🚧 Progress through chapter 1

chapter_1/examples.md

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+**Example 1.1.1**
+
+Page 24
+
+Use variables to rewrite the following sentences more formally.
+
+a. Are there numbers with the property that the sum of their squares equals the
+square of their sum?
+
+b. Given any real number, its square is nonnegative.
+
+**Solution**
+
+a. Are there numbers $a$ and $b$ with the property that $a^2 + b^2 = (a + b)^2$?
+
+_Or_: Are there numbers $a$$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
+
+_Or_: Do there exist any numbers $a$ and $b$ such that $a^2 + b^2 = (a + b)^2$?
+
+b. Given any real number $r$, $r^2$ is nonnegative.
+
+_Or_: For any real number $r$, $r^2 \geq 0$.
+
+_Or_: For every real number $r$, $r^2 \geq 0$.
+
+---
+
+**Example 1.1.2**
+
+Page 26
+
+Fill in the blanks to rewrite the following statement:
+
+For every real number $x$, if $x$ is nonzero then $x^2$ is positive.
+
+a. If a real number is nonzero, then its square ________.
+
+b. For every nonzero real number $x$, ________.
+
+c. If $x$ ________, then ________.
+
+d. The square of any nonzero real number is ________.
+
+e. All nonzero real numbers have ________.
+
+**Solution**.
+
+a. is positive.
+
+b. $x^2$ is positive.
+
+c. is a nonzero real number, $x^2$ is positive.
+
+d. positive.
+
+e. positive squares .
+
+---
+
+**Example 1.1.3**
+
+Page 27
+
+Fill in the blanks to rewrite the following statement: Every pot has a lid.
+
+a. All pots ________.
+
+b. For every pot $P$, there is ________.
+
+c. For every pot $P$, there is a lid $L$ such that ________.
+
+**Solution**
+
+a. have lids.
+
+b. a lid.
+
+c. $L$ is a lid for $P$..
+
+---
+
+**Example 1.1.4**
+
+Page 28
+
+Fill in the blanks to rewrite the following statement in three different ways:
+
+There is a person in my class who is at least as old as every person in my clas.
+
+a. Some ________ is at least as old as ________.
+
+b. There is a person $p$ in my class such that $p$ is ________.
+
+c. There is a person $p$ in my class with the property that for every person $q$
+in my class, $p$ is ________.
+
+**Solution**
+
+a. person; every person.
+
+b. at least as old as every person in my class.
+
+c. at least as old as $q$.
+
+---
+
+**Example 1.2.1**
+
+Page 30
+
+**Using the Set-Roster Notation**
+
+a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
+What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
+
+b. Is $\{0\} = 0$?
+
+c. How many elements are in the set $\{1, \{1}\}$?
+
+d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
+and $U_0$.
+
+**Solution**
+
+a. Let $A = \{1, 2, 3\}$, $B = \{3, 1, 2\}$, and $C = \{1, 1, 2, 3, 3, 3\}$.
+What are the elements of $A$, $B$, and $C$? How are $A$, $B$, and $C$ related?
+
+$A$, $B$, and $C$ have exactly the same three elements, $1$, $2$, and $3$.
+Therefore, $A$, $B$, and $C$ are simply different ways to represent the same
+set.
+
+b. Is $\{0\} = 0$?
+
+$\{0\} \neq 0$ because $\{0\}$ is a set with one element, namely $0$, whereas
+$0$ is just the symbol that represents the number zero.
+
+c. How many elements are in the set $\{1, \{1}\}$?
+
+The set $\{1, \{1\}\}$ has two elements. $1$ and the set whose only element is
+$1$.
+
+d. For each nonnegative integer $n$, let $U_n = \{n, -n\}$. Find $U_1$, $U_2$,
+and $U_0$.
+
+$U_1 = \{1, -1\}, \quad U_2 = \{2, -2\}, \quad U_0 = \{0, 0\} = \{0\}$
+
+---
+
+**Example 1.2.2**
+
+Page 31
+
+**Using the Set-Builder Notation**
+
+Given that $\mathbb{R}$ denotes the set of all real numbers, $\mathbb{Z}$ the
+set of all integers, and $\mathbb{Z}^+$ the set of all positive integers,
+describe each of the following sets.
+
+a. $\{x \in \mathbb{R} | -2 < x < 5\}$
+
+b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
+
+c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
+
+**Solution**
+
+a. $\{x \in \mathbb{R} | -2 < x < 5\}$
+
+$\{x \in \mathbb{R} | -2 < x < 5\}$ is the open interval of real numbers
+(strictly) between $-2$ and 5. It is pictured as follows (see page 31).
