🚧 Setup for 3.1

chapter_3/exercises.md

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+**Exercise Set 3.1**
+
+Page 142
+
+1. A menagerie consists of seven brown dogs, two black dogs, six gray cats, ten
+   black cats, five blue birds, six yellow birds, and one black bird. Determine
+   which of the following statements are true and which are false.
+
+a. There is an animal in the menagerie that is red.
+
+b. Every animal in the menagerie is a bird or a mammal.
+
+c. Every animal in the menagerie is brown or gray or black.
+
+d. There is an animal in the menagerie that is neither a cat nor a dog.
+
+e. No animal in the menagerie is blue.
+
+f. There are in the menagerie a dog, a cat, and a bird that all have the same
+color.
+
+2. Indicate which of the following statements are true and which are false.
+   Justify your answers as best you can.
+
+a. Every integer is a real number.
+
+b. $0$ is a positive real number.
+
+c. For every real number $r$, $-r$ is a negative real number.
+
+d. Every real number is an integer.
+
+3. Let $R(m, n)$ be the predicate "If $m$ is a factor of $n^2$ then $m$ is a
+   factor of $n$," with domain for both $m$ and $n$ being $\mathbb{Z}$ the set
+   of integers.
+
+a. Explain why $R(m, n)$ is false if $m = 25$ and $n = 10$.
+
+b. Give values different from those in part (a) for which $R(m, n)$ is false.
+
+c. Explain why $R(m, n)$ is true if $m = 5$ and $n = 10$.
+
+d. Give values different from those in part \(c\) for which $R(m, n)$ is true.
+
+4. Let $Q(x, y)$ be the predicate "If $x < y$ then $x^2 < y^2$" with the domain
+   for both $x$ and $y$ being $\mathbb{R}$ the set of real numbers.
+
+a. Explain why $Q(x, y)$ is false if $x = -2$ and $y = 1$.
+
+b. Give values different from those in part (a) for which $Q(x, y)$ is false.
+
+c. Explain why $Q(x, y)$ is true if $x = 3$ and $y = 8$.
+
+d. Give values different from those in part \(c\) for which $Q(x, y)$ is true.
+
+5. Find the truth set of each predicate.
+
+a. Predicate: $\dfrac{6}{d}$ is an integer, domain: $\mathbb{Z}$
+
+b. Predicate: $\dfrac{6}{d}$ is an integer, domain: $\mathbb{Z}^+$
+
+c. Predicate: $1 \leq x^2 \leq 4$, domain: $\mathbb{R}$
+
+d. Predicate: $1 \leq x^2 \leq 4$, domain: $\mathbb{Z}$
+
+6. Let $B(x)$ be "$-10 < x < 10$." Find the truth set of $B(x)$ for each of the
+   following domains.
+
+a. $\mathbb{Z}$
+
+b. $\mathbb{Z}^+$
+
+c. The set of all even integers
+
+7. Let $S$ be the set of all strings of length 3 consisting of _a_'s, _b_'s, and
+   _c_'s. List all the strings in $S$ that satisfy the following conditions:
+
+   1. Every string in $S$ begins with _b_.
+
+   2. No string in $S$ has more than one _c_.
+
+8. Let $T$ be the set of all strings of length 3 consisting of 0's and 1's. List
+   all the strings in $T$ that satisfy the following conditions:
+
+   1. For every string $s$ in $T$, the second character of $s$ is 1 or the first
+      two characters of $s$ are the same.
+
+   2. No string in $T$ has all three characters the same.
+
+Find counterexamples to show that the statements in 9-12 are false.
+
+9. $\forall x \in \mathbb{R}, x \geq \dfrac{1}{x}$.
+
+10. $\forall x \in \mathbb{Z}, \dfrac{(a - 1)}{a}$ is not an integer.
+
+11. $\forall$ positive integers $m$ and $n$, $m \cdot n \geq m + n$.
+
+12. $\forall$ real numbers $x$ and $y$, $\sqrt{x + y} = \sqrt{x} + \sqrt{y}$.
+
+13. Consider the following statement:
+
+$\forall$ basketball player $x$, $x$ is tall.
+
+Which of the following are equivalent ways of expressing the statement?
+
+a. Every basketball player is tall.
