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

- $x\ R\ y_1 \land x\ R\ y_2 \Rightarrow y_1 = y_2$. That is every $x$ in $X$ can be related to at most one $y \in Y$. This property is called determinism.
- $\forall x \in X \exists y \in Y : x\ R\ y$. 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.

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