Set theoretic notation

The basic objects we study will be specific sets. For example, the sets of integers, rationals, reals and complex numbers; denoted by $\mbox{${\Bbb Z}$ }$, $\mbox{${\Bbb Q}$ }$, $\mbox{${\Bbb R}$ }$and $\mbox{${\Bbb C}$ }$, respectively. The positive integers, rationals and reals are denoted by $\mbox{${\Bbb Z}$ }^+$, $\mbox{${\Bbb Q}$ }^+$ and $\mbox{${\Bbb R}$ }^+$; the nonzero integers, rationals, reals and complex numbers are denoted by $\mbox{${\Bbb Z}$ }^*$, $\mbox{${\Bbb Q}$ }^*$, $\mbox{${\Bbb R}$ }^*$ and $\mbox{${\Bbb C}$ }^*$. Most of our sets will be built up from these objects and the empty set ($\emptyset$). We will be using throughout the course the basic properties of the fields of real and complex numbers.

Sets will usually be denoted by capital letters, $R$, $S$, ..., ${\cal R}$, ${\cal S}$, ..., $\Delta$, $\Gamma$, ... . Lower case letters will usually denote elements of sets. The symbol $\in$ denotes set membership; while the symbol $\not\in$ denotes ``not a member of.'' The English sentences ``$1$ is an integer'' and `` $\frac{1}{2}$ is not an integer'' are written in symbols as `` $1 \in \mbox{${\Bbb Z}$ }$'' and `` $\frac{1}{2} \not\in \mbox{${\Bbb Z}$ }$''. Both statements are true. Whereas the statement `` $\frac{1}{2} \in \mbox{${\Bbb Z}$ }$'' is not true. In the most common form of logic, the one we use, most statements1 are either true or not true. This does not imply that we can always determine whether a given statement is true or not. Among other common symbols we use are $\cup$ union, $\cap$ intersection, $\exists$ there exists, $\exists !$ there exists a unique, $\forall$ for all. We will often describe sets by expressions like

\begin{displaymath}S = \{\mbox{points in some other set (that we already know)} T \mbox{ that satisfy some property} \} .\end{displaymath}

For example,

\begin{displaymath}\mbox{${\Bbb Z}$}^+ = \{ n \in \mbox{${\Bbb Z}$}; \ n > 0 \} \end{displaymath}

(the use of the semicolon ; in the above expression is equivalent to the words such that - that will often be abbreviated s.t.) and

\begin{displaymath}\mbox{${\Bbb Q}$}= \left \{ \frac{m}{n} \in \mbox{${\Bbb R}$}... ...nd } n \in \mbox{${\Bbb Z}$}\mbox{ with } n \not= 0 \right \} .\end{displaymath}

The same set may be described in more than one way:

\begin{displaymath}\mbox{${\Bbb Q}$}= \left \{ \frac{m}{n} \in \mbox{${\Bbb C}$}... ...x{ and } n \in \mbox{${\Bbb Z}$}\mbox{ with } n > 0 \right \} .\end{displaymath}

The formal definitions of $\cup$, set theoretic union, and $\cap$, set theoretic intersection, can now be given as

\begin{displaymath}S \cup T = \{x; x \in S \mbox{ or } x \in T \} \end{displaymath}

and

\begin{displaymath}S \cap T = \{x; x \in S \mbox{ and } x \in T \} = \{x \in S; x \in T \} .\end{displaymath}

Technically, the definition of $S \cup T$ and the first of our two definitions of $S \cap T$ are not correct since they do not specify from which set $x$ is chosen. All our operations must for technical foundational issues be restricted to some universal set. In practical terms, this issue seldom causes any problems for the topics that we will study. Two more symbols that are commonly used involve set inclusion: $\subset$, $\supset$. Both $A \subset B$ and $B \supset A$ mean that every element of $A$ is in $B$ ($B$ may coincide with $A$ or be a ``bigger'' set than $A$). The inclusion $A \subset B$ is proper if $A \not= B$. I2 will also us the symbol $-$ to denote differences of sets3:

\begin{displaymath}S - T = \{ x \in S; x \not\in T \} = S - (S \cap T) .\end{displaymath}