definition of set of equivalence classes of set by equivalence relation
Topics
About: set
The table of contents of this article
Starting Context
- The reader knows a definition of equivalence relation on set.
Target Context
- The reader will have a definition of set of equivalence classes of set by equivalence relation.
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:
\( S\): \(\in \{\text{ the sets }\}\)
\( R\): \(\in \{\text{ the equivalence relations on } S\}\)
\(*S / R\): \(\subseteq Pow (S)\), \(= \{S_s = \{s' \in S \vert s R s'\} \subseteq S \vert s \in S\}\)
//
Conditions:
//
2: Note
That does not mean that \(S / R\) has \(\vert S \vert\) elements, because \(S_{s_1}\) may equal \(S_{s_2}\).
Let us see that \(S / R\) is a division of \(S\), which means that \(S / R\) covers \(S\) and the elements of \(S / R\) are disjoint.
For each \(s \in S\), \(s \in S_s\), so, \(S / R\) covers \(S\).
Let us see that the elements of \(S / R\) are disjoint.
Let \(S_{s_1}, S_{s_2} \in S / R\) be any such that \(S_{s_1} \neq S_{s_2}\).
If \(S_{s_1} \cap S_{s_2} \neq \emptyset\), there would be an \(s \in S_{s_1} \cap S_{s_2}\), and for each \(s' \in S_{s_2}\), \(s_1 R s\), because \(s \in S_{s_1}\), \(s R s_2\), because \(s \in S_{s_2}\), \(s_2 R s'\), because \(s' \in S_{s_2}\), and so, \(s_1 R s'\), which would mean that \(s' \in S_{s_1}\), so, \(S_{s_2} \subseteq S_{s_1}\), and \(S_{s_1} \subseteq S_{s_2}\), by the symmetry, so, \(S_{s_1} = S_{s_2}\), a contradiction, so, \(S_{s_1} \cap S_{s_2} = \emptyset\).
So, the elements of \(S / R\) are disjoint.
So, \(S / R\) is a division of \(S\).
