diff --git a/appendix_b.txt b/appendix_b.txt
new file mode 100644
index 0000000..a2fa28f
--- /dev/null
+++ b/appendix_b.txt
@@ -0,0 +1 @@
+898
diff --git a/chapter_1/examples.md b/chapter_1/examples.md
new file mode 100644
index 0000000..0ef0a52
--- /dev/null
+++ b/chapter_1/examples.md
@@ -0,0 +1,424 @@
+**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.
diff --git a/chapter_1/exercises.md b/chapter_1/exercises.md
new file mode 100644
index 0000000..c8dac7c
--- /dev/null
+++ b/chapter_1/exercises.md
@@ -0,0 +1,755 @@
+**Exercise Set 1.1**
+
+Page 28
+
+In each of 1-6, fill in the blanks using a variable or variables to rewrite the
+given statement.
+
+1. Is there a real number whose square is $-1$?
+
+a. Is there a real number $x$ such that ______?
+
+b. Does there exist ______ such that $x^2 = -1$?
+
+**Solution**
+
+a. Is there a real number $x$ such that $x^2 = -1$?
+
+b. Does there exist a real number $x$ such that $x^2 = -1$?
+
+2. Is there an integer that has a remainder of $2$ when it is divided by $5$ and
+   a remainder of $3$ when it is divided by $6$?
+
+a. Is there an integer $n$ such that $n$ has ______?
+
+b. Does there exist ______ such that if $n$ is divided by $5$ the remainder is
+$2$ and if ______?
+
+_Note: There are integers with this property. Can you think of one?_
+
+**Solution**
+
+a. Is there an integer $n$ such that $n$ has a remainder of $2$ when $n$ is
+divided by $5$ and a remainder of $3$ when $n$ is divided by $6$?
+
+b. Does there exist a number $n$ such that if $n$ is divided by $5$ the
+remainder is $2$ and if $n$ is divided by $6$ the remainder is $3$?
+
+_Note: There are integers with this property. Can you think of one?_
+
+$$ 27 \mod 5 = 2 $$
+
+$$ 27 \mod 6 = 3 $$
+
+3. Given any two distinct real numbers, there is a real number in between them.
+
+a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
+such that $c$ is ______.
+
+b. For any two ______, ______ such that $c$ is between $a$ and $b$.
+
+**Solution**
+
+a. Given any two distinct real numbers $a$ and $b$, there is a real number $c$
+such that $c$ is $a \leq c \leq b$.
+
+b. For any two distinct real numbers $a$ and $b$, there exists a real number $c$
+such that $c$ is between $a$ and $b$.
+
+4. Given any real number, there is a real number that is greater.
+
+a. Given any real number $r$, there is ______ $s$ such that $s$ is ______
+
+b. For any ______, ______ such that $s > r$.
+
+**Solution**
+
+a. Given any real number $r$, there is a real number $s$ such that $s$ is
+greater than $r$.
+
+b. For any real number $r$, there exists a real number $s$ such that $s > r$.
+
+5. The reciprocal of any positive real number is positive.
+
+a. Given any positive real number $r$, the reciprocal of ______.
+
+b. For any real number $$, if $r$ is ______, then ______.
+
+c. If a real number $r$ ______, then ______.
+
+**Solution**
+
+a. Given any positive real number $r$, the reciprocal of $r$ is positive.
+
+b. For any real number $r$, if $r$ is positive, then the reciprocal of $r$ is
+positive.
+
+c. If a real number $r$ is positive, then the reciprocal of $r$ is positive.
+
+6. The cube root of any negative real number is negative.
+
+a. Given any negative real number $s$, the cube root of ______.
+
+b. For any real number $s$, if $s$ is ______, then ______.
+
+c. If a real number $s$ ______, then ______.
+
+**Solution**
+
+a. Given any negative real number $s$, the cube root of $s$ is negative.
+
+b. For any real number $s$, if $s$ is negative, then the cube root of $s$ is
+negative.
+
+c. If a real number $s$ is negative, then the cube root of $s$ is negative.
+
+7. Rewrite the following statements less formally, without using variables.
+   Determine, as best as you can, whether the statements are true or false.
+
+a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
+
+b. There is a real number $x$ such that $x^2 < x$.
