diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index a81b26a..aa87a54 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -8,95 +8,188 @@ Page 142 a. There is an animal in the menagerie that is red. +False + b. Every animal in the menagerie is a bird or a mammal. +True + c. Every animal in the menagerie is brown or gray or black. +False + d. There is an animal in the menagerie that is neither a cat nor a dog. +True + e. No animal in the menagerie is blue. +False + f. There are in the menagerie a dog, a cat, and a bird that all have the same color. +True + 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. +True, because the set of all integers, $\mathbb{Z}$ is a subset of the set of +all real numbers, $\mathbb{R}$. $\mathbb{Z} \in \mathbb{R}$. + b. $0$ is a positive real number. +False, $0$ is neither positive nor negative. + c. For every real number $r$, $-r$ is a negative real number. +False, if $r$ is negative, then $-r$ is positive. + d. Every real number is an integer. +False, $\dfrac{1}{2}$ is not an integer, but is a real number. + 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 $n = 10$. +The statement "If 25 is a factor of 100" is a true hypothesis, but the +conclusion "then 25 is a factor of 10" is false because 10 is not a product of +25 times any integer. Thus the hypothesis is true, but the conclusion is false, +making this predicate a false statement. + b. Give values different from those in part (a) for which $R(m, n)$ is false. +$m = 9$, $n = 3$ + c. Explain why $R(m, n)$ is true if $m = 5$ and $n = 10$. +Because 5 is a factor of 100, which is the hypothesis, and the conclusion is +that 5 is a factor of 10, which is also true. Thus the hypothesis and conclusion +are both true, so the statement as a whole is true. + d. Give values different from those in part \(c\) for which $R(m, n)$ is true. +$m = 4$, $n = 8$ + 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 $y = 1$. +The hypothesis becomes "If $-2 < 1$", which is true, the conclusion becomes +"then $4 < 1$", which is a false conclusion. Thus the hypothesis is true, but +the conclusion is false, making $Q(-2, 1)$ a false statement. + b. Give values different from those in part (a) for which $Q(x, y)$ is false. +$x = -5$, $y = 2$ + c. Explain why $Q(x, y)$ is true if $x = 3$ and $y = 8$. +The hypothesis becomes "If $3 < 8$", which is true, and the conclusion becomes +"then $9 < 64$", which is also true. This shows that the hypothesis and +conclusion true, making $Q(3, 8)$ a true statement. + d. Give values different from those in part \(c\) for which $Q(x, y)$ is true. +$x = 3$, $y = 4$ + 5. Find the truth set of each predicate. a. Predicate: $\dfrac{6}{d}$ is an integer, domain: $\mathbb{Z}$ +$$ \{-6, -3, -2, -1, 1, 2, 3, 6\} $$ + b. Predicate: $\dfrac{6}{d}$ is an integer, domain: $\mathbb{Z}^+$ +$$ \{1, 2, 3, 6\} $$ + c. Predicate: $1 \leq x^2 \leq 4$, domain: $\mathbb{R}$ +$$ \{x \in \mathbb{R} | -1 \leq x \leq -2 \text{ or } 1 \leq x \leq 2\} $$ + d. Predicate: $1 \leq x^2 \leq 4$, domain: $\mathbb{Z}$ +$$ \{-2, -1, 1, 2\} $$ + 6. Let $B(x)$ be "$-10 < x < 10$." Find the truth set of $B(x)$ for each of the following domains. a. $\mathbb{Z}$ +$$ \{-9, -8, -7, -6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9\} $$ + b. $\mathbb{Z}^+$ +$$ \{1, 2, 3, 4, 5, 6, 7, 8, 9\} $$ + c. The set of all even integers +$$ \{-8, -6, -4, -2, 2, 4, 6, 8\} $$ + 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_. + baa, bab, bac, bba, bbb, bbc, bca, bcb, bcc + 2. No string in $S$ has more than one _c_. + aaa, aab, aac, aba, abb, abc, aca, acb, baa, bab, bac, bba, bbb, bbc, bca, + bcb, caa, cab, cba, cbb + 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. + 000, 001, 010, 110, 111 + 2. No string in $T$ has all three characters the same. + 001, 010, 100, 101, 110 + 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. +$x = \dfrac{1}{2}$, so $\dfrac{1}{2} \geq \dfrac{1}{\dfrac{1}{2}}$ is false as: + +$$ \frac{1}{2} < 2 $$ + +10. $\forall a \in \mathbb{Z}, \dfrac{(a - 1)}{a}$ is not an integer. + +$$ a = -1 $$ + +$$ \frac{((-1) - 1)}{-1} = \frac{-2}{-1} = 2 $$ + +Since $2 \in \mathbb{Z}$, this statement is false. 