+
+b. Among all the basketball players, some are tall.
+
+c. Some of all the tall people are basketball players.
+
+d. Anyone who is tall is a basketball player.
+
+e. All people who are basketball players are tall.
+
+f. Anyone who is a basketball player is a tall person.
+
+14. Consider the following statement:
+
+$\exists x \in \mathbb{R}$ such that $x^2 = 2$.
+
+Which of the following are equivalent ways of expressing this statement
+
+a. The square of each real number is 2.
+
+b. Some real numbers have square 2.
+
+c. The number $x$ has square 2, for some real number $x$.
+
+d. If $x$ is a real number, then $x^2 = 2$.
+
+e. Some real number has square 2.
+
+f. There is at least one real number whose square is 2.
+
+15. Rewrite the following statements informally in at least two different ways
+    without using variables or quantifiers.
+
+a. $\forall$ rectangle $x$, $x$ is a quadrilateral.
+
+b. $\exists$ a set $A$ such that $A$ has 16 subsets.
+
+16. Rewrite each of the following statements in the form "$\forall$ ______ $x$,
+    ______."
+
+a. All dinosaurs are extinct.
+
+b. Every real number is positive, negative, or zero.
+
+c. No irrational numbers are integers.
+
+d. No logicians are lazy.
+
+e. The number 2,147,581,953 is not equal to the square of any integer.
+
+f. The number $-1$ is not equal to the square of any real number.
+
+17. Rewrite each of the following in the form "$\exists$ ______ $x$ such that
+    ______."
+
+a. Some exercises have answers.
+
+b. Some real numbers are rational.
+
+18. Let $D$ be the set of all students at your school, and let $M(s)$ be "$s$ is
+    a math major," let $C(s)$ be "$s$ is a computer science student," and let
+    $E(s)$ be "$s$ is an engineering student." Express each of the following
+    statements using quantifiers, variables, and the predicates $M(s)$, $C(s)$,
+    and $E(s)$.
+
+a. There is an engineering student who is a math major.
+
+b. Every computer science student is an engineering student.
+
+c. No computer science students are engineering students.
+
+d. Some computer science students are also math majors.
+
+e. Some computer science students are engineering students and some are not.
+
+19. Consider the following statement:
+
+$\forall$ integer $n$, if $n^2$ is even then $n$ is even.
+
+Which of the following are equivalent ways of expressing this statement?
+
+a. All integers have even squares and are even.
+
+b. Given any integer whose square is even, that integer is itself even.
+
+c. For all integers, there are some whose square is even.
+
+d. Any integer with an even square is even.
+
+e. If the square of an integer is even, then that integer is even.
+
+f. All even integers have even squares.
+
+20. Rewrite the following statement informally in at least two different ways
+    without using variables of the symbol $\forall$ or the words "for all."
+
+$\forall$ real numbers $x$, if $x$ is positive then the square root of $x$ is
+positive.
+
+21. Rewrite the following statements so that the quantifier trails the rest of
+    the sentence.
+
+a. For any graph $G$, the total degree of $G$ is even.
+
+b. For any isosceles triangle $T$, the base angles of $T$ are equal.
+
+c. There exists a prime number $p$ such that $p$ is even.
+
+d. There exists a continuous function $f$ such that $f$ is not differentiable.
+
+22. Rewrite each of the following statements in the form "$\forall$ ______ $x$,
+    if ______ then ______."
+
+a. All Java programs have at least 5 lines.
+
+b. Any valid argument with true premises has a true conclusion.
+
+23. Rewrite each of the following statements in the two forms "$\forall x$, if
+    ______ then ______" and "$\forall x$, ______" (without an if-then).
+
+a. All equilateral triangles are isosceles.
+
+b. Every computer science student needs to take data structures.
+
+24. Rewrite the following statements in the two forms "$\exists$ ______ $x$ such
+    that ______" and "$\exists x$ such that ______ and ______."
+
+a. Some hatters are mad.
+
+b. Some questions are easy.
