844 lines
23 KiB
Markdown
844 lines
23 KiB
Markdown
**Example 1.1.1**
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Page 24
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Use variables to rewrite the following sentences more formally.
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a. Are there numbers with the property that the sum of their squares equals the
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square of their sum?
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b. Given any real number, its square is nonnegative.
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**Solution**
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a. Are there numbers $a$ and $b$ with the property that $a^2 + b^2 = (a + b)^2$?
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_Or_: Are there numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
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_Or_: Do there exist any numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
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b. Given any real number $r$, $r^2$ is nonnegative.
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_Or_: For any real number $r$, $r^2 \geq 0$.
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_Or_: For every real number $r$, $r^2 \geq 0$.
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---
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**Example 1.1.2**
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Page 26
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Fill in the blanks to rewrite the following statement:
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For every real number $x$, if $x$ is nonzero then $x^2$ is positive.
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a. If a real number is nonzero, then its square ________.
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b. For every nonzero real number $x$, ________.
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c. If $x$ ________, then ________.
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d. The square of any nonzero real number is ________.
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e. All nonzero real numbers have ________.
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**Solution**.
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a. is positive.
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b. $x^2$ is positive.
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c. is a nonzero real number, $x^2$ is positive.
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d. positive.
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e. positive squares .
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---
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**Example 1.1.3**
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Page 27
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Fill in the blanks to rewrite the following statement: Every pot has a lid.
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a. All pots ________.
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b. For every pot $P$, there is ________.
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c. For every pot $P$, there is a lid $L$ such that ________.
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**Solution**
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a. have lids.
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b. a lid.
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c. $L$ is a lid for $P$..
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---
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**Example 1.1.4**
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Page 28
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Fill in the blanks to rewrite the following statement in three different ways:
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There is a person in my class who is at least as old as every person in my clas.
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a. Some ________ is at least as old as ________.
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b. There is a person $p$ in my class such that $p$ is ________.
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c. There is a person $p$ in my class with the property that for every person $q$
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in my class, $p$ is ________.
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**Solution**
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a. person; every person.
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b. at least as old as every person in my class.
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c. at least as old as $q$.
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---
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**Example 1.2.1**
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Page 30
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**Using the Set-Roster Notation**
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a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
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What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
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b. Is $\{0\} = 0$?
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c. How many elements are in the set $\{1, \{1}\}$?
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d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
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and $U_0$.
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**Solution**
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a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
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What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
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$A$, $B$, and $C$ have exactly the same three elements, $1$, $2$, and $3$.
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Therefore, $A$, $B$, and $C$ are simply different ways to represent the same
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set.
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b. Is $\{0\} = 0$?
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$\{0\} \neq 0$ because $\{0\}$ is a set with one element, namely $0$, whereas
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$0$ is just the symbol that represents the number zero.
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c. How many elements are in the set $\{1, \{1}\}$?
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The set $\{1, \{1\}\}$ has two elements. $1$ and the set whose only element is
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$1$.
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d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
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and $U_0$.
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$U_1 = \{1, -1\}, \quad U_2 = \{2, -2\}, \quad U_0 = \{0, 0\} = \{0\}$
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---
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**Example 1.2.2**
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Page 31
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**Using the Set-Builder Notation**
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Given that $\mathbb{R}$ denotes the set of all real numbers, $\mathbb{Z}$ the
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set of all integers, and $\mathbb{Z}^+$ the set of all positive integers,
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describe each of the following sets.
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a. $\{x \in \mathbb{R} | -2 < x < 5\}$
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b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
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c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
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**Solution**
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a. $\{x \in \mathbb{R} | -2 < x < 5\}$
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$\{x \in \mathbb{R} | -2 < x < 5\}$ is the open interval of real numbers
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(strictly) between $-2$ and 5. It is pictured as follows (see page 31).
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b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
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$\{x \in \mathbb{Z} | -2 < x < 5\}$ is the set of all integers (strictly)
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between $-2$ and $5$. It is equal to the set $\{-1, 0, 1, 2, 3, 4}$.
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c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
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Since all the integers in $\mathbb{Z}^+$ are positive,
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$\{x \in \mathbb{Z}^+ | -2 < x < 5\} = \{1, 2, 3, 4\}$.