+
+c. For every positive integer $n$, $n^2 \geq n$.
+
+d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
+
+**Solution**
+
+a. There are real numbers $u$ and $v$ with the property that $u + v < u - v$.
+
+There are two distinct real numbers where the sum of those two numbers is less
+than the difference of those two numbers.
+
+This is true if you consider our domain is all real numbers which include
+negatives. For example:
+
+$$ 1 + (-1) = 0 $$
+
+$$ 1 - (-1) = 2 $$
+
+$$ 0 < 2 $$
+
+b. There is a real number $x$ such that $x^2 < x$.
+
+There is a real number which is greater than it's square.
+
+This is true for any fraction/decimal. Consider:
+
+$$ \left(\frac{1}{4}\right)^2 = \frac{1}{16} $$
+
+$$ \frac{1}{16} < \frac{1}{4} $$
+
+c. For every positive integer $n$, $n^2 \geq n$.
+
+For all positive integers, an integer's square is always greater than or equal
+to the integer.
+
+This is true. Starting at $1$ we get $1^2 \geq 1$, which is true, $2^2 \geq 2$
+is true, and so on. We're essentially multiplying each side of the inequality by
+some positive integer, which we know from algebra does not change the direction
+of the inequality, so this statement holds true.
+
+d. For all real numbers $a$ and $b$, $|a + b| \leq |a| + |b|$.
+
+For any two distinct real numbers, the absolute value of their sum is less than
+or equal to the sum of the absolute values of each number.
+
+This is true, if both $a$ and $b$ are positive numbers or both $a$ and $b$ are
+negative integers, then the two statements are equal. If either $a$ or $b$ is
+negative and the other is positive, then the left statement will always be less
+than the right hand statement.
+
+---
+
+In each of 8-13, fill in the blanks to rewrite the given statement.
+
+8. For every object $J$, if $J$ is a square then $J$ has four sides.
+
+a. All squares ______.
+
+b. Every square ______.
+
+c. If an object is a square, then it ______.
+
+d. If $J$ ______, then $J$ ______.
+
+e. For every square $J$, ______.
+
+**Solution**
+
+a. All squares have four sides.
+
+b. Every square has four sides.
+
+c. If an object is a square, then it has four sides.
+
+d. If $J$ is a square, then $J$ has four sides.
+
+e. For every square $J$, $J$ has four sides.
+
+9. For every equation $E$, if $E$ is quadratic then $E$ has at most two real
+   solutions.
+
+a. All quadratic equations ______.
+
+b. Every quadratic equation ______.
+
+c. If an equation is quadratic, then it ______.
+
+d. If $E$ ______, then $E$ ______.
+
+e. For every quadratic equation $E$, ______.
+
+**Solution**
+
+a. All quadratic equations have at most two real solutions.
+
+b. Every quadratic equation has at most two real solutions.
+
+c. If an equation is quadratic, then it has at most two real solutions.
+
+d. If $E$ is a quadratic equation, then $E$ has at most two real solutions.
+
+e. For every quadratic equation $E$, $E$ has at most two real solutions.
+
+10. Every nonzero real number has a reciprocal.
+
+a. All nonzero real numbers ______.
+
+b. For every nonzero real number $r$, there is ______ for $r$.
+
+c. For every nonzero real number $r$, there is a real number $s$ such that
+______.
+
+**Solution**
+
+a. All nonzero real numbers have reciprocals.
+
+b. For every nonzero real number $r$, there is a reciprocal for $r$.
+
+c. For every nonzero real number $r$, there is a real number $s$ such that $s$
+is a reciprocal of $r$.
+
+11. Every positive number has a positive square root.
+
+a. All positive numbers ______.
+
+b. For every positive number $e$, there is ______ for $e$.
+
+c. For every positive number $e$, there is a positive number $r$ such that
+______.
+
+**Solution**
+
+a. All positive numbers have positive square roots.
+
+b. For every positive number $e$, there is a positive square root for $e$.
+
+c. For every positive number $e$, there is a positive number $r$ such that $r$
+is a positive square root for $e$.
+
+12. There is a real number whose product with every number leaves the number
+    unchanged.
+
+a. Some ______ has the property that its ______.
