A definition of relation
Topics
About: set
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: 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 \(dom R\), and its range is defined to be the set of the 2nd components of the pairs, denoted as \(ran R\), and \(p_1 R p_2\) means that \(\langle p_1, p_2 \rangle \in R\)
2: Note
\(dom R\) and \(ran R\) are indeed some sets: \(dom R = \{p_1 \in \cup \cup R \vert \exists p_2 \in \cup \cup R (\langle p_1, p_2 \rangle \in R)\}\) and \(ran R = \{p_2 \in \cup \cup R \vert \exists p_1 \in \cup \cup R (\langle p_1, p_2 \rangle \in R)\}\), which is because for any \(\langle p_1, p_2 \rangle = \{p_1, \{p_1, p_2\}\} \in R\), \(\{p_1, p_2\} \in \cup R\) and \(p_1, p_2 \in \cup \cup R\).