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---
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**Example 1.2.3**
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Page 32
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Let $A = \mathbb{Z}^+$, $B = \{n \in \mathbb{Z} | 0 \leq n \leq 100\}$, and
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$C = \{100, 200, 300, 400, 500\}$. Evaluate the truth and falsity of each of the
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following statements
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a. $B \subseteq A$
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b. $C$ is a proper subset of $A$.
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c. $C$ and $B$ have at least one element in common
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d. $C \subseteq B$
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e. $C \subseteq C$
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**Solution**
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a. $B \subseteq A$
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False. Zero is not a positive integer. Thus zero is in $B$ but zero is not in
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$A$, and so $B \nsubseteq A$
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b. $C$ is a proper subset of $A$.
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True. Each element in $C$ is a positive integer, and hence, is in $A$, but there
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are elements in $A$ that are not in $C$. For instance, $1$ is in $A$ and not in
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$C$.
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c. $C$ and $B$ have at least one element in common
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True. For example, $100$ is in both $C$ and $B$.
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d. $C \subseteq B$
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False. For example, $200$ is in $C$ but not in $B$.
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e. $C \subseteq C$
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True. Every element in $C$ is in $C$. In general, the definition of a subset
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implies that all sets are subsets of themselves.
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---
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**Example 1.2.4**
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Page 33
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**Distinction between $\in$ and $\subseteq$**
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Which of the following are true statements?
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a. $2 \in \{1, 2, 3\}$
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b. $\{2\} \in \{1, 2, 3\}$
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c. $2 \subseteq \{1, 2, 3\}$
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d. $\{2\} \subseteq \{1, 2, 3\}$
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e. $\{2\} \subseteq \{\{1\}, \{2\}\}$
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f. $\{2} \in \{\{1\}, \{2\}\}$
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**Solution**
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Only (a), (d), and (f) are true.
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For (b) to be true, the set $\{1, 2, 3\}$ would have to contain the element
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$\{2\}$. But the only elements of $\{1, 2, 3\}$ are $1$, $2$, and $3$, and $2$
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is not equal to $\{2\}$. Hence (b) is false.
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For \(c\) to be true, the number $2$ would have to be a set and every element in
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the set $2$ would have to be an element of $\{1, 2, 3}$. This is not the case,
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so \(c\) is false.
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For (e) to be true, every element in the set containing only the number $2$
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would have to be an element of the set whose elements are $\{1\}$ and $\{2\}$.
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But $2$ is not equal to either $\{1\}$ or $\{2\}$, and so (e) is false.
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---
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**Example 1.2.5 Ordered Pairs**
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Page 34
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a. Is $(1, 2) = (2, 1)$?
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b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
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c. What is the first element of $(1, 1)$?
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**Solution**
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a. Is $(1, 2) = (2, 1)$?
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No, By definition of equality of ordered pairs,
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$(1, 2) = (2, 1)$ if, and only if, 1 = 2, and 2 = 1.
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But $1 \neq 2$, and so the ordered pairs are not equal.
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b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
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Yes. By definition of equality of ordered pairs,
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$\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$ if, and
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only if, $3 = \sqrt{9}$ and $\dfrac{5}{10} = \dfrac{1}{2}$.
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Because these equations are both true, the ordered pairs are equal.
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c. What is the first element of $(1, 1)$?
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In the ordered pair $(1, 1)$, the first and second elements are both $1$.
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---
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**Example 1.2.6 Ordered $n$-tuples**
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Page 34
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a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
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b. Is
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$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
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**Solution**
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a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
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No. By definition of equality of ordered 4-tuples,
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$$ (1, 2, 3, 4) = (1, 2, 4, 3) \leftrightarrow 1 = 1, 2 = 2, 3 = 4, and 4 = 3 $$
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But $3 \neq 4$, and so the ordered 4-tuples are not equal.
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b. Is
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$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
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Yes. By definition of equality of ordered triples.
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$$ \left(3, (-2)^2, \frac{1}{2}\right) = \left(\sqrt{9}, 4, \frac{3}{6}\right) \leftrightarrow 3 = \sqrt{9} \text{ and } (-2)^2 = 4 \text{ and } \frac{1}{2} = \frac{3}{6} $$
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Because these equations are all true, the two ordered triples are equal.
