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\begin{document}
\title{Set Theory Preliminaries}
\section{Introduction}
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Since this is a course in real analysis and not set theory, we will not attempt to develop set theory from axioms. Instead, we will loosely define a \textit{set} as a collection of objects.  We will consider a set to be \textit{well-defined} if, given an arbitrary object, we can determine whether or not it belongs to the set.  
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By so doing, we avoid some issues with set theory that became apparent in the late $19^{th}$ and early $20^{th}$ centuries having to do with seemingly valid but clearly contradictory statements.
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Perhaps the most famous is the Russel paradox, which can be stated as follows:  Let $S$ be the set of all sets \textit{that do not contain themselves as an element}.  We then ask whether or not $S$ belongs to itself.
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If we say that $S$ does not belong to itself, then by definition it has to belong to itself.
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If we say that $S$ belongs to itself, then by definition it does not belong to itself.
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No matter how we answer the question we have a contradiction.
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It is understood today that to avoid this type of problem one must exclude from the definition of a set certain overly broad constructs like "the set of all sets". 
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The central idea behind a set is the idea of membership.  All we care about for a given object is whether it does or does not belong to the set in question.  We can conceptually represent a set as an unordered list of the elements that belong to the set.  A given object is either in the list, or it is not.
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\section{Notation and Definitions}
We will usually denote a set by an uppercase letter, and elements of a set by lowercase letters.
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For (sufficiently small) finite sets, we have the option of simply listing the elements of the set between curly brackets $\{\}$, for example,
\[
A=\{1,3,5,7,9\}
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\]
We can extend this a bit using \textit{elipses} "$\ldots$" symbol.  In this form, we list enough elements to make the pattern clear and indicate that the list goes on forever by the elipses.  The set of even natural numbers could be written this way as
\[
A=\{2,4,6,8,\ldots\}
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\]
More generally, we can use \textit{set-builder notation} which has the general form
\[
A=\{x:\mbox{ some condition is true}\}
\]
read "A is the set of all $x$ such that" followed by a statement of the condition.  we could have defined the set $\{3,5,7,9\}$ using set-builder notation as
\[
A=\{x:x=2n+1 \mbox{ for some } n\in\mathbb{N} \mbox{ with } n<5\}
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\]
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\begin{definition}[element of a set]
If an object $a$ belongs to the set $A$, we say that $a$ is an \textbf{element} of $A$ and write $a\in A$.
\end{definition}
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\begin{definition}[subset]
If every element of a set $B$ also belongs to the set $A$, we say that $B$ is a \textbf{subset} of $A$ and write $B\subseteq A$.
\end{definition}
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\begin{definition}[proper subset]
If $B\subseteq A$ and, in addition, some elements of $A$ do not belong to $B$, we say that $B$ is a \textbf{proper} subset of $A$ and write $B\subset A$.  
\end{definition}
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\begin{definition}[equality of sets]
We say that two sets $A$ and $B$ are \textbf{equal} and write $A=B$ if and only if $A\subseteq B$ and $B\subseteq A$.  Otherwise we say that $A$ and $B$ are not equal and write $A\neq B$.
\end{definition}
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\begin{definition}[empty set]
The set that has no elements is called the \textbf{empty set} or \textbf{null set} and denoted by the symbol $\emptyset$.  Any set that is not equal to $\emptyset$ is said to be \textbf{nonempty}.
\end{definition}
\section{Operations on Sets}
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\begin{definition}[set union]
If $A$ and $B$ are two sets, the set consisting of all elements that belong to $A$ or $B$ or both is called the \textbf{union} of $A$ and $B$ and denoted by $A\cup B$.  In set-builder notation,
\[
A\cup B=\{x:\;x\in A\mbox{ or } x\in B\;\mbox{(or both)}\}
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\]
\end{definition}
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\begin{definition}[set intersection]
If $A$ and $B$ are two sets, the set consisting of all elements that belong to both $A$ and $B$ is called the \textbf{intersection} of $A$ and $B$ and denoted by $A\cap B$.  In set-builder notation,
\[
A\cap B=\{x:\;x\in A\mbox{ and } x\in B\}
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\]
\end{definition}
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\begin{definition}[compliment of a set]
If $A$ and $B$ are two sets, the set consisting of all elements that belong to $A$ but not to $B$ is called the \textbf{compliment} of $B$ relative to $A$ and denoted by $A\setminus B$ or, if $A$ is understood from the context, by $B^c$.  In set-builder notation,
\[
A\setminus B=\{x:\;x\in A\mbox{ and } x\notin B\}
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\]
\end{definition}
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\begin{definition}[Cartesian product]
If $A$ and $B$ are two sets, the set consisting of all \textbf{ordered pairs} $(a,b)$ with the property that $a\in A$ and $b\in B$ is known as the \textbf{Cartesian product} of $A$ and $B$ and denoted by $A\times B$.  In set-builder notation,
\[
A\times B=\{(a,b):\;a\in A\mbox{ and } b\in B\}
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\]
\end{definition}
\section{Relations and Partial Orderings}
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\begin{definition}[relation]
If $A$ and $B$ are two sets, a subset $\mathcal{R}$ of $A\times B$ is called a \textbf{relation} from $A$ to $B$. If the ordered pair $(a,b)$ belongs to the subset that defines a relation $\mathcal{R}$, we write $a\mathcal{R}b$.  A relation defined on $A\times A$ is called a \textbf{relation on } $A$.