+
+25. The statement "The square of any rational number is rational" can be
+    rewritten formally as "For all rational numbers $x$, $x^2$ is rational" or
+    as "For all $x$, if $x$ is rational then $x^2$ is rational." Rewrite each of
+    the following statements in the two forms "$\forall$ ______ $x$, ______" and
+    "$\forall x$, if ______, then ______" or in the two forms "$\forall$ ______
+    $x$ and $y$, ______" and "$\forall x$ and $y$, if ______, then ______."
+
+a. The reciprocal of any nonzero function is a fraction.
+
+b. The derivative of any polynomial function is a polynomial function.
+
+c. The sum of the angles of any triangle is $180\degree$.
+
+d. The negative of any irrational number is irrational.
+
+e. The sum of any two even integers is even.
+
+f. The product of any two fractions is a fraction.
+
+26. Consider the statement "All integers are rational numbers but some rational
+    numbers are not integers."
+
+a. Write this statement in the form "$\forall x$, if ______ then ______, but
+$\exists$ ______ $x$, such that ______."
+
+b. Let $\text{Ratl}(x)$ be "$x$ is a rational number" and $\text{Int}(x)$ be
+"$x$ is an integer." Write the given statement formally using only the symbols
+$\text{Ratl}(x)$, $\text{Int}(x)$, $\forall$, $\exists$, $\wedge$, $\vee$,
+$\neg$, and $\to$.
+
+27. Refer to the picture of Tarski's world given in Example 3.1.1.3. Let
+    $\text{Above}(x, y)$ mean that $x$ is above $y$ (but possibly in a different
+    column). Determine the truth or falsity of each of the following statements.
+    Give reasons for your answers.
+
+a. $\forall u, \text{Circle}(u) \to \text{Gray(u)}$.
+
+b. $\forall u, \text{Gray}(u) \to \text{Circle}(u)$.
+
+c. $\exists y$ such that $\text{Square}(y) \wedge \text{Above}(y, d)$.
+
+d. $\exists z$ such that $\text{Triangle}(z) \wedge \text{Above}(f, z)$.
+
+In 28-30, rewrite each statement without using quantifiers or variables.
+Indicate which are true and which are false, and justify your answers as best as
+you can.
+
+28. Let the domain of $x$ be the set $D$ of objects discussed in mathematics
+    courses, and let $\text{Real}(x)$ be "$x$ is a real number," $\text{Pos}(x)$
+    be "$x$ is a positive real number," $\text{Neg}(x)$ be "$x$ is a negative
+    real number," and $\text{Int}(x)$ be "$x$ is an integer."
+
+a. $\text{Pos}(0)$
+
+b. $\forall x, \text{Real}(x) \wedge \text{Neg}(x) \to \text{Pos}(-x)$
+
+c. $\forall x, \text{Int}(x) \to \text{Real}(x)$
+
+d. $\exists x$ such that $\text{Real}(x) \wedge \neg \text{Int}(x)$
+
+29. Let the domain of $x$ be the set of geometric figures in the plane, and let
+    $\text{Square}(x)$ be "$x$ is a square" and $\text{Rect}(x)$ be "$x$ is a
+    rectangle."
+
+a. $\exists x$ such that $\text{Rect}(x) \wedge \text{Square}(x)$
+
+b. $\exists x$ such that $\text{Rect}(x) \wedge \neg \text{Square}(x)$
+
+c. $\forall x, \text{Square}(x) \to \text{Rect}(x)$
+
+30. Let the domain of $x$ be $\mathbb{Z}$, the set of integers, and let
+    $\text{Odd}(x)$ be "$x$ is odd," $\text{Prime}(x)$ be "$x$ is prime," and
+    $\text{Square}(x)$ be "$x$ is a perfect square." (An integer $n$ is said to
+    be a **perfect square** if, and only if, it equals the square of some
+    integer. For example, $25$ is a perfect square because $25 = 5^2$.)
+
+a. $\exists x$ such that $\text{Prime}(x) \wedge \neg \text{Odd}(x)$
+
+b. $\forall x, \text{Prime}(x) \to \neg \text{Square}(x)$
+
+c. $\exists x$ such that $\text{Odd}(x) \wedge \text{Square}(x)$
+
+31. In any mathematics or computer science text other than this book, find an
+    example of a statement that is universal but is implicitly quantified. Copy
+    the statement as it appears and rewrite it making the quantification
+    explicit. Give a complete citation for your example, including title,
+    author, publisher, year, and page number.