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---
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**Example 1.2.7 Cartesian Products**
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Page 35
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Let $A = \{x, y\}$, $B = \{1, 2, 3\}$, and $C = \{a, b\}$.
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a. Find $A \times B$.
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b. Find $B \times A$.
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c. Find $A \times A$.
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d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
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e. Find $(A \times B) \times C$
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f. Find $A \times B \times C$
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g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
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$\mathbb{R} times \mathbb{R}$.
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**Solution**
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a. Find $A \times B$.
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$$ A \times B = \{(x, 1), (y, 1), (x, 2), (y, 2), (x, 3), (y, 3)\} $$
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b. Find $B \times A$.
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$$ B \times A = \{(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)\} $$
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c. Find $A \times A$.
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$$ A \times A = \{(x, x), (x, y), (y, x), (y, y)\} $$
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d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
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$A \times B$ has 6 elements. Note that this is the number of elements in $A$
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times the number of elements in $B$. $B \times A$ has 6 elements, the number of
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elements in $B$ times the number of elements in $A$. $A \times A$ has 4
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elements, the number of elements in $A$ times the number of elements in $A$.
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e. Find $(A \times B) \times C$
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$$ (A \times B) \times C = \{(u, v) | u \in A \times B \text{ and } v \in C\} $$
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By definition of Cartesian product.
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$$ (A \times B) \times C = \{((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a), ((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b), ((y, 2), b), ((y, 3), b)\} $$
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f. Find $A \times B \times C$
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The Cartesian product $A \times B \times C$ is superficially similar to but is
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not quite the same mathematical object as $(A \times B) \times C$.
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$(A \times B) \times C$ is a set of ordered pairs of which one element is itself
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an ordered pair, whereas $A \times B \times C$ is a set of ordered triples. By
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definition of Cartesian product,
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$$ A \times B \times C = \{(u, v, w) | u \in A, v \in B, \text{ and } w \in C\} $$
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$$ A \times B \times C = \{(x, 1, a), (x, 2, a), (x, 3, a), (y, 1, a), (y, 2, a), (y, 3, a), (x, 1, b), (x, 2, b), (x, 3, b), (y, 1, b), (y, 2, b), (y, 3, b)\} $$
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g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
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$\mathbb{R} times \mathbb{R}$.
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$\mathbb{R} \times \mathbb{R}$ is the set of all ordered pairs $(x, y)$ where
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both $x$ and $y$ are real numbers. If horizontal and vertical axes are drawn on
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a plane and a unit length is marked off, then each ordered pair in
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$\mathbb{R} \times \mathbb{R}$ corresponds to a unique point in the plane, with
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the first and second elements o the pair indicating, respectively, the
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horizontal and vertical positions of the point. The term **Cartesian plane** is
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often used to refer to a plane with this coordinate system, as illustrated in
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Figure 1.2.1 (see page 36).
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---
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**Example 1.2.8 Strings**
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Page 36
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Let $A = \{a, b\}$. List all the strings of length 3 over $A$ with at least two
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characters that are the same.
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**Solution**
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_aab, aba, baa, aaa, bba, bab, abb, bbb_
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In computer programming it is important to distinguish among different kinds of
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data structures and to respect the notations that are used for them. Similarly
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in mathematics, it is important to distinguish among, say, _{a, b, c}, {{ab},
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c}, (a, b, c), (a, (b, c)), abc_ and so forth, because these are all
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significantly different objects.
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---
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**Example 1.3.1 A Relation as a Subset**
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Page 39
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Let $A = \{1, 2\}$ and $B = \{1, 2, 3\}$ and define a relation $R$ from $A$ to
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$B$ as follows: Given any $(x, y) \in A \times B$.
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$(x, y) \in R$ means that $\dfrac{x - y}{2}$ is an integer.
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a. State explicitly which ordered pairs are in $A \times B$ and which are in
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$R$.
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b. Is 1 _R_ 3? Is 2 _R_ 3? Is 2 _R_ 2?
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c. What are the domain and co-domain of _R_?
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**Solution**
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a. State explicitly which ordered pairs are in $A \times B$ and which are in
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$R$.
