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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 .
(a) Example(b) Example(c) Example
(d) Example(e) Example(f) Example
(g) Example(h) Example
Figure 5.1.1. Each of these is a graph
(a) Example(b) Example(c) Example
(d) Example(e) Example(f) Example (g) Example
Figure 5.1.2. None of these is a graph
Checkpoint 5.1.3 .
(a) Question(b) Question(c) Question
(d) Question(e) Question(f) Question
(g) Question(h) Question
Figure 5.1.4. Determine which of these are graphs
\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*}
(a) Example
\begin{align*}
P \amp =\{p_1,p_2,p_3,p_4\}\\
Q \amp =\emptyset
\end{align*}
(b) Example
\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*}
(c) Example
\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*}
(d) Example
\begin{align*}
V \amp =\{c,d,e,f\}\\
E \amp =\{\{c,d\}, \{d,e\}, \{e,f\},\\
\amp \{c,f\}\}
\end{align*}
(e) Example
\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*}
(f) Example
\begin{align*}
P \amp =\{p,q,r\}\\
Q \amp = \{ \{p,q\} \}
\end{align*}
(g) Example
\begin{align*}
P \amp =\{q,r,s,t\}\\
Q \amp =\{\{q,r\}, \{r,t\}, \{q,t\}, \{r,s\}\}
\end{align*}
(h) Example
Figure 5.1.5. Each 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*}
(a) Example
\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*}
(b) Example
\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*}
(a) Example
\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*}
(b) Example
\begin{align*}
P \amp =\emptyset\\
Q \amp =\emptyset
\end{align*}
(a) Example
\begin{align*}
P \amp =\{q,r,s,t\}\\
Q \amp =\{\{q,r\}, \{r,t\}, \{s\} \}
\end{align*}
(b) Example
\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*}
(a) Example
\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*}
(b) Example Figure 5.1.6. None of these is a graph.
Checkpoint 5.1.7 .
\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 5.1.8 .
In terms of the diagrams in
Figure 5.1.1 list properties required and properties not allowed in graphs.
Checkpoint 5.1.9 .
In terms of the sets in
Figure 5.1.5 list properties required and properties not allowed in graphs.
Checkpoint 5.1.10 .
Checkpoint 5.1.11 .
Write the set form of the fourth graph (bottom left) in
Figure 5.1.1
