definition of power set of set
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a definition of power set of 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:
\( S\): \(\in \{\text{ the sets }\}\)
\(*Pow (S)\): \(= \{S' \subseteq S \}\)
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Conditions:
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2: Natural Language Description
For any set, \(S\), the set of all the subsets of \(S\), denoted as \(Pow (S)\)
3: Note
\(Pow (S)\) exists by the power set axiom.
It is important to know that \(Pow (S)\) is not guaranteed to be the collection of all the subcollections of \(S\): as the ZFC set theory guarantees only the subcollections with legitimate formulas to be sets, there may be a subcollection without any legitimate formula, which may not be any set (so, not any subset) and not be contained in \(Pow (S)\), because \(Pow (S)\) contains only the subsets.