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$$ A \times B = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)\} $$
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$$ (x, y) \in R = \{(A \times B) | \left(\frac{x - y}{2}\right) \in \mathbb{Z}\} $$
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$$ R = \{(1, 1), (1, 3), (2, 2)\} $$
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b. Is 1 _R_ 3? Is 2 _R_ 3? Is 2 _R_ 2?
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Is 1 _R_ 3?: Yes, because $(1, 3) \in R$.
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Is 2 _R_ 3? No, because $(2, 3) \notin R$.
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Is 2 _R_ 2? Yes, because $(2, 2) \in R$.
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c. What are the domain and co-domain of _R_?
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The domain of _R_ is $\{1, 2\}$ and the co-domain of _R_ is $\{1, 2, 3\}$
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---
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**Example 1.3.2 The Circle Relation**
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Page 40
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Define a relation $C$ from $\mathbb{R}$ to $\mathbb{R}$ as follows: For any
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$(x, y) \in \mathbb{R} \times \mathbb{R}$.
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$(x, y) \in C$ means that $x^2 + y^2 = 1$.
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a. Is $(1, 0) \in C$? Is $(0, 0) \in C$? Is
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$\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$? Is -2 _C_ 0? Is 0 _C_
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(-1)? Is 1 _C_ 1?
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b. What are the domain and co-domain of _C_?
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c. Draw a graph for _C_ by plotting the points of _C_ in the Cartesian plane.
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**Solution**
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a. Is $(1, 0) \in C$? Is $(0, 0) \in C$? Is
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$\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$? Is -2 _C_ 0? Is 0 _C_
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(-1)? Is 1 _C_ 1?
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Is $(1, 0) \in C$?
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Yes, $(1)^2 + (0)^2 = 1$
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Is $(0, 0) \in C$?
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No, $(0)^2 + (0)^2 = 0 \neq 1$
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Is $\left(-\dfrac{1}{2}, \dfrac{\sqrt{3}}{2}\right) \in C$?
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Yes,
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$\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1$
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Is -2 _C_ 0?
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No because $(-2)^2 + (0)^2 = 4 \neq 1$
|
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Is 0 _C_ (-1)?
|
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Yes because $(0)^2 + (-1)^2 = 1$.
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Is 1 _C_ 1?
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No, because $(1)^2 + (1)^2 = 2 \neq 1$.
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b. What are the domain and co-domain of _C_?
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The domain of _C_ is $\mathbb{R}$ and the co-domain of _C_ is also $\mathbb{R}$.
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c. Draw a graph for _C_ by plotting the points of _C_ in the Cartesian plane.
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This is just the circle formula, so:
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---
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**Example 1.3.3 Arrow Diagrams and Relations**
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Page 41
|
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|
Let $A = \{1, 2, 3\}$ and $B = \{1, 2, 3\}$ and define relations $S$ and $T$
|
|
from $A$ to $B$ as follows:
|
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For every $(x, y) \in A \times B$,
|
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$(x, y) \in S$ means that $x < y$ ($S$ is a "less than" relation).
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|
$T = \{(2, 1), (2, 5)\}$.
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Draw arrow diagrams for $S$ and $T$.
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|
**Solution**
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|
|
|
|
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These example relations illustrate that it is possible to have several arrows
|
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coming out of the same element of $A$ pointing in different directions. Also, it
|
|
is quite possible to have an element of $A$ that does not have an arrow coming
|
|
out of it.
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|
|
|
---
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|
**Example 1.3.4 Functions and Relations on Finite Sets**
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|
Page 42
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|
Let $A = \{2, 4, 6\}$ and $B = \{1, 3, 5\}$. Which of the relations $R$, $S$,
|
|
and $T$ defined below are functions from $A$ to $B$?
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a. $R = \{(2, 5), (4, 1), (4, 3), (6, 5)\}$
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b. For every $(x, y) \in A \times B$, $(x, y) \in S$ means that $y = x + 1$.
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c. $T$ is defined by the arrow diagram
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|
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|
**Solution**
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|
a. $R = \{(2, 5), (4, 1), (4, 3), (6, 5)\}$
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$R$ is not a function because it does not satisfy property (2). The ordered
|
|
pairs $(4, 1)$ and $(4, 3)$ have the same first element but different second
|
|
elements. You can see this graphically if you draw the arrow diagram for $R$.
|
|
There are two arrows coming out of 4: One point to 1 and the other points to 3.
