functions are a special type of relation

tags : [[category theory]]

source : [[ACT4E - Session 2 - Connection]]

A function is a special type of [[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.