636: Equivalence Relation on Set

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definition of equivalence relation on set

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Topics

About: set

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of relation.

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Target Context

• The reader will have a definition of equivalence relation on set.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S$: $\in \{\text{ the sets }\}$
$S\times S$:
$\ast \sim$: $\subseteq S\times S$, $\in \{\text{ the relations }\}$
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Conditions:
1) $\mathrm{\forall }p\in S(p\sim p)$: reflexivity
$\wedge$
2) $\mathrm{\forall }{p}_1,p_2\in S(p_1\sim p_2⟹p_2\sim p_1)$: symmetry
$\wedge$
3) $\mathrm{\forall }{p}_1,p_2,p_3\in S((p_1\sim p_2\wedge p_2\sim p_3)⟹p_1\sim p_3)$: transitivity
//

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2: Natural Language Description

For any set, $S$, and $S\times S$, any relation, $\sim \subseteq S\times S$, such that 1) $\mathrm{\forall }p\in S(p\sim p)$: reflexivity; 2) $\mathrm{\forall }{p}_1,p_2\in S(p_1\sim p_2⟹p_2\sim p_1)$: symmetry; 3) $\mathrm{\forall }{p}_1,p_2,p_3\in S((p_1\sim p_2\wedge p_2\sim p_3)⟹p_1\sim p_3)$: transitivity

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References

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