+
+b. There is a real number $r$ such that the product of $r$ ______.
+
+c. There is a real number $r$ with the property that for every real number $s$,
+______.
+
+**Solution**
+
+a. Some real number has the property that its product with every number leaves
+the number unchanged.
+
+b. There is a real number $r$ such that the product of $r$ with every number
+leaves $r$ unchanged.
+
+c. There is a real number $r$ with the property that for every real number $s$,
+such that $rs = s$.
+
+13. There is a real number whose product with every real number equals zero.
+
+a. Some _____ has the property that its ______.
+
+b. There is a real number $a$ such that the product of $a$ ______.
+
+c. There is a real number $a$ with the property that for every real number $b$,
+______.
+
+**Solution**
+
+a. Some real number has the property that its product with every real number
+equals zero.
+
+b. There is a real number $a$ such that the product of $a$ with every real
+number equals zero.
+
+c. There is a real number $a$ with the property that for every real number $b$,
+$ab = 0$.
+
+---
+
+**Exercise Set 1.2**
+
+Page 37
+
+1. Which of the following sets are equal?
+
+$$ A = \{a, b, c, d\} $$
+
+$$ B = \{d, e, a, c\} $$
+
+$$ C = \{d, b, a, c\} $$
+
+$$ D = \{a, a, d, e, c, e\} $$
+
+**Solution**
+
+$$ A = C $$
+
+$$ B = D $$
+
+2. Write in words how to read each of the following out loud.
+
+a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
+
+b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
+
+c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
+
+d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
+
+**Solution**
+
+a. $\{x \in \mathbb{R}^+ | 0 < x < 1\}$
+
+The set of all positive real numbers $x$ such that $x$ is greater than $0$ and
+less than $1$.
+
+b. $\{x \in \mathbb{R} | x \leq 0 \text{ or } x \geq 1\}$
+
+The set of all real numbers $x$ such that $x$ is less than or equal to $0$ or
+$x$ is greater than or equal to $1$.
+
+c. $\{n \in \mathbb{Z} | n \text{ is a factor of } 6\}$
+
+The set of all integers $n$ such that $n$ is a factor of $6$.
+
+d. $\{n \in \mathbb{Z}^+ | n \text{ is a factor of } 6\}$
+
+The set of all positive integers $n$ such that $n$ is a factor of $6$.
+
+3.
+
+a. Is $4 = \{4\}$?
+
+b. How many elements are in the set $\{3, 4, 3, 5\}$?
+
+c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
+
+**Solution**
+
+a. Is $4 = \{4\}$?
+
+No, the symbol $4$, which represents the number four, does not equal the set
+that contains an element that is the number $4$.
+
+b. How many elements are in the set $\{3, 4, 3, 5\}$?
+
+There are 3 elements in the set $\{3, 4, 3, 5\}$. Repeated elements are not
+counted as more than 1 element in a set.
+
+c. How many elements are in the set $\{1, \{1\}, \{1, \{1\}\}\}$ ?
+
+There are three elements in the set, namely the symbol $1$, the set $\{1\}$, and
+the set $\{1, \{1\}\}$.
+
+4.
+
+a. Is $2 \in \{2}$ ?
+
+b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
+
+c. How many elements are in the set $\{0, \{0\}\}$ ?
+
+d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
+
+e. Is $0 \in \{\{0\}, \{1\}\}$ ?
+
+**Solution**
+
+a. Is $2 \in \{2}$ ?
+
+No, the symbol $2$ which represents the number two, is not equal to the set
+$\{2\}$, which is a set that contains the element $2$.
+
+b. How many elements are in the set $\{2, 2, 2, 2\}$ ?
+
+There is one element in the set $\{2, 2, 2, 2\}$, namely the element $2$.
+
+c. How many elements are in the set $\{0, \{0\}\}$ ?
+
+There are two elements in the set, namely the symbol $0$, and the set $\{0\}$.
+
+d. Is $\{0\} \in \{\{0\}, \{1\}\}$ ?
+
+Yes, the set of $\{0\}$ is in the set $\{\{0\}, \{1\}\}$, as the set contains
+both the sets $\{0\}$ and $\{1\}$.
+
+e. Is $0 \in \{\{0\}, \{1\}\}$ ?