11. $\forall$ positive integers $m$ and $n$, $m \cdot n \geq m + n$. +$m = 1$, $n = 2$ + +$$ 1 \cdot 2 = 2 \geq 3 = 1 + 2 $$ + +But $2$ is not greater than or equal to $3$, so this statement is false. + 12. $\forall$ real numbers $x$ and $y$, $\sqrt{x + y} = \sqrt{x} + \sqrt{y}$. +$x = 4$, $y = 9$ + +$$ \sqrt{4 + 9} = \sqrt{13} \approx 3.605551275 $$ + +$$ \sqrt{4} + \sqrt{9} = 2 + 3 = 5 $$ + +Since $\sqrt{13} \neq 5$, this statement is false. + 13. Consider the following statement: $\forall$ basketball player $x$, $x$ is tall. @@ -105,16 +198,28 @@ Which of the following are equivalent ways of expressing the statement? a. Every basketball player is tall. +Yes. + b. Among all the basketball players, some are tall. +No. + c. Some of all the tall people are basketball players. +No. + d. Anyone who is tall is a basketball player. +No. + e. All people who are basketball players are tall. +Yes. + f. Anyone who is a basketball player is a tall person. +Yes. + 14. Consider the following statement: $\exists x \in \mathbb{R}$ such that $x^2 = 2$. @@ -123,45 +228,81 @@ Which of the following are equivalent ways of expressing this statement a. The square of each real number is 2. +No. + b. Some real numbers have square 2. +Yes. + c. The number $x$ has square 2, for some real number $x$. +Yes. + d. If $x$ is a real number, then $x^2 = 2$. +No. + e. Some real number has square 2. +Yes. + f. There is at least one real number whose square is 2. +Yes. + 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. +If a shape is a rectangle, then the shape is a quadrilateral. + +For any given rectangle, that rectangle is a quadrilateral. + b. $\exists$ a set $A$ such that $A$ has 16 subsets. +There must be a set that has 16 subsets. + +At least one set has 16 subsets. + 16. Rewrite each of the following statements in the form "$\forall$ ______ $x$, ______." a. All dinosaurs are extinct. +$\forall$ dinosaurs, $x$, $x$ is extinct. + b. Every real number is positive, negative, or zero. +$\forall$ real numbers $x$, $x$ is positive, negative, or zero. + c. No irrational numbers are integers. +$\forall$ irrational numbers $x$, $x$ is not an integer. + d. No logicians are lazy. +$\forall$ logicians $x$, $x$ is not lazy. + e. The number 2,147,581,953 is not equal to the square of any integer. +$\forall$ integer $x$, $x^2$ does not equal 2,147,581,953. + f. The number $-1$ is not equal to the square of any real number. +$\forall$ real numbers $x$, $x^2$ does not equal -1. + 17. Rewrite each of the following in the form "$\exists$ ______ $x$ such that ______." a. Some exercises have answers. +$\exists$ some exercise, $x$, such that $x$ has an answer. + b. Some real numbers are rational. +$\exists$ some real number $x$, such that $x$ is 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 @@ -170,14 +311,25 @@ b. Some real numbers are rational. a. There is an engineering student who is a math major. +$\exists s$ such that $E(s)$ and $M(s)$. + b. Every computer science student is an engineering student. +$\forall s \in D, C(s) \to E(s)$ + c. No computer science students are engineering students. +$\forall s \in D, C(s) \to \neg E(s)$ + d. Some computer science students are also math majors. +$\exists s$ such that $C(s) \wedge M(s)$ + e. Some computer science students are engineering students and some are not. +$\exists s$ such that $C(s) \wedge E(s)$ or $\exists$ such that +$C(s) \wedge \neg E(s)$ + 19. Consider the following statement: $\forall$ integer $n$, if $n^2$ is even then $n$ is even. @@ -186,54 +338,101 @@ Which of the following are equivalent ways of expressing this statement? a. All integers have even squares and are even. +No + b. Given any integer whose square is even, that integer is itself even. +Yes + c. For all integers, there are some whose square is even. +No + d. Any integer with an even square is even. +Yes + e. If the square of an integer is even, then that integer is even. +Yes + f. All even integers have even squares. +No + 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. +If a number is a positive real number, then the square root of that number is +positive. + +Any positive