A description/proof of that subset minus union of sequence of subsets is intersection of subsets each of which is 1st subset minus partial union of sequence
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 that for any set, any subset (the 1st subset) minus the union of any sequence of subsets is the intersection of the subsets each of which is the 1st subset minus a partial union of the sequence.
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, \(S\), any subset, \(S_1 \subseteq S\), and any sequence of subsets, \(S_{2, 1}, S_{2, 2}, ...\) where \(S_{2, i} \subseteq S\) but not necessarily \(S_{2, i} \subseteq S_1\), \(S_1 \setminus \cup_{i} S_{2, i} = \cap_j (S_1 \setminus \cup_{i = 1 \sim j} S_{2, i})\).
2: Proof
For any \(p \in S_1 \setminus \cup_{i} S_{2, i}\), \(p \in S_1\) and \(p \notin S_{2, i}\) for each \(i\), \(p \in S_1 \setminus \cup_{i = 1 \sim j} S_{2, i}\) for each \(j\), so, \(p \in \cap_j (S_1 \setminus \cup_{i = 1 \sim j} S_{2, i})\).
For any \(p \in \cap_j (S_1 \setminus \cup_{i = 1 \sim j} S_{2, i})\), \(p \in S_1 \setminus \cup_{i = 1 \sim j} S_{2, i}\) for each \(j\), \(p \in S_1\) and \(p \notin S_{2, i}\) for each \(i\), so, \(p \in S_1 \setminus \cup_{i} S_{2, i}\).
