diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index e69de29..a81b26a 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -0,0 +1,347 @@ +**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. $a < b \text{ and } c < d \Rightarrow ac < bd$ diff --git a/chapter_3/notes.md b/chapter_3/notes.md index e69de29..0dacfd8 100644 --- a/chapter_3/notes.md +++ b/chapter_3/notes.md @@ -0,0 +1,60 @@ +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, + $\forall x, P(x) \leftrightarrow Q(x)$. diff --git a/chapter_3/test_yourself.md b/chapter_3/test_yourself.md index e69de29..36c9719 100644 --- a/chapter_3/test_yourself.md +++ b/chapter_3/test_yourself.md @@ -0,0 +1,27 @@ +**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 $D$.