description/proof of that for set and \(2\) disjoint subsets, 1st subset is contained in complement of 2nd 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 \(2\) disjoint subsets, the 1st subset is contained in the complement of the 2nd 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:
\(S_1 \cap S_2 = \emptyset\)
\(\implies\)
\(S_1 \subseteq S \setminus S_2\)
//
2: Note
Of course, also \(S_2 \subseteq S \setminus S_1\) holds, because \(S_2 \cap S_1 = \emptyset\).
3: Proof
Whole Strategy: Step 1: see that for each \(s_1 \in S_1\), \(s_1 \in S \setminus S_2\).
Step 1:
Let \(s_1 \in S_1\) be any.
\(s_1 \notin S_2\), because \(S_1 \cap S_2 = \emptyset\).
So, \(s_1 \in S \setminus S_2\).
So, \(S_1 \subseteq S \setminus S_2\).
