definition of equivalence relation on collection
Topics
About: collection
The table of contents of this article
Starting Context
- The reader knows a definition of relation.
Target Context
- The reader will have a definition of equivalence relation on collection.
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:
\( C\): \(\in \{\text{ the collections }\}\)
\(*\sim\): \(\in \{\text{ the relations }\}\), such that \(Dom (\sim) \subseteq C\) and \(Ran (\sim) \subseteq C\)
//
Conditions:
1) \(\forall p \in C (p \sim p)\): reflexivity
\(\land\)
2) \(\forall p_1, p_2 \in C (p_1 \sim p_2 \implies p_2 \sim p_1)\): symmetry
\(\land\)
3) \(\forall p_1, p_2, p_3 \in C ((p_1 \sim p_2 \land p_2 \sim p_3)\implies p_1 \sim p_3)\): transitivity
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2: Note
As for the distinction between 'collection' and 'set', refer to Are You Talking About Sets or About Collections?.
Compare with the definition of equivalence relation on set: there is no necessity for \(S\) to be a set for the definition.
