definition of subset 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 subset 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:
\( S'\): \(\in \{\text{ the sets }\}\), with any equivalence relation, \(\sim'\)
\( S\): \(\subseteq S'\)
\(*\sim\): \(\in \{\text{ the equivalence relations on } S\}\)
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Conditions:
\(\forall s_1, s_2 \in S (s_1 \sim s_2 \iff s_1 \sim' s_2)\)
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2: Note
Let us see that \(\sim\) is indeed an equivalence relation on \(S\).
\(\sim\) is indeed a relation, because \(\sim = \{\langle s_1, s_2 \rangle \in S \times S \vert s_1 \sim s_2\}\), which is well-defined, because \(s_1 \sim s_2\) if and only if \(s_1 \sim' s_2\), which equals \(\langle s_1, s_2 \rangle \in \sim'\), so, \(\sim = \{\langle s_1, s_2 \rangle \in S \times S \vert \langle s_1, s_2 \rangle \in \sim'\}\).
1) \(\forall s \in S (s \sim s)\): reflexivity: \(s \sim' s\).
2) \(\forall s_1, s_2 \in S (s_1 \sim s_2 \implies s_2 \sim s_1)\): symmetry: \(s_1 \sim s_2 \implies s_1 \sim' s_2 \implies s_2 \sim' s_1 \implies s_2 \sim s_1\).
3) \(\forall s_1, s_2, s_3 \in S ((s_1 \sim s_2 \land s_2 \sim s_3) \implies s_1 \sim s_3)\): transitivity: \((s_1 \sim s_2 \land s_2 \sim s_3) \implies (s_1 \sim' s_2 \land s_2 \sim' s_3) \implies s_1 \sim' s_3 \implies s_1 \sim s_3\).
So, \(\sim\) is an equivalence relation on \(S\).