+
+No, the symbol $0$, representing the number zero, is not in the set, which holds
+two sets with the symbols in them.
+
+5. Which of the following sets are equal?
+
+$$
+A = \{0, 1, 2\} \\
+B = \{x \in \mathbb{R} | -1 \leq x < 3\} \\
+C = \{x \in \mathbb{R} | -1 < x < 3\} \\
+D = \{x \in \mathbb{Z} | -1 < x < 3\} \\
+E = \{x \in \mathbb{Z}^+ | -1 < x < 3\}
+$$
+
+**Solution**
+
+None of these sets are equal. $A = E$ might have worked had $A$ not included
+$0$, but $E$ essentially evaluates to $E = \{1, 2\}$, and does not include $0$.
+
+6. For each integer $n$, let $T_n = \{n, n^2\}$. How many elements are in each
+   of $T_2, T_{-3}, T_1$, and $T_0$? Justify your answers.
+
+**Solution**
+
+$$ T_2 = \{2, 2^2\} = \{2, 4\} \quad \text{ two elements } $$
+
+$$ T_{-3} = \{-3, (-3)^2\} = \{-3, 9\} \quad \text{ two elements }$$
+
+$$ T_1  = \{1, 1^2\} = \{1, 1\} = \{1\} \quad \text{ one element } $$
+
+$$ T_0 = \{0, 0^2\} = \{0, 0\} = \{0\} \quad \text{ one element } $$
+
+7. Use the set-roster notation to indicate the elements in each of the following
+   sets.
+
+a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
+
+b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
+
+c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
+
+d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
+
+e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
+
+f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
+
+**Solution**
+
+a. $S = \{n \in \mathbb{Z} | n = (-1)^k \text{, for some integer } k\}$
+
+$$ \{-1, 1\} $$
+
+b. $T = \{m \in \mathbb{Z}| m = 1 + (-1)^i \text{, for some integer } i\}$
+
+$$ \{0, 2\} $$
+
+c. $U = \{r \in \mathbb{Z} | 2 \leq r \leq -2\}$
+
+$$ \emptyset $$
+
+d. $V = \{s \in \mathbb{Z} | s > 2 \text{ or }x < 3\}$
+
+$$ \mathbb{Z} $$
+
+e. $W = \{t \in \mathbb{Z} | 1 < t < -3\}$
+
+$$ \emptyset $$
+
+f. $X = \{u \in \mathbb{Z} | u \leq 4 \text{ or } u \geq 1\}$
+
+$$ \mathbb{Z}^+ $$
+
+8. Let $A = \{c, d, f, g\}$, $B = \{f, j\}$, and $C = \{d, g\}$. Answer each of
+   the following questions. Give reasons for your answers.
+
+a. Is $B \subseteq A$?
+
+b. Is $C \subseteq A$?
+
+c. Is $C \subseteq C$?
+
+d. Is $C$ a proper subset of $A$?
+
+**Solution**
+
+a. Is $B \subseteq A$?
+
+No, because every element of $B$ must be an element of $A$ by definition of a
+subset, but $j \in B$, but $j \notin A$.
+
+b. Is $C \subseteq A$?
+
+Yes, every element of $C$ is an element of $A$.
+
+c. Is $C \subseteq C$?
+
+Yes, every element of $C$ is an element of $C$. By implication, every set is a
+subset of itself.
+
+d. Is $C$ a proper subset of $A$?
+
+Yes, $C \subset A$, but $C \neq A$. Every element of $C$ is an element of $A$,
+but $C$ does not equal $A$, which is the definition of a proper subset.
+
+9.
+
+a. Is $3 \in \{1, 2, 3\}$?
+
+b. Is $1 \subseteq \{1}$?
+
+c. Is $\{2\} \in \{1, 2\}$?
+
+d. Is $\{3\} \in \{1, \{2\}, \{3\}\}$?
+
+e. Is $1 \in \{1\}$?
+
+f. Is $\{2\} \subseteq \{1, \{2\}, \{3\}\}$?
+
+g. Is $\{1\} \subseteq \{1, 2\}$?
+
+h. Is $1 \in \{\{1\}, 2\}$?
+
+i. Is $\{1\} \subseteq \{1, \{2\}\}$?