real number's square root 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. +The total degree of $G$ is even, for any graph $G$. + b. For any isosceles triangle $T$, the base angles of $T$ are equal. +The base angles of $T$ are equal, for any isosceles triangle $T$. + c. There exists a prime number $p$ such that $p$ is even. +$p$ is even for some prime number $p$. + d. There exists a continuous function $f$ such that $f$ is not differentiable. +$f$ is not differentiable for some continuous function $f$. + 22. Rewrite each of the following statements in the form "$\forall$ ______ $x$, if ______ then ______." a. All Java programs have at least 5 lines. +$\forall$ programs $x$, if $x$ is a Java program, then it has at least 5 lines. + b. Any valid argument with true premises has a true conclusion. +$\forall$ arguments $x$, if $x$ is a valid argument with a true premise, then it +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. +$\forall x$, if $x$ is equilateral, then $x$ is isosceles. + +$\forall$ equilateral triangles $x$, $x$ is isosceles. + b. Every computer science student needs to take data structures. +$\forall x$ if $x$ is a computer science student, then $x$ needs to take data +structures. + +$\forall$ computer science students $x$, $x$ 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. +$\exists$ a hatter $x$ such that $x$ is mad. + +$\exists x$ such that $x$ is a hatter and $x$ is mad. + b. Some questions are easy. +$\exists$ a question $x$ such that $x$ is easy. + +$\exists x$ such that $x$ is a question and $x$ is 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 @@ -243,27 +442,63 @@ b. Some questions are easy. a. The reciprocal of any nonzero function is a fraction. +$\forall$ nonzero function $x$, the reciprocal of $x$ is a fraction. + +$\forall x$, if $x$ is a nonzero fraction, then the reciprocal of $x$ is a +fraction. + b. The derivative of any polynomial function is a polynomial function. +$\forall$ derivatives of any polynomial function $x$, $x$ is a polynomial +function. + +$\forall x$, if $x$ is a derivative of any polynomial function, then $x$ is a +polynomial function. + c. The sum of the angles of any triangle is $180\degree$. +$\forall$ triangles $x$, the sum of the angles of $x$ is $180\degree$. + +$\forall x$ if $x$ is a triangle, then the sum of the angles of $x$ is +$180\degree$. + d. The negative of any irrational number is irrational. +$\forall$ negative of any irrational number, $x$, $x$ is irrational. + +$\forall x$ if $x$ is a negative of any irrational number, then $x$ is +irrational. + e. The sum of any two even integers is even. +$\forall$ even integers $x$, and $y$, the sum of $x$ and $y$ is even. + +$\forall x, y$ if $x$ and $y$ are even integers, then the sum of $x$ and $y$ is +even. + f. The product of any two fractions is a fraction. +$\forall$ fractions $x$ and $y$, the product of $x$ and $y$ is a fraction. + +$\forall x, y$ if $x$ and $y$ are fractions, then the product of $x$ and $y$ 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 ______." +$\forall x$, if $x$ is an integer, then $x$ is a rational number, but $\exists$ +a rational number $x$, such that $x$ is not an integer. + 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 $\to$. +$$ \forall x (\text{Int}(x) \to \text{Ratl}(x)) \wedge \exists x (\text{Ratl(x)} \wedge \neg \text{Int}(x))$$ + 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. @@ -271,12 +506,20 @@ $\neg$, and $\to$. a. $\forall u, \text{Circle}(u) \to \text{Gray(u)}$. +This is false, b is a circle and is black. + b. $\forall u, \text{Gray}(u) \to \text{Circle}(u)$. +This is true, all gray shapes are circles. + c. $\exists y$ such that $\text{Square}(y) \wedge \text{Above}(y, d)$. +This is false, there is no shape that is a square and is above shape d. + d. $\exists z$ such that $\text{Triangle}(z) \wedge \text{Above}(f, z)$. +This is true, shape g is a triangle where shape f is above shape g. + 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. @@ -288,22 +531,56 @@ you can. a. $\text{Pos}(0)$ +"0 is a positive real number." + +This is a false statement, as 0 is neither positive nor