Skip to main content ☰ Contents Index You! < Prev ^ Up Next > \(\newcommand{\Z}{\mathbb{Z}}
\newcommand{\E}{\mathbb{E}}
\DeclareMathOperator{\ch}{char}
\newcommand{\N}{\mathbb{N}}
\newcommand{\W}{\mathbb{W}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\J}{\mathbb{J}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\M}{\mathcal{M}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\NE}{\mathcal{E}}
\newcommand{\Mn}[1]{\mathcal{M}_{#1 \times #1}}
\renewcommand{\P}{\mathcal{P}}
\newcommand{\Rn}{\R^n}
\newcommand{\Mat}{\mathbf}
\newcommand{\Seq}{\boldsymbol}
\newcommand{\seq}[1]{\boldsymbol{#1}}
\newcommand{\set}[1]{\mathcal{#1}}
\newcommand{\abs}[1]{\left\lvert{}#1\right\rvert}
\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
\newcommand{\cq}{\scalebox{.34}{\pscirclebox{ \textbf{?}}}}
\newcommand{\cqup}{\,$^{\text{\scalebox{.2}{\pscirclebox{\textbf{?}}}}}$}
\newcommand{\cqupmath}{\,^{\text{\scalebox{.2}{\pscirclebox{\textbf{?}}}}}}
\newcommand{\cqupmathnospace}{^{\text{\scalebox{.2}{\pscirclebox{\textbf{?}}}}}}
\newcommand{\uspace}[1]{\underline{}}
\newcommand{\muspace}[1]{\underline{\mspace{#1 mu}}}
\newcommand{\bspace}[1]{}
\newcommand{\ie}{\emph{i}.\emph{e}.}
\newcommand{\nq}[1]{\scalebox{.34}{\pscirclebox{\textbf{#1}}}}
\newcommand{\equalwhy}{\stackrel{\cqupmath}{=}}
\newcommand{\notequalwhy}{\stackrel{\cqupmath}{\neq}}
\newcommand{\Ker}{\text{Ker}}
\newcommand{\Image}{\text{Im}}
\newcommand{\polyp}{p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots a_2x^2 + a_1x + a_0}
\newcommand{\GL}{\text{GL}}
\newcommand{\SL}{\text{SL}}
\newcommand{\lcm}{\text{lcm}}
\newcommand{\Aut}{\text{Aut}}
\newcommand{\Inn}{\text{Inn}}
\newcommand{\Hol}{\text{Hol}}
\newcommand{\cl}{\text{cl}}
\newcommand{\bsq}{\hfill $\blacksquare$}
\newcommand{\NIMdot}{{ $\cdot$ }}
\newcommand{\eG}{e_{\scriptscriptstyle{G}}}
\newcommand{\eGroup}[1]{e_{\scriptscriptstyle{#1}}}
\newcommand{\Gdot}[1]{\cdot_{\scriptscriptstyle{#1}}}
\newcommand{\rbar}{\overline{r}}
\newcommand{\nuG}[1]{\nu_{\scriptscriptstyle{#1}}}
\newcommand{\rightarray}[2]{\left[ \begin{array}{#1} #2 \end{array} \right]}
\newcommand{\polymod}[1]{\mspace{5 mu}(\text{mod} #1)}
\newcommand{\ts}{\mspace{2 mu}}
\newcommand{\ds}{\displaystyle}
\newcommand{\adj}{\text{adj}}
\newcommand{\pol}{\mathbb{P}}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\CB}{\mathcal{B}}
\renewcommand{\CD}{\mathcal{D}}
\newcommand{\CP}{\mathcal{P}}
\newcommand{\CQ}{\mathcal{Q}}
\newcommand{\CW}{\mathcal{W}}
\newcommand{\rank}{\text{rank}}
\newcommand{\nullity}{\text{nullity}}
\newcommand{\trace}{\text{trace}}
\newcommand{\Area}{\text{Area}}
\newcommand{\Vol}{\text{Vol}}
\newcommand{\cov}{\text{cov}}
\newcommand{\var}{\text{var}}
\newcommand{\la}{\left|}
\newcommand{\ra}{\right|}
\newcommand{\Int}{\text{Int}}
\newcommand{\Bdry}{\text{Bdry}}
\newcommand{\lub}{\text{lub}}
\newcommand{\glb}{\text{glb}}
\newcommand{\ssim}{\sim}
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\vd}{\mathbf{d}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vf}{\mathbf{f}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\vh}{\mathbf{h}}
\newcommand{\vm}{\mathbf{m}}