\end{definition}
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\begin{example}
We may define a relation $\mathcal{R}$ on the real numbers $\mathbb{R}$ by
\[
\mathcal{R}=\{(x,y): x<y,\quad x,y\in\mathbb{R}\}
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\]
so that $x\mathcal{R}y$ whenever $x<y$.  In this case, we simply write "$<$" in place of $\mathcal{R}$.  This is known as the "less than" relation.
\end{example}
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\begin{definition}[antisymmetric]
A relation $\mathcal{R}$ on $\mathbb{R}$ is called \textbf{antisymmetric} if
\[
x\mathcal{R}y \;\mbox{and} \; y\mathcal{R}x\quad\Rightarrow\quad x=y\quad\forall x,y\in\mathbb{R}
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\]
\end{definition}
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\begin{definition}[transitive]
A relation $\mathcal{R}$ on $\mathbb{R}$ is called \textbf{transitive} if
\[
x\mathcal{R}y \;\mbox{and} \; y\mathcal{R}z\quad\Rightarrow\quad x\mathcal{R}z\quad\forall x,y,z\in\mathbb{R}
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\]
\end{definition}
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\begin{definition}[partial ordering]
A relation $\mathcal{R}$ on $\mathbb{R}$ is transitive and antisymmetric is called a \textbf{partial ordering} of $\mathbb{R}$
\end{definition}
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\begin{definition}[transitive]
A relation $\mathcal{R}$ on $\mathbb{R}$ is called \textbf{transitive} if
\[
x\mathcal{R}y \;\mbox{and} \; y\mathcal{R}z\quad\Rightarrow\quad x\mathcal{R}z\quad\forall x,y,z\in\mathbb{R}
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\]
\end{definition}
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\begin{definition}[inverse relation]
If $\mathcal{R}$ is a relation from $A$ to $B$, then the \textbf{inverse} relation $\mathcal{R}^{-1}$ is a relation from $B$ to $A$ defined by
\[
\mathcal{R}^{-1}=\{(b,a): (a,b)\in\mathcal{R}\}
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\]
\end{definition}
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\begin{definition}[image under a relation]
If $\mathcal{R}$ is a relation from $A$ to $B$ and $A_1\subseteq A$, then the set defined by
\[
\mathcal{R}(A_1)=\{b\in B: (a,b)\in\mathcal{R}\mbox{  for some } a\in A_1\}
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\]
\end{definition}
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\begin{definition}[domain of a relation]
If $\mathcal{R}$ is a relation from $A$ to $B$ the \textbf{domain} of $\mathcal{R}$ is the set
\[
D_{\mathcal{R}}=\{a\in A: (a,b)\in\mathcal{R}\mbox{  for some } b\in B\}
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\]
\end{definition}
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\begin{definition}[range of a relation]
If $\mathcal{R}$ is a relation from $A$ to $B$ the \textbf{domain} of $\mathcal{R}$ is the set
\[
R_{\mathcal{R}}=\{b\in B: (a,b)\in\mathcal{R}\mbox{  for some } a\in A\}
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\]
\end{definition}
\section{Functions}
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\begin{definition}[function]
A \textbf{function} from a set $X$ into a set $Y$, denoted by $f:X\rightarrow Y$ is a relation on $X\times Y$ with the property that for each element $x$ of the domain of $f$, there is exactly one element of $Y$ such that $(x,y)\in f$ or, in the notation usually used with functions, $f(x)=y$.  In this case we say that $f$ is \textbf{defined} on $X$ and has \textbf{codomain} $Y$.  
\end{definition}
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\begin{definition}[preimage]
If $f:X\rightarrow Y$ is a function and $(x,y)\in f$ (or, in the usual notation, $f(x)=y$), we say that $x$ is a \textbf{preimage} of $y$ under $f$.
\end{definition}
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\begin{definition}[range of a function]
If $f:X\rightarrow Y$ is a function the \textbf{range} of $f$ is the set of all function values,
\[
R_f=\{y\in Y: f(x)=y\mbox{ for some } x\in X\}
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\]
\end{definition}
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\begin{definition}[inverse of a function]
If $f:X\rightarrow Y$ is a function with the property that the range of $f$ is $Y$ and each element of $Y$ has a unique preimage in $X$, then $f$ has an \textbf{inverse} $f^{-1}:Y\rightarrow X$ defined by
\[
f^{-1}(y)=x\quad\mbox{ where } f(x)=y
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\]
\end{definition}
\end{document}



