63 lines
2.3 KiB
TeX
63 lines
2.3 KiB
TeX
% File: 20-01-2022.tex
|
|
% Created: 12:28:35 Thu, 20 Jan 2022 EST
|
|
% Last Change: 12:28:35 Thu, 20 Jan 2022 EST
|
|
%
|
|
\documentclass[letterpaper]{article}
|
|
\usepackage{amsmath}
|
|
\usepackage{graphicx}
|
|
\usepackage{cancel}
|
|
\usepackage{amssymb}
|
|
\usepackage{listings}
|
|
\usepackage[shortlabels]{enumitem}
|
|
\usepackage{soul}
|
|
%\usepackage[smartEllipses,hashEnumerators,hybrid]{markdown}
|
|
\usepackage{geometry}
|
|
\usepackage{dirtytalk}
|
|
\usepackage{mathtools}
|
|
\usepackage{lplfitch}
|
|
|
|
\geometry{portrait, margin=1in}
|
|
|
|
%\begin{minted}[linenos,bgcolor=LightGray]{[language]}
|
|
|
|
\date{01/20/2022}
|
|
\title{%
|
|
Set Theory: Functions\\
|
|
\large MATH--190--01 : Discrete Mathematics for Computing}
|
|
\author{Blizzard MacDougall}
|
|
\begin{document}
|
|
\maketitle
|
|
\pagenumbering{arabic}
|
|
Functions work just like in math before.
|
|
Its denoted as:
|
|
\begin{equation}
|
|
f:X\to Y
|
|
\end{equation}
|
|
This is an assignment of X to Y, such that every element of $X$ is assigned to
|
|
exactly one element $Y$. This is also written as $X\mapsto Y$, or $x\xrightarrow{f}y$.
|
|
If $X\mapsto Y$ is a function, then $f(x)$, is called \say{f of x} or \say{the image of x under f}.
|
|
|
|
X is the domain, Y is the \emph{codomain}. The range is $\{y\in Y | y=f(x)\ for\ some\ x\in X\}$.
|
|
Note that \verb|range|$(f)\subseteq Y$, but it is not necessarily equal.
|
|
|
|
For a function $f: X \mapsto Y$, given a $y\in Y$ there \emph{may} be a function such that
|
|
$f(x)=y$. If there is such an $x$, it is called the \emph{preimage} or \emph{inverse image} of $y$.
|
|
There may be multiple of these $x$s.
|
|
The inverse image of y is $\{x\in X|f(x)=y\}$
|
|
|
|
A function is one-to-one or \emph{injective} if there is only one $x$ mapped to every $y$.
|
|
Functions that are not injective collapse points together.
|
|
|
|
To prove injectivity, either pick $x_1$ and $x_2$ and prove that $[f(x_1)=f(x_2)]\to (x_1=x_2)$,
|
|
or pick two arbitrary elements $x_1\neq x_2$ and show that $f(x_1)\neq f(x_2)$
|
|
|
|
A function $f: X\mapsto Y$ is onto or \emph{surjective} if every $y$ is the image of some $x$.
|
|
That is, the range is the same as the codomain.
|
|
|
|
A function $f:X\mapsto Y$ is a bijection if $f$ is \emph{both} injective and surjective.
|
|
This is also called one-to-one correspondence.
|
|
|
|
$f^{-1}(x)=y \leftrightarrow f(y)=x$
|
|
|
|
If $f$ is a bijection, then $f^{-1}$ is a bijection.
|
|
\end{document}
|