+
+j. Is $\{1\} \subseteq \{1\}$?
+
+**Solution**
+
+a. Is $3 \in \{1, 2, 3\}$?
+
+Yes, the symbol $3$, representing the number three, is in the set $\{1, 2, 3\}$.
+
+b. Is $1 \subseteq \{1}$?
+
+No, the number $1$ is not a set, and therefore cannot be a subset of $\{1\}$.
+
+c. Is $\{2\} \in \{1, 2\}$?
+
+No, the subset $\{2\}$ is not in the set $\{1, 2\}$, the number $2$ is in the
+subset, but not the set $\{2\}$.
+
+d. Is $\{3\} \in \{1, \{2\}, \{3\}\}$?
+
+Yes, the set $\{3\}$ is an element of $\{1, \{2\}, \{3\}\}$.
+
+e. Is $1 \in \{1\}$?
+
+Yes, the number $1$ is in the set $\{1\}$.
+
+f. Is $\{2\} \subseteq \{1, \{2\}, \{3\}\}$?
+
+No, the set $\{2\}$ holds the element $2$, and $2$ is not an element in
+$\{1, \{2\}, \{3\}\}$.
+
+g. Is $\{1\} \subseteq \{1, 2\}$?
+
+Yes, the set $\{\1}$ holds the element $1$, and $1$ is an element of $\{1, 2\}$.
+
+h. Is $1 \in \{\{1\}, 2\}$?
+
+No, the element $1$ is not in $\{\{1\}, 2\}$.
+
+i. Is $\{1\} \subseteq \{1, \{2\}\}$?
+
+Yes, the set $\{1\}$ holds the element $1$, which is an element of
+$\{1, \{2\}\}$.
+
+j. Is $\{1\} \subseteq \{1\}$?
+
+Yes $\{1\}$ holds the element $1$, which is an element of $\{1\}$. They are
+equal and it is implied that any set is a subset of itself.
+
+10.
+
+a. Is $((-2)^2, -2^2) = (-2^2, (-2)^2)$?
+
+b. Is $(5, -5) = (-5, 5)$?
+
+c. Is $(8 - 9, \sqrt[3]{-1}) = (-1, -1)$?
+
+d. Is $\left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)$
+
+**Solution**
+
+a. Is $((-2)^2, -2^2) = (-2^2, (-2)^2)$?
+
+$$ ((-2)^2, -2^2) = (4, -4) \neq (-4, 4) = (-2^2, (-2)^2) $$
+
+So no, they are not equal. For ordered pair tuples to be equal, the order
+matters and so each entry into the tuple must match the other for them to be
+equal.
+
+b. Is $(5, -5) = (-5, 5)$?
+
+No, ordered pair tuples require that the entries be equal to each other _in
+order_.
+
+c. Is $(8 - 9, \sqrt[3]{-1}) = (-1, -1)$?
+
+$$ (8 - 9, \sqrt[3]{-1}) = (-1, -1) = (-1, -1) $$
+
+So yes, these two ordered pair tuples are equal.
+
+d. Is $\left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\dfrac{3}{6}, -8\right)$
+
+$$ \left(\dfrac{-2}{-4}, (-2)^3\right) = \left(\frac{1}{2}, -8\right) = \left(\frac{1}{2}, -8\right) = \left(\dfrac{3}{6}, -8\right) $$
+
+So yes, these two ordered pair tuples are equal.
+
+11. Let $A = \{w, x, y, z\}$ and $B = \{a, b\}$. Use set-roster notation to
+    write each of the following sets, and indicate the number of elements that
+    are in each set.
+
+a. $A \times B$
+
+b. $B \times A$
+
+c. $A \times A$
+
+d. $B \times B$
+
+**Solution**
+
+a. $A \times B$
+
+$$ A \times B = \{(w, a), (w, b), (x, a), (x, b), (y, a), (y, b), (z, a), (z, b)\} $$
+
+There are 8 elements in $A \times B$.
+
+b. $B \times A$
+
+$$ B \times A = \{(a, w), (b, w), (a, x), (b, x), (a, y), (b, y), (a, z), (b, z)\} $$
+
+There are 8 elements in $B \times A$.