+
+b. $\{x \in \mathbb{Z} | -2 < x < 5\}$
+
+$\{x \in \mathbb{Z} | -2 < x < 5\}$ is the set of all integers (strictly)
+between $-2$ and $5$. It is equal to the set $\{-1, 0, 1, 2, 3, 4}$.
+
+c. $\{x \in \mathbb{Z}^+ | -2 < x < 5\}$
+
+Since all the integers in $\mathbb{Z}^+$ are positive,
+$\{x \in \mathbb{Z}^+ | -2 < x < 5\} = \{1, 2, 3, 4\}$.
+
+---
+
+**Example 1.2.3**
+
+Page 32
+
+Let $A = \mathbb{Z}^+$, $B = \{n \in \mathbb{Z} | 0 \leq n \leq 100\}$, and
+$C = \{100, 200, 300, 400, 500\}$. Evaluate the truth and falsity of each of the
+following statements
+
+a. $B \subseteq A$
+
+b. $C$ is a proper subset of $A$.
+
+c. $C$ and $B$ have at least one element in common
+
+d. $C \subseteq B$
+
+e. $C \subseteq C$
+
+**Solution**
+
+a. $B \subseteq A$
+
+False. Zero is not a positive integer. Thus zero is in $B$ but zero is not in
+$A$, and so $B \nsubseteq A$
+
+b. $C$ is a proper subset of $A$.
+
+True. Each element in $C$ is a positive integer, and hence, is in $A$, but there
+are elements in $A$ that are not in $C$. For instance, $1$ is in $A$ and not in
+$C$.
+
+c. $C$ and $B$ have at least one element in common
+
+True. For example, $100$ is in both $C$ and $B$.
+
+d. $C \subseteq B$
+
+False. For example, $200$ is in $C$ but not in $B$.
+
+e. $C \subseteq C$
+
+True. Every element in $C$ is in $C$. In general, the definition of a subset
+implies that all sets are subsets of themselves.
+
+---
+
+**Example 1.2.4**
+
+Page 33
+
+**Distinction between $\in$ and $\subseteq$**
+
+Which of the following are true statements?
+
+a. $2 \in \{1, 2, 3\}$
+
+b. $\{2\} \in \{1, 2, 3\}$
+
+c. $2 \subseteq \{1, 2, 3\}$
+
+d. $\{2\} \subseteq \{1, 2, 3\}$
+
+e. $\{2\} \subseteq \{\{1\}, \{2\}\}$
+
+f. $\{2} \in \{\{1\}, \{2\}\}$
+
+**Solution**
+
+Only (a), (d), and (f) are true.
+
+For (b) to be true, the set $\{1, 2, 3\}$ would have to contain the element
+$\{2\}$. But the only elements of $\{1, 2, 3\}$ are $1$, $2$, and $3$, and $2$
+is not equal to $\{2\}$. Hence (b) is false.
+
+For \(c\) to be true, the number $2$ would have to be a set and every element in
+the set $2$ would have to be an element of $\{1, 2, 3}$. This is not the case,
+so \(c\) is false.
+
+For (e) to be true, every element in the set containing only the number $2$
+would have to be an element of the set whose elements are $\{1\}$ and $\{2\}$.
+But $2$ is not equal to either $\{1\}$ or $\{2\}$, and so (e) is false.
+
+---
+
+**Example 1.2.5 Ordered Pairs**
+
+Page 34
+
+a. Is $(1, 2) = (2, 1)$?
+
+b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
+
+c. What is the first element of $(1, 1)$?
+
+**Solution**
+
+a. Is $(1, 2) = (2, 1)$?
+
+No, By definition of equality of ordered pairs,
+
+$(1, 2) = (2, 1)$ if, and only if, 1 = 2, and 2 = 1.
+
+But $1 \neq 2$, and so the ordered pairs are not equal.
+
+b. Is $\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$?
+
+Yes. By definition of equality of ordered pairs,
+
+$\left(3, \dfrac{5}{10}\right) = \left(\sqrt{9}, \dfrac{1}{2}\right)$ if, and
+only if, $3 = \sqrt{9}$ and $\dfrac{5}{10} = \dfrac{1}{2}$.
+
+Because these equations are both true, the ordered pairs are equal.
+
+c. What is the first element of $(1, 1)$?
+
+In the ordered pair $(1, 1)$, the first and second elements are both $1$.
+
+---
+
+**Example 1.2.6 Ordered $n$-tuples**
+
+Page 34
+
+a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
+
+b. Is
+$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
+
+**Solution**
+
+a. Is $(1, 2, 3, 4) = (1, 2, 4, 3)$?
