description/proof of that union of complements of subsets is complement of intersection 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 union of the complements of any possibly uncountable number of subsets is the complement of the intersection 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:
\(S\): \(\in \{\text{ the sets }\}\)
\(\{S_\beta \subseteq S \vert \beta \in B\}\): \(B \in \{\text{ the possibly uncountable index sets }\}\)
//
Statements:
\(\cup_{\beta \in B} (S \setminus S_\beta) = S \setminus \cap_{\beta \in B} S_\beta\)
//
2: Proof
Whole Strategy: Step 1: see that \(\cup_{\beta \in B} (S \setminus S_\beta) \subseteq S \setminus \cap_{\beta \in B} S_\beta\); Step 2: see that \(S \setminus \cap_{\beta \in B} S_\beta \subseteq \cup_{\beta \in B} (S \setminus S_\beta)\).
Step 1:
For any element, \(p \in \cup_{\beta \in B} (S \setminus S_\beta)\), \(p \in S \setminus S_\beta\) for a \(\beta\), so, \(p \notin S_\beta\) for a \(\beta\), so, \(p \notin \cap_{\beta \in B} S_\beta\), so, \(p \in S \setminus \cap_{\beta \in B} S_\beta\).
Step 2:
For any element, \(p \in S \setminus \cap_{\beta \in B} S_\beta\), \(p \notin \cap_{\beta \in B} S_\beta\), so, \(p \notin S_\beta\) for a \(\beta\), so, \(p \in S \setminus S_\beta\) for a \(\beta\), so, \(p \in \cup_{\beta \in B} (S \setminus S_\beta)\).