+
+c. $A \times A$
+
+$$ A \times A = \{(w, w), (w, x), (w, y), (w, z), (x, w), (x, x), (x, y), (x, z), (y, w), (y, x), (y, y), (y, z), (z, w), (z, x), (z, y), (z, z)\} $$
+
+There are 16 elements in $A \times A$.
+
+d. $B \times B$
+
+$$ B \times B = \{(a, a), (a, b), (b, a), (b, b)\} $$
+
+There are 4 elements in $B \times B$.
+
+12. Let $S = \{2, 4, 6\}$ and $T = \{1, 3, 5\}$. Use the set-roster notation to
+    write each of the following sets, and indicate the number of elements that
+    are in each set.
+
+a. $S \times T$
+
+b. $T \times S$
+
+c. $S \times S$
+
+d. $T \times T$
+
+**Solution**
+
+a. $S \times T$
+
+$$ S \times T = \{(2, 1), (2, 3), (2, 5), (4, 1), (4, 3), (4, 5), (6, 1), (6, 3), (6, 5)\} $$
+
+There are 9 elements in $S \times T$.
+
+b. $T \times S$
+
+$$ T \times S = \{(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)\} $$
+
+There are 9 elements in $T \times S$.
+
+c. $S \times S$
+
+$$ S \times S = \{(2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6)\} $$
+
+There are 9 elements in $S \times S$.
+
+d. $T \times T$
+
+$$ T \times T = \{(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)\} $$
+
+There are 9 elements in $T \times T$.
+
+13. Let $A = \{1, 2, 3\}$, $B = \{u\}$, and $C = \{m, n\}$. Find each of the
+    following sets.
+
+a. $A \times (B \times C)$
+
+b. $(A \times B) \times C$
+
+c. $A \times B \times C$
+
+**Solution**
+
+a. $A \times (B \times C)$
+
+$$ B \times C = \{(u, m), (u, n)\} $$
+
+$$ A \times (B \times C) = \{(1, (u, m)), (1, (u, n)), (2, (u, m)), (2, (u, n)), (3, (u, m)), (3, (u, n))\} $$
+
+b. $(A \times B) \times C$
+
+$$ A \times B = \{(1, u), (2, u), (3, u)\} $$
+
+$$ (A \times B) \times C = \{((1, u), m), ((1, u), n), ((2, u), m), ((2, u), n), ((3, u), m), ((3, u), n)\} $$
+
+c. $A \times B \times C$
+
+$$ A \times B \times C = \{(1, u, m), (1, u, n), (2, u, m), (2, u, n), (3, u, m), (3, u, n)\} $$
+
+14. Let $R = \{a\}$, $S = \{x, y\}$, and $T = \{p, q, r\}$. Find each of the
+    following sets.
+
+a. $R \times (S \times T)$
+
+$$ S \times T = \{(x, p), (x, q), (x, r), (y, p), (y, q), (y, r)\} $$
+
+$$ R \times (S \times T) = \{(a, (x, p)), (a, (x, q)), (a, (x, r)), (a, (y, p)), (a, (y, q)), (a, (y, r))\} $$
+
+b. $(R \times S) \times T$
+
+$$ R \times S = \{(a, x), (a, y)\} $$
+
+$$ (R \times S) \times T = \{((a, x), p), ((a, x), q), ((a, x), r), ((a, y), p), ((a, y), q), ((a, y), r)\} $$
+
+c. $R \times S \times T$
+
+**Solution**
+
+$$ R \times S \times T = \{(a, x, p), (a, x, q), (a, x, r), (a, y, p), (a, y, q), (a, y, r)\} $$
+
+a. $R \times (S \times T)$
+
+b. $(R \times S) \times T$
+
+c. $R \times S \times T$
+
+15. Let $S = \{0, 1\}$. List all the strings of length 4 over $S$ that contain
+    three or more $0$'s.
+
+**Solution**
+
+0000, 0001, 0010, 0100, 1000
+
+16. Let $T = \{x, y\}$. List all the strings of length 5 over $T$ that have
+    exactly one $y$.
