# 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.