definition of well-ordered set. So, \(p_1 \subseteq p_2\) or \(p_2 \subseteq p_1\), which means that \(S_1\) is a chain in \(S\), by the definition of chain in set.