+
+**Solution**
+
+xxxxy, xxxyx, xxyxx, xyxxx, yxxxx
diff --git a/chapter_1/notes.md b/chapter_1/notes.md
new file mode 100644
index 0000000..807fb9b
--- /dev/null
+++ b/chapter_1/notes.md
@@ -0,0 +1,152 @@
+Page 25
+
+A **universal statement** says that a certain property is true for all elements
+in a set. (For example: _All positive numbers are greater than zero_.)
+
+A **conditional statement** says that if one thing is true then some other thing
+also has to be true. (For example: _If 378 is divisible by 18, then 378 is
+divisible by 6_.)
+
+Given a property that may or may not be true, an **existential statement** says
+that there is at least one thing for which the property is true. (For example:
+_There is a prime number that is even_.)
+
+---
+
+Page 30
+
+**Set Roster Notation**
+
+If $S$ is a set, the notation $x \in S$ means that $x$ is an element of $S$. The
+notation $x \notin S$ means that $x$ is not an element of $S$. A set may be
+specified using the **set-roster notation** by writing all of its elements
+between braces. For example, $\{1, 2, 3\}$ denotes the set whose elements are
+$1$, $2$, and $3$. A variation of the notation is sometimes used to describe a
+very large set, as when we write $\{1, 2, 3, \dots, 100\}$ to refer to the set
+of all integers from $1$ to $100$. A similar notation can also describe an
+infinite set, as when we write $\{1, 2, 3, \dots\}$ to refer to the set of all
+positive integers. (The symbol $\dots$ is called an **ellipsis** and is read
+"and so forth.")
+
+The **axiom of extension** says that a set is completely determined by what its
+elements are - not the order in which they might be listed or the fact that some
+elements might be listed more than once.
+
+---
+
+Page 30
+
+| Symbol     | Set                                                       |
+| ---------- | --------------------------------------------------------- |
+| \mathbb{R} | the set of all real numbers                               |
+| \mathbb{Z} | the set of all integers                                   |
+| \mathbb{Q} | the set of all rational numbers, or quotients of integers |
+
+---
+
+Page 31
+
+**Set-Builder Notation**
+
+Let $S$ denote a set and let $P(x)$ be a property that elements of $S$ may or
+may not satisfy. We may define a new set to be **the set of all elements $x$ in
+$S$ such that $P(x)$ is true**. We denote this set as follows:
+
+$$ \{x \in S | P(x)\} $$
+
+Where $x$ is "the set of all" and $|$ is "such that."
+
+---
+
+Page 32
+
+**Definition**
+
+If $A$ and $B$ are sets, then $A$ is called a **subset** of $B$, written
+$A \subseteq B$, if and only if, every element of $A$ is also an element of $B$.
+
+Symbolically:
+
+$A \subseteq B$ means that for every element $x$, if $x \in A$ then $x \in B$.
+
+The phrases $A$ _is contained in_ $B$ and $B$ _contains_ $A$ are alternative
+ways of saying that $A$ is a subset of $B$.
+
+It follows from the definition of subset that for a set $A$ not to be a subset
+of a set $B$ means that there is at least one element of $A$ that is not an
+element of $B$. Symbolically:
+
+$A \nsubseteq B$ means that there is at least one element $x$ such that
+$x \in A$ and $x \notin B$.
+
+**Definition**
+
+Let $A$ and $B$ be sets. $A$ is a **proper subset** of $B$ if, and only if,
+every element of $A$ is in $B$ but there is at least one element of $B$ that is
+not in $A$.
+
+---
+
+Page 33
+
+**Notation**
+
+Given elements $a$ and $b$, the symbol $(a, b)$ denotes the **ordered pair**
+consisting of $a$ and $b$ together with the specification that $a$ is the first
+element of the pair and $b$ is the second element. Two ordered pairs $(a, b)$
+and $(c, d)$ are equal if, and only if, $a = c$ and $b = d$. Symbolically:
+
+$(a, b) = (c, d)$ means that $a = c$ and $b = d$.
+
+---
+
+Page 34
+
+**Definition**
+
+Let $n$ be a positive integer and let $x_1, x_2, \dots, x_n$ be (not necessarily
+distinct) elements. The **ordered $n$-tuple, $(x_1, x_2, \dots, x_n)$**,
+consists of $x_1$, $x_2$, $\dots$, $x_n$ together with the ordering: first
+$x_1$, then $x_2$, and so forth up to $x_n$. An ordered 2-tuple is called an
+**ordered pair**, and an ordered 3-tuple is called an **ordered triple.**
+
+Two ordered $n$-tuples $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ are
+**equal** if, and only if, $x_1 = y^1$, $x^2 = y^2$, $\dots$, and $x_n = y_n$.