negative. + b. $\forall x, \text{Real}(x) \wedge \text{Neg}(x) \to \text{Pos}(-x)$ +"For any number, if that number is both real and negative, then the negative of +that number is positive."" + +This is true, if you take the negative of a negative of any real number, then it +is positive. + c. $\forall x, \text{Int}(x) \to \text{Real}(x)$ +"For any number, if that number is an integer, then that number is a real +number." + +This is true, the set of all integers is a subset of all real numbers. + d. $\exists x$ such that $\text{Real}(x) \wedge \neg \text{Int}(x)$ +"There is at least one number that is both a real number and not an integer." + +This is true, an example would be $\dfrac{1}{2}$, which is a real number but not +an integer. + 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 $\text{Rect}(x) \wedge \text{Square}(x)$ +"There exists a shape that is both a rectangle and a square." + +This is true since any shape that is a square is also a rectangle. + b. $\exists x$ such that $\text{Rect}(x) \wedge \neg \text{Square}(x)$ +"There exists a shape that is both a rectangle and not a square." + +This is true, as any shape that is a rectangle that has unequal length and width +is not a square. + c. $\forall x, \text{Square}(x) \to \text{Rect}(x)$ +"For any shape, if that shape is a square, then that shape is a rectangle." + +This is true, for all shapes, any shape that is a square, that shape is then a +rectangle. + 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 @@ -312,36 +589,85 @@ c. $\forall x, \text{Square}(x) \to \text{Rect}(x)$ a. $\exists x$ such that $\text{Prime}(x) \wedge \neg \text{Odd}(x)$ +"There exists some number that is both prime and not odd." + +This is true, for example $2$ is a prime number (cannot be divided except by 1 +and itself), but $2$ is also not odd. + b. $\forall x, \text{Prime}(x) \to \neg \text{Square}(x)$ +"For any number, if that number is prime, then that number is not a perfect +square." + +This is true, since a prime number is only divisible by 1 and itself, it cannot +equal the square of some integer, since that square would also be the product of +two smaller positive integers. + c. $\exists x$ such that $\text{Odd}(x) \wedge \text{Square}(x)$ +"There exists some number that is both odd and is a perfect square." + +This is true, take $9$ as an example, $9$ is an odd number, but is also a +perfect square as $9 = 3^2$. + 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. +Omitted. + 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. $x > 2 \Rightarrow x > 1$ +This is true, for any real number that is greater than 2, that same real number +is greater than 1. + b. $x > 2 \Rightarrow x^2 > 4$ +This is true, for any real number that is greater than 2, that same real number +squared is greater than 4. + c. $x^2 > 4 \Rightarrow x > 2$ +This is false, as $x = -3$ would mean $(-3)^2 > 4$, which is true as that is +$9 > 4$, but then $(-3) > 2$ is false. Since the hypothesis is true, but the +conclusion is false for at least one example, this predicate is therefore false. + d. $x^2 > 4 \Leftrightarrow |x| > 2$ +This is true. For all numbers $x$, if $x^2 > 4$, then $|x| > 2$ is true. + +Additionally, for all numbers $x$, if $|x| > 2$, then $x^2 > 4$ is true. + +Since both directions of this universal "if and only if" statement are true, +this is a true statement. + 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. $a > 0 \text{ and } b > 0 \Rightarrow ab > 0$ +This is true. If both $a$ and $b$ are positive, then their product is also +positive. + b. $a < 0 \text{ and } b < 0 \Rightarrow ab < 0$ +This is false, If both $a$ and $b$ are negative, then their product is positive, +not negative. Take $-1$ and $-2$ for example, whose product is $2$. + c. $ab = 0 \Rightarrow a = 0 \text{ or } b = 0$ +This is true, for all real numbers $a$ and $b$, if their product, $ab$ is equal +to $0$, then either $a$ or $b$ must be $0$. + d. $a < b \text{ and } c < d \Rightarrow ac < bd$ + +This is false. Say $a = -1$, $b = 2$, $c = -8$ and $d = 3$. This would make +$-1 < 2$ and $-8 < 3$, which is true, but then $(-1)(-8) < (2)(3)$ would be +$8 < 6$, which is false.