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|
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|
b. For every $(x, y) \in A \times B$, $(x, y) \in S$ means that $y = x + 1$.
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|
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|
$S$ is not a function because it does not satisfy property (1). It is not true
|
|
that every element of $A$ is the first element of an ordered pair in $S$. For
|
|
example $6 \in A$ but there is no $y$ in $B$ such that $y = 6 + 1 = 7$. You can
|
|
also see this graphically by drawing the arrow diagram for $S$.
|
|
|
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|
c. $T$ is defined by the arrow diagram
|
|
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|
$T$ is a function: Each element in $\{2, 4, 6\}$ is related to some element in
|
|
$\{1, 3, 5\}$, and no element in $\{2, 4, 6\}$ is related to more than one
|
|
element in $\{1, 3, 5\}$. When these properties are stated in terms of the arrow
|
|
diagram, they become (1) there is an arrow coming out of each element of the
|
|
domain, and (2) no element of the domain has more than one arrow coming out of
|
|
it. So you can write $T(2) = 5$, $T(4) = 1$, $T(6) = 1$.
|
|
|
|
---
|
|
|
|
**Example 1.3.5 Functions and Relations on Sets of Strings**
|
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|
|
Page 43
|
|
|
|
Let $A = \{a, b\}$ and let $S$ be the set of all strings over $A$.
|
|
|
|
a. Define a relation $L$ from $S$ to $\mathbb{Z}^{\text{nonneg}}$ as follows:
|
|
For every string $s$ in $S$ and for every nonnegative integer $n$,
|
|
|
|
$$ (s, n) \in L \text{ means that the length of } s \text{ is } n $$
|
|
|
|
Observe that $L$ is a function because every string in $S$ has one and only one
|
|
length. Find _L(abaaba)_ and _L(bbb)_.
|
|
|
|
b. Define a relation $C$ from $S$ to $S$ as follows: For all strings $s$ and $t$
|
|
in $S$,
|
|
|
|
$$ (s, t) \in C \text{ means that } t = as $$
|
|
|
|
where $as$ is the string obtained by appending $a$ on the left of the characters
|
|
in $s$. ($C$ is called **concatenation** by $a$ on the left.) Observe that $C$
|
|
is a function because every string in $S$ consists entirely of $a$'s and $b$'s
|
|
and adding an additional $a$ on the left creates a new string that also consists
|
|
of $a$'s and $b$'s and thus is also in $S$. Find _C(abaaba)_ and _C(bbb)_.
|
|
|
|
**Solution**
|
|
|
|
a. Define a relation $L$ from $S$ to $\mathbb{Z}^{\text{nonneg}}$ as follows:
|
|
For every string $s$ in $S$ and for every nonnegative integer $n$,
|
|
|
|
$$ (s, n) \in L \text{ means that the length of } s \text{ is } n $$
|
|
|
|
Observe that $L$ is a function because every string in $S$ has one and only one
|
|
length. Find _L(abaaba)_ and _L(bbb)_.
|
|
|
|
_L(abaaba)_ = 6
|
|
|
|
_L(bbb)_ = 3
|
|
|
|
b. Define a relation $C$ from $S$ to $S$ as follows: For all strings $s$ and $t$
|
|
in $S$,
|
|
|
|
$$ (s, t) \in C \text{ means that } t = as $$
|
|
|
|
where $as$ is the string obtained by appending $a$ on the left of the characters
|
|
in $s$. ($C$ is called **concatenation** by $a$ on the left.) Observe that $C$
|
|
is a function because every string in $S$ consists entirely of $a$'s and $b$'s
|
|
and adding an additional $a$ on the left creates a new string that also consists
|
|
of $a$'s and $b$'s and thus is also in $S$. Find _C(abaaba)_ and _C(bbb)_.