+
+Symbolically:
+
+$$ (x_1, x_2, \dots x_n) = (y_1, y_2, \dots, y_n) \leftrightarrow x_1 = y_1, x^2 = y_2, \dots , x_n = y_n $$
+
+---
+
+Page 35
+
+**Definition**
+
+Given sets $A_1, A_2, \dots, A_n$, the **Cartesian product** of
+$A_1, A_2, \dots, A_n$ denoted $A_1 \times A_2 \times \dots \times A_n$, is the
+set of all ordered $n$-tuples $(a_1, a_2, \dots, a_n)$ where
+$a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n$.
+
+Symbolically:
+
+$$ A_1 \times A_2 \times \dots \times A_n = \{(a_1, a_2, \dots, a_n) | a_1 \in A_1, a_2 \in A_2, \dots , a_n \in A_n\} $$
+
+In particular,
+
+$$ A_1 \times A_2 = \{(a_1, a_2) | a_1 \in A_1 \text{ and } a_2 \in A_2\} $$
+
+is the Cartesian product of $A_1$ and $A_2$.
+
+---
+
+Page 36
+
+**Definition**
+
+Let $n$ be a positive integer. Given a finite set $A$, a **string of length $n$
+over $A$** is an ordered $n$-tuple of elements of $A$ written without
+parentheses or commas. The elements of $A$ are called the **characters** of the
+string. The **null string** over $A$ is defined to be the "string" with no
+characters. It is often denoted $\lambda$ and is said to have length $0$. If
+$A = \{0, 1\}$, then a string over $A$ is called a **bit string**.
diff --git a/chapter_1/test_yourself.md b/chapter_1/test_yourself.md
new file mode 100644
index 0000000..55efaf4
--- /dev/null
+++ b/chapter_1/test_yourself.md
@@ -0,0 +1,83 @@
+**Test Yourself**
+
+Page 28
+
+1. A universal statement asserts that a certain property is _______ for _______.
+
+2. A conditional statement asserts that if one thing _______ then some other
+   thing _______.
+
+3. Given a property that may or may not be true, an existential statement
+   asserts that _______ for which the property is true.
+
+**Solutions**:
+
+1. true, for all elements of a set.
+
+2. is true, also has to be true.
+
+3. there is at least one thing
+
+---
+
+**Test Yourself**
+
+Page 37
+
+1. When the elements of a set are given using the set-roster notation, the order
+   in which they are listed _______.
+
+**Solution**
+
+does not matter.
+
+2. The symbol $\mathbb{R}$ denotes _______.
+
+**Solution**
+
+The set of all real numbers.
+
+3. The symbol $\mathbb{Z}$ denotes _______.
+
+**Solution**
+
+The set of all integers.
+
+4. The symbol $\mathbb{Q}$ denotes _______.
+
+**Solution**
+
+The set of all rational numbers.
+
+5. The notation $\{x | P(x)\}$ is read _______.
+
+**Solution**
+
+The set of all $x$ such that $P(x)$ is true.
+
+6. For a set $A$ to be a subset of a set $B$ means that _______.
+
+**Solution**
+
+Every element in $A$ is an element in $B$.
+
+7. Given sets $A$ and $B$, the Cartesian product $A \times B$ is _______.
+
+**Solution**
+
+The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
+
+8. Given sets $A$, $B$, and $C$, the Cartesian product $A \times B \times C$ is
+   _______.
+
+The set of all ordered triples, $(a, b, c)$ where $a \in A$ and $b \in B$ and
+$c \in C$.
+
+**Solution**
+
+9. A string of length $n$ over a set $S$ is an ordered $n$-tuple of elements
+   $S$, written without _______ or _______.
+
+**Solution**
+
+parentheses; commas
diff --git a/leftoff.txt b/leftoff.txt
index 573541a..e522732 100644
--- a/leftoff.txt
+++ b/leftoff.txt
@@ -1 +1 @@
-0
+38