A 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: Description
For any set, S, the intersection of the compliments of any possibly uncountable number of subsets, \(\cap_\alpha (S \setminus S_\alpha)\) where \(S_\alpha \subseteq S\) where \({\alpha}\) is any possibly uncountable indices set, is the complement of the union of the subsets, \(S \setminus \cup_\alpha S_\alpha\).
2: Proof
For any element, \(p \in \cap_\alpha (S \setminus S_\alpha)\), \(p \in S \setminus S_\alpha\) for each \(\alpha\), which means \(p \notin S_\alpha\) for each \(\alpha\), so, \(p \notin \cup_\alpha S_\alpha\), so, \(p \in S \setminus \cup_\alpha S_\alpha\).
For any element, \(p \in S \setminus \cup_\alpha S_\alpha\), \(p \notin \cup_\alpha S_\alpha\), so, \(p \notin S_\alpha\) for each \(\alpha\), so, \(p \in S \setminus S_\alpha\) for each \(\alpha\), so, \(p \in \cap_\alpha (S \setminus S_\alpha)\).