+
+No. By definition of equality of ordered 4-tuples,
+
+$$ (1, 2, 3, 4) = (1, 2, 4, 3) \leftrightarrow 1 = 1, 2 = 2, 3 = 4, and 4 = 3 $$
+
+But $3 \neq 4$, and so the ordered 4-tuples are not equal.
+
+b. Is
+$\left(3, (-2)^2, \dfrac{1}{2}\right) = \left(\sqrt{9}, 4, \dfrac{3}{6}\right)$?
+
+Yes. By definition of equality of ordered triples.
+
+$$ \left(3, (-2)^2, \frac{1}{2}\right) = \left(\sqrt{9}, 4, \frac{3}{6}\right) \leftrightarrow 3 = \sqrt{9} \text{ and } (-2)^2 = 4 \text{ and } \frac{1}{2} = \frac{3}{6} $$
+
+Because these equations are all true, the two ordered triples are equal.
+
+---
+
+**Example 1.2.7 Cartesian Products**
+
+Page 35
+
+Let $A = \{x, y\}$, $B = \{1, 2, 3\}$, and $C = \{a, b\}$.
+
+a. Find $A \times B$.
+
+b. Find $B \times A$.
+
+c. Find $A \times A$.
+
+d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
+
+e. Find $(A \times B) \times C$
+
+f. Find $A \times B \times C$
+
+g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
+$\mathbb{R} times \mathbb{R}$.
+
+**Solution**
+
+a. Find $A \times B$.
+
+$$ A \times B = \{(x, 1), (y, 1), (x, 2), (y, 2), (x, 3), (y, 3)\} $$
+
+b. Find $B \times A$.
+
+$$ B \times A = \{(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)\} $$
+
+c. Find $A \times A$.
+
+$$ A \times A = \{(x, x), (x, y), (y, x), (y, y)\} $$
+
+d. How many elements are in $A \times B$, $B \times A$, and $A \times A$?
+
+$A \times B$ has 6 elements. Note that this is the number of elements in $A$
+times the number of elements in $B$. $B \times A$ has 6 elements, the number of
+elements in $B$ times the number of elements in $A$. $A \times A$ has 4
+elements, the number of elements in $A$ times the number of elements in $A$.
+
+e. Find $(A \times B) \times C$
+
+$$ (A \times B) \times C = \{(u, v) | u \in A \times B \text{ and } v \in C\} $$
+
+By definition of Cartesian product.
+
+$$ (A \times B) \times C = \{((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a), ((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b), ((y, 2), b), ((y, 3), b)\} $$
+
+f. Find $A \times B \times C$
+
+The Cartesian product $A \times B \times C$ is superficially similar to but is
+not quite the same mathematical object as $(A \times B) \times C$.
+$(A \times B) \times C$ is a set of ordered pairs of which one element is itself
+an ordered pair, whereas $A \times B \times C$ is a set of ordered triples. By
+definition of Cartesian product,
+
+$$ A \times B \times C = \{(u, v, w) | u \in A, v \in B, \text{ and } w \in C\} $$
+
+$$ A \times B \times C = \{(x, 1, a), (x, 2, a), (x, 3, a), (y, 1, a), (y, 2, a), (y, 3, a), (x, 1, b), (x, 2, b), (x, 3, b), (y, 1, b), (y, 2, b), (y, 3, b)\} $$
+
+g. Let $\mathbb{R}$ denote the set of all real numbers. Describe
+$\mathbb{R} times \mathbb{R}$.
+
+$\mathbb{R} \times \mathbb{R}$ is the set of all ordered pairs $(x, y)$ where
+both $x$ and $y$ are real numbers. If horizontal and vertical axes are drawn on
+a plane and a unit length is marked off, then each ordered pair in
+$\mathbb{R} \times \mathbb{R}$ corresponds to a unique point in the plane, with
+the first and second elements o the pair indicating, respectively, the
+horizontal and vertical positions of the point. The term **Cartesian plane** is
+often used to refer to a plane with this coordinate system, as illustrated in
+Figure 1.2.1 (see page 36).
+
+---
+
+**Example 1.2.8 Strings**
+
+Page 36
+
+Let $A = \{a, b\}$. List all the strings of length 3 over $A$ with at least two
+characters that are the same.
+
+**Solution**
+
+_aab, aba, baa, aaa, bba, bab, abb, bbb_
+
+In computer programming it is important to distinguish among different kinds of
+data structures and to respect the notations that are used for them. Similarly
+in mathematics, it is important to distinguish among, say, _{a, b, c}, {{ab},
+c}, (a, b, c), (a, (b, c)), abc_ and so forth, because these are all
+significantly different objects.