671: Power Set of Set

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definition of power set of set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of set.

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Target Context

• The reader will have a definition of power set of set.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S$: $\in \{\text{ the sets }\}$
$\ast Pow(S)$: $=\{S^{\prime }\subseteq S\}$
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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 $Pow(S)$

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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.

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References

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