|
|
|
|
_C(abaaba)_ = aabaaba
|
|
|
|
_C(bbb)_ = abbb
|
|
|
|
---
|
|
|
|
**Example 1.3.6 Functions Defined by Formulas**
|
|
|
|
Page 44
|
|
|
|
The **squaring function** $f$ from $\mathbb{R}$ to $\mathbb{R}$ is defined by
|
|
the formula $f(x) = x^2$ for every real number $x$. This means that no matter
|
|
what real number input is substituted for $x$, the output of $f$ will be the
|
|
square of that number. The idea can be represented by writing
|
|
$f(\Box) = \Box^2$. In other words, $f$ sends each real number $x$ to $x^2$, or
|
|
symbolically, $f: x \to x^2$. Note that the variable $x$ is a dummy variable;
|
|
any other symbol could replace it, as long as the replacement is made everywhere
|
|
the $x$ appears.
|
|
|
|
The **successor function** $g$ from $\mathbb{Z}$ to $\mathbb{Z}$ is defined by
|
|
the formula $g(n) = n + 1$. Thus, no matter what integer is substituted for $n$,
|
|
the output of $g$ will be that number plus $1$: $g(\Box) = \Box + 1$. In other
|
|
words, $g$ sends each integer $n$ to $n + 1$, or, symbolically,
|
|
$g: n \to n + 1$.
|
|
|
|
An example of a **constant function** is the function $h$ from $\mathbb{Q}$ to
|
|
$\mathbb{Z}$ defined by the formula $h(r) = 2$ for all rational numbers $r$.
|
|
This function sends each rational number $r$ to $2$. In other words, no matter
|
|
what the input, the output is always $2$: $h(\Box) = 2$ or $h: r \to 2$.
|
|
|
|
The functions $f$, $g$, and $h$, are represented by the function machines in
|
|
Figure 1.3.2 (see page 44).
|
|
|
|
A function is an entity in its own right. It can be thought of as a certain
|
|
relationship between sets or as an input/output machine that operates according
|
|
to a certain rule. This is the reason why a function is generally denoted by a
|
|
single symbol or string of symbols, such as $f$, $G$, or $\log$, or $\sin$.
|
|
|
|
A relation is a subset of a Cartesian product and a function is a special kind
|
|
of relation. Specifically, if $f$ and $g$ are functions from a set $A$ to a set
|
|
$B$, then
|
|
|
|
$$ f = \{(x, y) \in A \times B | y = f(x)\} \quad \text{ and } g(x) = \{(x, y) \in A \times B | y = g(x)\} $$
|
|
|
|
It follows that
|
|
|
|
$$ f \text{ equals } g, \quad \text{ written } f = g, \quad \text{ if, and only if, } f(x) = g(x) \text{ for all } x \text{ in } A $$
|
|
|
|
---
|
|
|
|
**Example 1.3.7 Equality of Functions**
|
|
|
|
Page 44
|
|
|
|
Define functions $f$ and $g$ from $\mathbb{R}$ to $\mathbb{R}$ by the following
|
|
formulas:
|
|
|
|
$$ f(x) = |x| \quad \text{ for every } x \in \mathbb{R} $$
|
|
|
|
$$ g(x) = \sqrt{x^2} \quad \text{ for every } x \in \mathbb{R} $$
|
|
|
|
Does $f = g$?
|
|
|
|
**Solution**
|
|
|
|
Yes. Because the absolute value of any real number equals the square root of its
|
|
square, $|x| = \sqrt{x^2}$ for all $x \in \mathbb{R}$. Hence $f = g$.
|
|
|
|
---
|
|
|
|
**Example 1.4.1 Terminology**
|
|
|
|
Consider the following graph:
|
|
|
|
|
|
|
|
a. Write the vertex set and edge set, and give a table showing the edge-endpoint
|
|
function.
|
|
|
|
b. Find all edges that are incident on $v_1$, all vertices that are adjacent to
|
|
$v_1$, all edges that are adjacent to $e_1$, all loops, all parallel edges, all
|
|
vertices that are adjacent to themselves, and all isolated vertices.
|
|
|
|
**Solution**
|
|
|
|
a.
|
|
|
|
$$ \text{vertex set } = \{v_1, v_2, v_3, v_4, v_5, v_6\} $$
|
|
|
|
$$ \text{edge set } = \{e_1, e_2, e_3, e_4, e_5, e_6, e_7\} $$
|
|
|
|
| Edge | Endpoints |
|
|
| ----- | -------------- |
|
|
| $e_1$ | $\{v_1, v_2\}$ |
|
|
| $e_2$ | $\{v_1, v_3\}$ |
|
|
| $e_3$ | $\{v_1, v_3\}$ |
|
|
| $e_4$ | $\{v_2, v_3\}$ |
|
|
| $e_5$ | $\{v_5, v_6\}$ |
|
|
| $e_6$ | $\{v_5\}$ |
|
|
| $e_7$ | $\{v_6\}$ |
|
|
|
|
b.
