description/proof of that set with ordering as containment is partially-ordered set
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of partially-ordered set.
Target Context
- The reader will have a description and a proof of the proposition that any set with the ordering as containment is a partially-ordered 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: Structured Description
Here is the rules of Structured Description.
Entities:
\(S\): \(\in \{\text{ the sets }\}\), with the ordering, \(\lt\), such that \(\forall s_1, s_2 \in S (s_1 \lt s_2 \iff s_1 \subset s_2)\)
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Statements:
\(S \in \{\text{ the partially ordered sets }\}\)
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2: Proof
Whole Strategy: Step 1: see that \(S\) with \(\lt\) satisfies the conditions to be a partially-ordered set.
Step 1:
1) \(\lt\) is irreflexive: for each \(s \in S\), \(s \subset s\) does not hold, so, \(s \lt s\) does not hold.
2) \(\lt\) is transitive: for each \(s_1, s_2, s_3 \in S\) such that \(s_1 \lt s_2\) and \(s_2 \lt s_3\), \(s_1 \subset s_2 \subset s_3\), which implies that \(s_1 \subset s_3\), so, \(s_1 \lt s_3\).
So, \(S\) with \(\lt\) is a partially-ordered set.