+
+32. Let $\mathbb{R}$ be the domain of the predicate variable $x$. Which of the
+    following are true and which are false? Give counter examples for the
+    statements that are false.
+
+a. $x > 2 \Rightarrow x > 1$
+
+b. $x > 2 \Rightarrow x^2 > 4$
+
+c. $x^2 > 4 \Rightarrow x > 2$
+
+d. $x^2 > 4 \Leftrightarrow |x| > 2$
+
+33. Let $\mathbb{R}$ be the domain of the predicate variables $a$, $b$, $c$, and
+    $d$. Which of the following are true and which are false? Give
+    counterexamples for the statements that are false.
+
+a. $a > 0 \text{ and } b > 0 \Rightarrow ab > 0$
+
+b. $a < 0 \text{ and } b < 0 \Rightarrow ab < 0$
+
+c. $ab = 0 \Rightarrow a = 0 \text{ or } b = 0$
+
+d. $a < b \text{ and } c < d \Rightarrow ac < bd$

chapter_3/notes.md

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+Page 132
+
+**Definition**
+
+A **predicate** is a sentence that contains a finite number of variables and
+becomes a statement when specific values are substituted for the variables. The
+**domain** of a predicate variable is the set of all values that may be
+substituted in place of the variable.
+
+---
+
+Page 132
+
+**Definition**
+
+If $P(x)$ is a predicate and $x$ has domain $D$, the **truth set** of $P(x)$ is
+the set of all elements of $D$ that make $P(x)$ true when they are substituted
+for $x$. The truth set of $P(x)$ is denoted
+
+$$ \{x \in D | P(x)\} $$
+
+---
+
+Page 133
+
+**Definition**
+
+Let $Q(x)$ be a predicate and $D$ the domain of $x$. A **universal statement**
+is a statement of the form "$\forall x \in D, Q(x)$." It is defined to be true
+if, and only if, $Q(x)$ is true for each individual $x$ in $D$. It is defined to
+be false if, and only if, $Q(x)$ is false for at least one $x$ in $D$. A value
+for $x$ for which $Q(x)$ is false is called a **counterexample** to the
+universal statement.
+
+---
+
+Page 134
+
+**Definition**
+
+Let $Q(x)$ be a predicate and $D$ the domain of $x$. An **existential
+statement** is a statement of the form "$\exists x \in D$ such that $Q(x)$." It
+is defined to be true if, and only if, $Q(x)$ is true for at least one $x$ in
+$D$. It is false if, and only if, $Q(x)$ is false for all $x$ in $D$.
+
+---
+
+Page 140
+
+**Notation**
+
+Let $P(x)$ and $Q(x)$ be predicates and suppose the domain of $x$ is $D$.
+
+- The notation $P(x) \Rightarrow Q(x)$ means that every element in the truth set
+  of $P(x)$ is in the truth set of $Q(x)$, or, equivalently,
+  $\forall x, P(x) \to Q(x)$.
+
+- The notation $P(x) \Leftrightarrow Q(x)$ means that $P(x)$ and $Q(x)$ have
+  identical truth sets, or, equivalently,
+  $\forall x, P(x) \leftrightarrow Q(x)$.

chapter_3/test_yourself.md

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+**Test Yourself**
+
+Page 141
+
+1. If $P(x)$ is a predicate with domain $D$, the truth set of $P(x)$ is denoted
+   _______. We read these symbols out loud as _______.
+
+$\{x \in D | P(x)\}$; "the set of all $x$ in $D$ such that $P(x)$."
+
+2. Some ways to express the symbol $\forall$ in words are _______.
+
+for every for all, for any, for each, for arbitrary, given any
+
+3. Some ways to express the symbol $\exists$ in words are _______.
+
+there exists, there exist, there exists at least one, for some, for at least
+one, we can find a
+
+4. A statement of the form $\forall x \in D$, $Q(x)$ is true if, and only if,
+   $Q(x)$ is _______ for _______.
+
+true; every $x$ in $D$.
+
+5. A statement of the form $\exists x \in D$ such that $Q(x)$ is true if, and
+   only if, $Q(x)$ is _______ for _______.
+
+true; at least one $x$ in $D$.