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\begin{document}
\title{Set Theory Preliminaries (continued)}
\section{Introduction}
\par\vspace{0.3 cm}
The following notation is used for unions and intersections:  
\par\vspace{0.3 cm}
Suppose $A_1,A_2,\ldots,A_n$ are sets.  The \textbf{union} of all of these sets, that is, the set consisting of all elements that belong to \textit{any one} of $A_1-A_n$, is denoted by
\[
\bigcup_{i=1}^n A_i
\vspace{0.2 cm}
\] 
The \textbf{intersection} of all of these sets, namely the set consisting of all elements that belong to \textit{every one} of $A_1-A_n$ is denoted by 
\[
\bigcap_{i=1}^n A_i
\vspace{0.2 cm}
\] 
We will find that we need to extend this notation, due to the fact that it is possible to have collections of sets that are too numerous to represent in this way.  
\par\vspace{0.3 cm}
To that end, we introduce the idea of an \textit{index set}.  An index set is just a set that contains the subscripts of the $A$'s.  Now if $I$ is our index set, the notation for the union and intersection of the collection is
\[
\bigcup_{\alpha\in I}^n A_{\alpha}\quad\mbox{and}\quad\bigcap_{\alpha\in I}^n A_{\alpha}
\vspace{0.2 cm}
\] 
\section{Equivalence Relations}
We defined a relation on $X\times Y$ as a subset of the Cartesian product $X\times Y$, namely 
\[
\mathcal{R}\subseteq\{(x,y):x\in X\mbox{  and  } y\in Y\}
\vspace{0.2 cm}
\]
If $X$ and $Y$ are the same set, we say that $\mathcal{R}\subseteq X\times X$ is a relation on $X$.
\par\vspace{0.2 cm}
We have previously defined the following two properties:
\par\vspace{0.2 cm}
\begin{definition}[reflexive]
A relation on $X$ with the property that $x\mathcal{R}x$ for every $x\in X$ is called \textbf{reflexive}.  That is, for every $x\in X$, $(x,x)\in\mathcal{R}$.
\end{definition}
\par\vspace{0.2 cm}
\begin{definition}[symmetric]
A relation on $X$ with the property that $y\mathcal{R}x$ whenever $x\mathcal{R}y$ for every $x,y\in X$ is called \textbf{symmetric}.  That is, for every $x,y\in X$,
\[
(x,y)\in\mathcal{R}\quad\Leftrightarrow\quad(y,x)\in\mathcal{R}
\vspace{0.2 cm}
\]
\end{definition}
Now we add the following:
\par\vspace{0.2 cm}
\begin{definition}[transitive]
A relation on $X$ with the property that together $x\mathcal{R}y$ and $y\mathcal{R}z$ imply $x\mathcal{R}z$ for every $x,y,z\in X$ is called \textbf{transitive}.  That is, for every $x,y,z\in X$, 
\[
(x,y)\in\mathcal{R}\mbox{  and  }(y,z)\in\mathcal{R}\quad\Rightarrow\quad(x,z)\in\mathcal{R}
\]
\end{definition}
\par\vspace{0.2 cm}
A relation on $X$ with these three properties is called an \textbf{equivalence relation}. On the set of real numbers, 
the relation defined on $\mathbb{N}$ by
\[
a\mathcal{R}b\mbox{  if  } a\mbox{ is congruent to } b\mbox{  modulo  } 7
\vspace{0.2 cm}
\]
is an equivalence.  An important characteristic of equivalence ralations is that they partition $X$ in to disjoint sets called \textit{equivalance classes}.  In the above example, $\mathcal{R}$ partitions the natural numbers into 7 equivalence classes, depending on the value of $x\mod 7$.
\section{The Field Axioms}
We will not attempt to construct the real numbers, but will simply assume that we understand what is meant by $\mathbb{R}=(-\infty,\infty)$.
\par\vspace{0.2 cm}
We will also suppose that we are familiar with two operations, addition $+$ and multiplication $\cdot$ defined on $\mathbb{R}$.
