definition of representatives set of quotient set
Topics
About: set
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of quotient set.
Target Context
- The reader will have a definition of representatives set of quotient set.
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:
\( S\): \(\in \{\text{ the sets }\}\)
\( \sim\): \(\in \{\text{ the equivalence relations on } S\}\)
\( S / \sim\): \(= \text{ the quotient set }\)
\( f\): \(S / \sim \to S\)
\(*\overline{S / \sim - f}\): \(= ran f\)
//
Conditions:
\(\forall p \in S / \sim (f (p) \in p)\)
//
The above condition, of course, does not really determine \(f\): that is just a necessarily condition. So, \(f\) needs to be determined somehow: by really specifying the mapping or by assuming to have chosen an arbitrary mapping, which is possible by the axiom of choice.
In fact, the term, "representatives set of quotient set", or the notation, \(\overline{S / \sim - f}\), is not particularly standard: as one has not seen any standard term or notation in the literature, one has concocted them for one's own use.
Let such any \(f\) be called "representatives map".
2: Natural Language Description
For any set, \(S\), any equivalence relation on \(S\), \(\sim\), the quotient set, \( S / \sim\), and any map, \(f: S / \sim \to S\), such that for each \(p \in S / \sim\), \(f (p) \in p\), \(\overline{S / \sim - f} = ran f\)
