description/proof of that intersection of subsets is complement of union of complements of subsets
Topics
About: set
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition for any set, the intersection of any possibly uncountable number of subsets is the complement of the union of the complements of the subsets.
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:
\(S\): \(\in \{\text{ the sets }\}\)
\(\{S_\beta \subseteq S \vert \beta \in B\}\): \(B \in \{\text{ the possibly uncountable index sets }\}\)
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Statements:
\(\cap_{\beta \in B} (S_\beta) = S \setminus \cup_{\beta \in B} (S \setminus S_\beta)\)
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2: Proof
Whole Strategy: Step 1: take \(S \setminus S_\beta\) as \(S_\beta\) in the proposition for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets.
Step 1:
\(S \setminus (S \setminus S_\beta) = S_\beta\).
As \(S_\beta\) in the proposition for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets, let us take \(S \setminus S_\beta\), which makes \(\cap_{\beta \in B} (S \setminus (S \setminus S_\beta)) = S \setminus \cup_{\beta \in B} (S \setminus S_\beta)\), but \(\cap_{\beta \in B} (S \setminus (S \setminus S_\beta)) = \cap_{\beta \in B} S_\beta\), so, \(\cap_{\beta \in B} S_\beta = S \setminus \cup_{\beta \in B} (S \setminus S_\beta)\).