\par\vspace{0.2 cm}
In fact, we will consider $+$ and $\cdot$ as real-valued functions defined on $\mathbb{R}\times\mathbb{R}$:
\[
+:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\quad\mbox{and}\quad\cdot:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}
\vspace{0.2 cm}
\]
We will assume that the set $\mathbb{R}$ together with the functions $+$ and $\cdot$ constitute a \textbf{field}, which means that the following properties hold for every $a,b,c\in\mathbb{R}$:
\par\vspace{0.2 cm}\noindent
\begin{tabular}{|l|r|}
\hline
Closure & $a+b\in\mathbb{R}$ and $a\cdot b\in\mathbb{R}$\\
&\\
Associative & $a+(b+c)=(a+b)+c$ and $a\cdot(b\cdot c)=(a\cdot b)\cdot c$\\
&\\
Commutative & $a+b=b+a$ and $a\cdot b=b\cdot a$\\
&\\
Distributive & $a\cdot(b+c)=a\cdot b+a\cdot c$\\
&\\
Additive Identity & There is a unique element $0\in\mathbb{R}$\\
                  &  with the property $0+a=a$\\
&\\
Multiplicative Identity & There is a unique element $1\in\mathbb{R}$\\
& with the properties that $1\neq0$ and $1\cdot a=a$\\
&\\
Additive Inverse & For each $a\in\mathbb{R}$ there is\\
& a unique element $-a\in\mathbb{R}$ such that $a+(-a)=0$\\
&\\
Multiplicative Inverse & For each $a\in\mathbb{R}\setminus {0}$ there is\\
& a unique element $a^{-1}\in\mathbb{R}$ such that $a\cdot a^{-1}=1$\\
\hline
\end{tabular}
\section{Order Axioms}
The field axioms define the algebraic properties of the real numbers, but there are other examples of fields, some of which do not have all of the features we associate with $\mathbb{R}$.  To complete the picture, we need to define an order relation on $\mathbb{R}$.  We suppose therre is a relation $<$ on $\mathbb{R}\times\mathbb{R}$ with the following properties:
\par\vspace{0.2 cm}
\begin{tabular}{|l|r|}
\hline
Trichotomy & For $a,b\in\mathbb{R}$ only one of $a<b$, $a=b$, and $a>b$ holds\\
&\\
Transitive & For $a,b,c\in\mathbb{R}$, $a<b$ and $b<c$ imply $a>c$\\
&\\
Additive & For $a,b,c\in\mathbb{R}$, $a<b$ implies $a+c>b+c$\\
&\\
Multiplicative & For $a,b,c\in\mathbb{R}$, $a<b$ and $c>0$ imply $ac<bc$\\
& and $a,b,c\in\mathbb{R}$, $a<b$ and $c<0$ imply $bc<ac$\\
&\\
\hline
\end{tabular}
\section{Subsets of the Real Numbers}
\begin{definition}[nonnegative, positive]
We will say that $x\in\mathbb{R}$ is \textbf{nonnegative} if $x\geq0$ and \textbf{positive} if $x>0$.
\end{definition}
as well as the following subsets:
\par\vspace{0.2 cm} The natural numbers:
\[
\mathbb{N}:=\{1,2,3,\ldots\}
\vspace{0.2 cm}
\]
\par\vspace{0.2 cm} The integers:
\[
\mathbb{Z}:=\{\ldots,-2,-1,0,1,2,\ldots\}
\vspace{0.2 cm}
\]
\par\vspace{0.2 cm} The rational numbers:
\[
\mathbb{Q}:=\left\{\frac{m}{n}:m,n\in\mathbb{Z}\mbox{ and }n\neq0\right\}
\vspace{0.2 cm}
\]
\par\vspace{0.2 cm} The irrationals:
\[
\mathbb{Q}^c:=\mathbb{R}\setminus\mathbb{Q}
\vspace{0.2 cm}
\]
\par\vspace{0.2 cm}
We will assume the following properties of these subsets without proof:
\begin{itemize}
\item i) $\mathbb{Z}$ is closed under addition, subtraction, and multiplication.
\item ii) If $n\in\mathbb{Z}$ then $n\in\mathbb{N}$ if and only if $n\geq 1$.
\item iii) There is no element of $z\in\mathbb{Z}$ such that $0<z<1$.
\end{itemize}
 


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