|
|
|
|
$e_1$, $e_2$, and $e_3$ are incident on $v_1$.
|
|
|
|
$v_1$ and $v_3$ are adjacent to $v_1$.
|
|
|
|
$e_2$, $e_3$, and $e_4$ are adjacent to $e_1$.
|
|
|
|
$e_6$ and $e_7$ are loops.
|
|
|
|
$e_2$ and $e_3$ are parallel.
|
|
|
|
$v_5$ and $v_6$ are adjacent to themselves.
|
|
|
|
$v_4$ is an isolated vertex.
|
|
|
|
---
|
|
|
|
**Example 1.4.2 Drawing More Than One Picture for a Graph**$adjacent
|
|
|
|
Page 49
|
|
|
|
Consider the graph specified as follows:
|
|
|
|
$$ \text{vertex set } = \{v_1, v_2, v_3, v_4\} $$
|
|
|
|
$$ \text{edge set } = \{e_1, e_2, e_3, e_4\} $$
|
|
|
|
edge-endpoint function:
|
|
|
|
| Edge | Endpoints |
|
|
| ----- | -------------- |
|
|
| $e_1$ | $\{v_1, v_3\}$ |
|
|
| $e_2$ | $\{v_2, v_4\}$ |
|
|
| $e_3$ | $\{v_2, v_4\}$ |
|
|
| $e_4$ | $\{v_3\}$ |
|
|
|
|
Both drawings (a) and (b) shown below are pictorial representations of this
|
|
graph (see Page 50).
|
|
|
|
---
|
|
|
|
**Example 1.4.3 Labeling Drawings to Show They Represent the Same Graph**
|
|
|
|
Page 50
|
|
|
|
Consider the two drawings shown in Figure 1.4.1. Label vertices and edges in
|
|
such a way that both drawings represent the same graph.
|
|
|
|
(see page 50)
|
|
|
|
**Solution**
|
|
|
|
Imagine putting one end of a piece of string at the top vertex of Figure
|
|
1.4.1(a) (call this vertex $v_1$), then laying the string to the next adjacent
|
|
vertex on the lower right (call this vertex $v_2$), then laying it to the next
|
|
adjacent vertex on the upper left ($v_3$), and so forth, returning finally to
|
|
the top vertex $v_1$. Call the first edge $e_1$, the second edge $e_2$, and so
|
|
forth, as shown below.
|
|
|
|
(see page 50)
|
|
|
|
Now imagine picking up the piece of string, together with its labels, and
|
|
repositioning it as follows:
|
|
|
|
(see page 50)
|
|
|
|
This is the same as Figure 1.4.1(b), so both drawings represent the graph with
|
|
vertex set $\{v_1, v_2, v_3, v_4, v_5\}$, edge set
|
|
$\{e_1, e_2, e_3, e_4, e_5\}$, and edge-endpoint function as follows:
|
|
|
|
| Edge | Endpoints |
|
|
| ----- | -------------- |
|
|
| $e_1$ | $\{v_1, v_2\}$ |
|
|
| $e_2$ | $\{v_2, v_3\}$ |
|
|
| $e_3$ | $\{v_3, v_4\}$ |
|
|
| $e_4$ | $\{v_4, v_5\}$ |
|
|
| $e_5$ | $\{v_5, v_1\}$ |
|
|
|
|
---
|
|
|
|
**Example 1.4.4 Using a Graph to Represent a Network**
|
|
|
|
Page 51
|
|
|
|
Telephone, electric power, gas pipeline, and air transport systems can all be
|
|
represented by graphs, as can computer networks - from small local area networks
|
|
to the global Internet system that connects millions of computers worldwide.
|
|
Questions that arise in the design of such systems involve choosing connecting
|
|
edges to minimize cost, optimize a certain type of service, and so forth. A
|
|
typical network, called a _hub-ad-spoke model_, is shown below.
|
|
|
|
(see page 51)
|