Skip to main content
Contents
Dark Mode Prev Up Next
\(\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Pow}{\mathcal{P}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section 5.1 Discovering Graphs
Definitions are typically constructed after we work with new objects. The definition is constructed to match the properties we have observed and need. In this section we will practice this by developing a definition for a type of object known as a
graph .
Figure 5.1.1. Each of these is a graph
Figure 5.1.2. None of these is a graph
Checkpoint 5.1.3 .
Example 5.1.4 .
Each of these is a graph.
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1,q_2,q_3\}\\
q_1 \amp = \{p_1,p_2\}\\
q_2 \amp = \{p_1,p_3\}\\
q_3 \amp = \{p_2,p_3\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3,p_4\}\\
Q \amp =\emptyset
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3,p_4\}\\
Q \amp =\{q_1,q_2\}\\
q_1 \amp = \{p_1,p_3\}\\
q_2 \amp = \{p_2,p_4\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3,p_4,p_5\}\\
L \amp =\{\ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\}\\
\ell_1 \amp = \{p_2,p_3\}\\
\ell_2 \amp = \{p_1,p_4\}\\
\ell_3 \amp = \{p_4,p_5\}\\
\ell_4 \amp = \{p_1,p_5\}\\
\ell_5 \amp = \{p_2,p_4\}\\
\ell_6 \amp = \{p_2,p_5\}
\end{align*}
\begin{align*}
V \amp =\{c,d,e,f\}\\
E \amp =\{\{c,d\}, \{d,e\}, \{e,f\},\\
\amp \{c,f\}\}
\end{align*}
\begin{align*}
V \amp =\{v_1,v_2,v_3,v_4\}\\
E \amp =\{\{v_1,v_2\}, \{v_1,v_3\}, \\
\amp \{v_2,v_3\}, \{v_2,v_4\}\}
\end{align*}
\begin{align*}
P \amp =\{p\}\\
Q \amp =\emptyset
\end{align*}
\begin{align*}
P \amp =\{q,r,s,t\}\\
Q \amp =\{\{q,r\}, \{r,t\}, \{q,t\}, \{r,s\}\}
\end{align*}
Example 5.1.5 .
None of these is a graph.
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1,q_2\}\\
q_1 \amp = \{p_1,p_2\}\\
q_2 \amp = \{p_3,p_4\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2\}\\
Q \amp =\{q_1,q_2\}\\
q_1 \amp = \{p_1,p_2\}\\
q_2 \amp = \{p_1,p_2\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1,q_2,q_3\}\\
q_1 \amp = (p_1,p_2)\\
q_2 \amp = (p_1,p_3)\\
q_3 \amp = (p_2,p_3)
\end{align*}
\begin{align*}
V \amp =\{x,y,z\}\\
E \amp =\{f,g,h\}\\
f \amp = (x,y)\\
g \amp = (y,x)\\
h \amp = (y,z)
\end{align*}
\begin{align*}
P \amp =\emptyset\\
Q \amp =\emptyset
\end{align*}
\begin{align*}
P \amp =\{q,r,s,t\}\\
Q \amp =\{\{q,r\}, \{r,t\}, \{s\} \}
\end{align*}
\begin{align*}
V \amp =\{p_1,p_2,\ldots,p_n,\ldots\}\\
E \amp =\{\{p_1,p_2\}, \{p_2,p_4\}, \{p_1,p_3\}, \{p_4,p_6\}\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1,q_2,q_3\}\\
q_1 \amp = \{p_1,p_2\}\\
q_2 \amp = \{p_1,p_3\}
\end{align*}
Checkpoint 5.1.6 .
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1\}\\
q_1 \amp = \{p_1,p_3\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\{q_1,q_2\}\\
q_1 \amp = \{p_1,p_2\}
\end{align*}
\begin{align*}
V \amp =\{v_1,v_2,\ldots,v_n,\ldots\}\\
E \amp =\{e_1,e_2,\ldots,e_n,\ldots\}\\
e_1 \amp = (v_1,v_2)\\
\vdots\\
e_n \amp = (v_n,v_{n+1})\\
\vdots
\end{align*}
\begin{align*}
P \amp =\{p,q,r\}\\
Q \amp =\{t,u,v\}\\
t \amp = \{p,q\}\\
u \amp = \{p,r\}\\
v \amp = \{q,r\}
\end{align*}
\begin{align*}
P \amp =\{p_1,p_2,p_3\}\\
Q \amp =\emptyset
\end{align*}
\begin{align*}
V \amp =\{u_1,u_2,u_3,u_4\}\\
E \amp =\{(u_1,u_2), (u_2,u_1),\\
\amp (u_3,u_4)\}
\end{align*}
\begin{align*}
V \amp =\{v_1,v_2,v_3\}\\
E \amp =\{\{v_1,v_2\}, \{v_1,v_3\}, \{v_3\}\}
\end{align*}
\begin{align*}
P \amp =\{p,q,r,s\}\\
Q \amp =\{\{p,q\}, \{p,r\}, \{p,s\},\\
\amp \{q,r\}, \{q,s\}, \{r,s\}\}
\end{align*}
Checkpoint 5.1.7 .
In terms of the diagrams in
FigureΒ 5.1.1 list properties required and properties not allowed in graphs.
Checkpoint 5.1.8 .
In terms of the sets in
ExampleΒ 5.1.4 list properties required and properties not allowed in graphs.
Checkpoint 5.1.9 .
Checkpoint 5.1.10 .
Write the set form of the fourth graph (bottom left) in
FigureΒ 5.1.1