description/proof of that intersection of complements of subsets is complement of union of subsets
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of set.
Target Context
- The reader will have a description and a proof of 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.
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:
\(\cap_{\beta \in B} (S \setminus S_\beta) = S \setminus \cup_{\beta \in B} S_\beta\)
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2: Proof
Whole Strategy: Step 1: see that \(\cap_{\beta \in B} (S \setminus S_\beta) \subseteq S \setminus \cup_{\beta \in B} S_\beta\); Step 2: see that \(S \setminus \cup_{\beta \in B} S_\beta \subseteq \cap_{\beta \in B} (S \setminus S_\beta)\).
Step 1:
For any element, \(p \in \cap_{\beta \in B} (S \setminus S_\beta)\), \(p \in S \setminus S_\beta\) for each \(\beta\), which means \(p \notin S_\beta\) for each \(\beta\), so, \(p \notin \cup_{\beta \in B} S_\beta\), so, \(p \in S \setminus \cup_{\beta \in B} S_\beta\).
Step 2:
For any element, \(p \in S \setminus \cup_{\beta \in B} S_\beta\), \(p \notin \cup_{\beta \in B} S_\beta\), so, \(p \notin S_\beta\) for each \(\beta\), so, \(p \in S \setminus S_\beta\) for each \(\beta\), so, \(p \in \cap_{\beta \in B} (S \setminus S_\beta)\).
