820: Intersection of Set

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definition of intersection 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: Note

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Starting Context

• The reader knows a definition of union of set.

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Target Context

• The reader will have a definition of intersection 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:
$S$: $\in \{\text{ the sets }\}$
$\ast \cap S$: $=\{p\in \cup S|\mathrm{\forall }s\in S(p\in s)\}$, $\in \{\text{ the sets }\}$
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Conditions:
//

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2: Note

$\cap S$ is indeed a set in the ZFC set theory, by the union axiom and the subset axiom.

Some frequently seen expressions like $S_1\cap S_2$ and ${\cap }_{\alpha \in A}{S}_{\alpha }$ are indeed $S_1\cap S_2:=\cap \{S_1,S_2\}$ and ${\cap }_{\alpha \in A}{S}_{\alpha }:=\cap \{S_{\alpha }|\alpha \in A\}$: when $A$ is a set, $\{S_{\alpha }|\alpha \in A\}$ is a set by the replacement axiom.

The special case of ${\cap }_{\alpha \in A}{S}_{\alpha }$ where $A=\{p\}$ cannot be expressed as $\cap S_p$, because ${\cap }_{\alpha \in A}{S}_{\alpha }$ is $\cap \{S_p\}=S_p$ but not $\cap S_p$.

$\cap \mathrm{\varnothing }=\mathrm{\varnothing }$, because $\cup \mathrm{\varnothing }=\mathrm{\varnothing }$.

We could not define like $p\in \cap S⟺\mathrm{\forall }{p}^{\prime }\in S(p\in p^{\prime })$, because when $S=\mathrm{\varnothing }$, $\mathrm{\forall }{p}^{\prime }\in S(p\in p^{\prime })$ would be vacuously satisfied, but where would $p$ come from?

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References

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