# 3.Functions as relations

## 08.06.2020

### Contents/Index

1. Intro
2. Properties and Special Relations
@3. Functions as relations

A function is a binary relation $F$ on the set $X \times Y$ where for each element we have that

1. $x\ R\ y_1 \land x\ R\ y_2 \Rightarrow y_1 = y_2$. That is every $x$ in $X$ can be related to at most one $y \in Y$. This property is called determinism.
2. $\forall x \in X \exists y \in Y : x\ R\ y$. That is fore every $x$ in $X$ there must exists one $y$ in $Y$ that is related to this $x$.

If only 1. is satisfied, we call the function partial. If both we call the function total or well defined. Normally for a function $f$ we define it as $$f : A \rightarrow B$$ instead of using the cartesian product notation.

We can expand on the definition, we say that

• A function is injective if $$x_1\ R\ y \land x_2\ R\ y \Rightarrow x_1 = x_2$$ That is for every $y$ in $Y$ there is at most one $x$ in $X$ that is related to this $y$.
• A function is surjective if $$\forall y \in Y \exists x \in X : x\ R\ y$$ That is for every $y$ in $Y$ there must exists a $x$ in $X$ that is related to this $y$.
• A function is bijective if it is both surjective and injective.