#+title: functions are a special type of relation
- tags :: [[file:20200708122545-category_theory.org][category theory]]
- source :: [[file:literature/20210117123933-act4e_session_2_connection.org][ACT4E - Session 2 - Connection]]

A function is a special type of [[file:20210117142702-relations.org][relation]].

A relation is a function if it satisfies the two following conditions:
1. $\forall x \in X \exists y \in Y : \langle x,y \rangle \in R$
   - every element of the source X gets mapped by f to some element of the target Y
2. $\exists \langle x_1,y_1 \rangle, \langle x_2,y_2 \rangle \in R$ holds: $x_1 = x_2 \Rightarrow y_1 = y_2$

Like categories and subsequently relations, functions can also be composed.