A description/proof of that subset minus subset is complement of 2nd subset minus complement of 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, any subset (the 1st subset) minus any subset (the 2nd subset) is the complement of the 2nd subset minus the complement of 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: Description
For any set, \(S\), and any subsets, \(S_1, S_2 \subseteq S\), \(S_1 \setminus S_2 = (S \setminus S_2) \setminus (S \setminus S_1)\).
2: Proof
For any element, \(p \in S_1 \setminus S_2\), \(p \in S_1\) and \(p \notin S_2\), \(p \in S \setminus S_2\) and \(p \notin S \setminus S_1\), so, \(p \in (S \setminus S_2) \setminus (S \setminus S_1)\).
For any element, \(p \in (S \setminus S_2) \setminus (S \setminus S_1)\), \(p \in S \setminus S_2\) and \(p \notin S \setminus S_1\), \(p \in S_1\) and \(p \notin S_2\), so, \(p \in S_1 \setminus S_2\).
