description/proof of that union of intersection of subsets and subset is intersection of unions of each of subsets and latter subset
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, the union of the intersection of any possibly uncountable number of subsets and any subset is the intersection of the unions of each of the subsets and the latter subset.
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:
\(S'\): \(\in \{\text{ the sets }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(\{S_j \subseteq S' \vert j \in J\}\):
\(S\): \(S \subseteq S'\)
//
Statements:
\((\cap_{j \in J} S_j) \cup S = \cap_{j \in J} (S_j \cup S)\).
//
2: Proof
Whole Strategy: Step 1: for each \(s \in (\cap_{j \in J} S_j) \cup S\), see that \(s \in \cap_{j \in J} (S_j \cup S)\); Step 2: for each \(s \in \cap_{j \in J} (S_j \cup S)\), see that \(s \in (\cap_{j \in J} S_j) \cup S\).
Step 1:
Let \(s \in (\cap_{j \in J} S_j) \cup S\) be any.
1) \(s \in \cap_{j \in J} S_j\) or 2) \(s \in S\).
Let us suppose 1).
For each \(j \in J\), \(s \in S_j\). So, \(s \in S_j \cup S\). So, \(s \in \cap_{j \in J} (S_j \cup S)\).
Let us suppose 2).
For each \(j \in J\), \(s \in S_j \cup S\). So, \(s \in \cap_{j \in J} (S_j \cup S)\).
So, anyway, \(s \in \cap_{j \in J} (S_j \cup S)\).
Step 2:
Let \(s \in \cap_{j \in J} (S_j \cup S)\) be any.
For each \(j \in J\), \(s \in S_j \cup S\). So, 1) \(s \in S_j\) or 2) \(s \in S\).
Let us suppose 2).
\(s \in (\cap_{j \in J} S_j) \cup S\).
Let us suppose that 2) does not hold.
Then, for each \(j \in J\), \(s \in S_j\).
Then, \(s \in (\cap_{j \in J} S_j) \cup S\).
So, anyway, \(s \in (\cap_{j \in J} S_j) \cup S\).