\newcommand{\vn}{\mathbf{n}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vq}{\mathbf{q}}
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vs}{\mathbf{s}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vz}{\mathbf{z}}
\newcommand{\vzero}{\mathbf{0}}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\newcommand{\vF}{\mathbf{F}}
\newcommand{\vP}{\mathbf{P}}
\newcommand{\nin}{}
\newcommand{\Dom}{\text{Dom}}
\newcommand{\tr}{\mathsf{T}}
\newcommand{\proj}{\text{proj}}
\newcommand{\comp}{\text{comp}}
\newcommand{\Row}{\text{Row }}
\newcommand{\Col}{\text{Col }}
\newcommand{\Nul}{\text{Nul }}
\newcommand{\Span}{\text{Span}}
\newcommand{\Range}{\text{Range}}
\newcommand{\Domain}{\text{Domain}}
\newcommand{\gr}{\varphi}
\newcommand{\grc}{\overline{\varphi}}
\newcommand{\hthin}{\hlinewd{.1pt}}
\newcommand{\hthick}{\hlinewd{.7pt}}
\newcommand{\lint}{\underline{}
\int}
\newcommand{\uint}{ \underline{}
\int}
\newcommand{\be}{\begin{enumerate}}
\newcommand{\ee}{\end{enumerate}}
\newcommand{\bei}{\begin{numlist2}}
\newcommand{\eei}{\end{numlist2}}
\newcommand{\ba}{\begin{itemize}}
\newcommand{\ea}{\end{itemize}}
\newcommand{\bal}{\begin{alphalist2}}
\newcommand{\eal}{\end{alphalist2}}
\newcommand{\bi}{\begin{itemize}}
\newcommand{\ei}{\end{itemize}}
\newcommand{\btl}{\begin{thmlist}}
\newcommand{\etl}{\end{thmlist}}
\renewcommand{\be}{\begin{enumerate}}
\renewcommand{\ee}{\end{enumerate}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section Summary
Important ideas that we discussed in this section include the following.
We can consider a set to be a well-defined collection of elements.
A subset of a set is any collection of elements from that set. That is, a subset \(S\) of a set \(X\) is a set with the property that if \(s \in S\text{,}\) then \(s \in X\text{.}\)
If \(X\) and \(Y\) are sets, then the union \(X \cup Y\) is the set
\begin{equation*}
X \cup Y = \{z \mid z \in X \text{ or } z \in Y\}\text{.}
\end{equation*}
The union of an arbitrary collection \(\{X_{\alpha}\}\) of sets for \(\alpha\) in some indexing set \(I\) is the set
\begin{equation*}
\bigcup_{\alpha \in I} X_{\alpha} = \{z \mid z \in X_{\beta} \text{ for some } \beta \in I\}\text{.}
\end{equation*}
If \(X\) and \(Y\) are sets, then the intersection \(X \cap Y\) is the set
\begin{equation*}
X \cap Y = \{z \mid z \in X \text{ and } z \in Y\}\text{.}
\end{equation*}
The intersection of an arbitrary collection \(\{X_{\alpha}\}\) of sets for \(\alpha\) in some indexing set \(I\) is the set
\begin{equation*}
\bigcap_{\alpha \in I} X_{\alpha} = \{z \mid z \in X_{\beta} \text{ for all } \beta \in I\}\text{.}
\end{equation*}
If \(X\) is a set and \(A\) is a subset of \(X\text{,}\) then the complement of \(A\) in \(X\) is the set
\begin{equation*}
A^c = \{x \in X \mid x \notin A\}\text{.}
\end{equation*}
If \(\{X_{i}\}\) is a collection of sets with \(i\) in some indexing set \(I\text{,}\) where \(I\) is finite or \(I\) is the set of positive integers, the Cartesian product \(\Pi_{i \in I} X_i\) of the sets \(X_{i}\) as the set of all ordered tuples of the form \((x_i)\) where \(i \in I\text{.}\)
