definition of equivalence relation on 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 relation.
Target Context
- The reader will have a definition of equivalence relation on 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:
\( S\): \(\in \{\text{ the sets }\}\)
\( S \times S\):
\(*\sim\): \(\subseteq S \times S\), \(\in \{\text{ the relations }\}\)
//
Conditions:
1) \(\forall p \in S (p \sim p)\): reflexivity
\(\land\)
2) \(\forall p_1, p_2 \in S (p_1 \sim p_2 \implies p_2 \sim p_1)\): symmetry
\(\land\)
3) \(\forall p_1, p_2, p_3 \in S ((p_1 \sim p_2 \land p_2 \sim p_3)\implies p_1 \sim p_3)\): transitivity
//
2: Natural Language Description
For any set, \(S\), and \(S \times S\), any relation, \(\sim \subseteq S \times S\), such that 1) \(\forall p \in S (p \sim p)\): reflexivity; 2) \(\forall p_1, p_2 \in S (p_1 \sim p_2 \implies p_2 \sim p_1)\): symmetry; 3) \(\forall p_1, p_2, p_3 \in S ((p_1 \sim p_2 \land p_2 \sim p_3)\implies p_1 \sim p_3)\): transitivity
