\part{Functions and Maps}


\chapter{Functions}

Authors may differentiate between a 'function' and a 'map', but they are really the same thing; 'map' tends to be used for special types of functions but in this text the definitions are synonymous. Generally the term 'function' will be used, unless the function in question is conventionally referred to as a map (fugayzi, fugazi).




\begin{definition}
A \emph{map}
\end{definition}

\begin{definition}
A \emph{function}
\end{definition}




\begin{definition}
The \emph{domain of a function} is the set of all well defined inputs for the function.
\[ f : X \to Y \implies \mathrm{Dom}(f)= \{ x \in X : f(x) \text{ is well defined}\} \]
\end{definition}

\begin{definition}
The \emph{predomain of a function} is the set representing the 'space' of which the domain is a subset of.
\[ f : X \to Y \implies X \text{ is the predomain of } f \]
\end{definition}

\begin{definition}
The \emph{codomain of a function}
\[ f : X \to Y \implies \mathrm{Codom}(f)= Y \]
\end{definition}

\begin{definition}
The \emph{image of a subset} is the set of all outputs the function has mapped to some subset of its domain.
\[ U \subseteq \mathrm{dom}(f) \implies f(U) = \{ f(u) : u \in U\} \]
The \emph{image of a function} is the set of all outputs the function has mapped to its domain elements.
\[ \mathrm{Im}(f)= \{ f(x) : x \in \mathrm{Dom}(f) \} =f [\mathrm{dom}(f)] \]
\end{definition}

The term 'range' is used ambiguously to denote either the codomain or image; from my experience it is used informally in contexts where the image is equal to the codomain. Consider the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x)=x^3 -3x$, indeed $\mathrm{Im}(f)=\mathrm{Codom}(f)= \mathbb{R}$, hence no confusion can occur on the function's 'range'.


\chapter{Types of functions}

\begin{definition}
An \emph{injective function}
\end{definition}



\begin{definition}
A \emph{surjective function}
\end{definition}


\begin{theorem}
If a function is surjective then its image has the same cardinality as its codomain.
\[ f \text{ is surjective } \implies |\mathrm{Im}(f)|=|\mathrm{Codom}(f)| \]
\end{theorem}




\begin{definition}
A \emph{bijective function}
\end{definition}


\begin{theorem}
If a function is surjective then its domain the same cardinality as its range.
\[ f \text{ is bijective } \implies |\mathrm{Im}(f)|=|\mathrm{Dom}(f)| \]
\end{theorem}



\begin{definition}
The \emph{composition operator} $\circ$ is a $(g \circ f)=g(f(x))$
\end{definition}



\begin{definition}
An \emph{inverse function} of $f$
\end{definition}


\begin{theorem}
A function is invertible iff it is bijective.
\end{theorem}



\chapter{Indexed families}

Sets are powerful objects; we've been able to develop much of mathematics through their grace. However, sets are limited in that they are unordered and can only hold one 'instance' of some mathematical object (i.e sets don't allow repetition of elements). The theory of functions can build upon the notion of a set to allow more complicated container structures.

\begin{definition}
An \emph{indexed family}
\end{definition}

\begin{definition}
A \emph{sequence}
\end{definition}

\begin{definition}
A \emph{tuple}

A tuple of 2 elements is an ordered pair
A tuple of $n$ elements is an $n$-tuple
\end{definition}

