description/proof of that for set and \(2\) subsets, if complement of 1st subset is contained in 2nd subset, complement of 2nd subset is contained in 1st 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 and any \(2\) subsets, if the complement of the 1st subset is contained in the 2nd subset, the complement of the 2nd subset is contained in the 1st 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:
\(S\): \(\in \{\text{ the sets }\}\)
\(S_1\): \(\subseteq S\)
\(S_2\): \(\subseteq S\)
//
Statements:
\(S \setminus S_1 \subseteq S_2\)
\(\implies\)
\(S \setminus S_2 \subseteq S_1\)
//
2: Proof
Whole Strategy: Step 1: see that for each \(s \in S \setminus S_2\), \(s \in S_1\).
Step 1:
Let \(s \in S \setminus S_2\) be any.
\(s \notin S_2\).
\(s \notin S \setminus S_1\), because \(S \setminus S_1 \subseteq S_2\).
So, \(s \in S_1\).
