500: Relation

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A definition of relation

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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: Definition
• 2: Note

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Starting Context

• The reader knows a definition of ordered pair.

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Target Context

• The reader will have a definition of relation.

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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: Definition

The set, $R$, of any ordered pairs, where its domain is defined to be the set of the 1st components of the pairs, denoted as $domR$, and its range is defined to be the set of the 2nd components of the pairs, denoted as $ranR$, and $p_{1}R{p}_2$ means that $⟨p_1,p_2⟩\in R$

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2: Note

$domR$ and $ranR$ are indeed some sets: $domR=\{p_1\in \cup \cup R|\mathrm{\exists }{p}_2\in \cup \cup R(⟨p_1,p_2⟩\in R)\}$ and $ranR=\{p_2\in \cup \cup R|\mathrm{\exists }{p}_1\in \cup \cup R(⟨p_1,p_2⟩\in R)\}$, which is because for any $⟨p_1,p_2⟩=\{p_1,\{p_1,p_2\}\}\in R$, $\{p_1,p_2\}\in \cup R$ and $p_1,p_2\in \cup \cup R$.

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References

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