definition of relation
Topics
About: collection
The table of contents of this article
Starting Context
- The reader knows a definition of ordered pair.
Target Context
- The reader will have a definition of 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:
\(*R\): \(\in \{\text{ the collections of some ordered pairs }\}\)
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Conditions:
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\(Dom (R)\) is the collection of the 1st components of the pairs, called "domain of \(R\)".
\(Ran (R)\) is the collection of the 2nd components of the pairs, called "range of \(R\)".
\(s_1 R s_2\) means that \(\langle s_1, s_2 \rangle \in R\).
2: Note
As for the distinction between 'collection' and 'set', refer to Are You Talking About Sets or About Collections?.
\(Dom (R)\) and \(Ran (R)\) are indeed some collections: for each object, \(o\), whether \(o \in Dom (R)\) or not is unambiguous and whether \(o \in Ran (R)\) or not is unambiguous.
When \(R\) is any set, \(Dom (R)\) and \(Ran (R)\) are indeed some sets: \(Dom (R) = \{s_1 \in \cup \cup R \vert \exists s_2 \in \cup \cup R (\langle s_1, s_2 \rangle \in R)\}\) and \(Ran (R) = \{s_2 \in \cup \cup R \vert \exists s_1 \in \cup \cup R (\langle s_1, s_2 \rangle \in R)\}\), which is because for any \(\langle s_1, s_2 \rangle = \{s_1, \{s_1, s_2\}\} \in R\), \(\{s_1, s_2\} \in \cup R\) and \(s_1, s_2 \in \cup \cup R\).
