diff --git a/chapter_3/exercises.md b/chapter_3/exercises.md index 31ad120..56cae21 100644 --- a/chapter_3/exercises.md +++ b/chapter_3/exercises.md @@ -1547,3 +1547,392 @@ This is: $$ F \to S $$ This is the converse statement, and is not logically equivalent to the original. + +--- + +**Exercise Set 3.3** + +Page 166 + +1. Let $C$ be the set of cities in the world, let $N$ be the set of nations in + the world, and let $P(c, n)$ be "$c$ is the capital city of $n$." Determine + the truth values of the following statements. + +a. $P(\text{Tokyo}, \text{Japan})$ + +b. $P(\text{Athens}, \text{Egypt})$ + +c. $P(\text{Paris}, \text{France})$ + +d. $P(\text{Miami}, \text{Brazil})$ + +2. Let $G(x, y)$ be "$x^2 > y$." Indicate which of the following statements are + true and which are false. + +a. $G(2, 3)$ + +b. $G(1, 1)$ + +c. $G(\dfrac{1}{2}, \dfrac{1}{2})$ + +d. $G(-2, 2)$ + +3. The following statement is true: "$\forall$ nonzero number $x$, $\exists$ a + real number $y$ such that $xy = 1$." For each $x$ given below, find a $y$ to + make the predicate "$xy = 1$" true. + +a. $x = 2$ + +b. $x = -1$ + +c. $x = \dfrac{3}{4}$ + +4. The following statement is true: "$\forall$ real number $x$, $\exists$ an + integer $n$ such that $n > x$.". For each $x$ given below, find an $n$ to + make the predicate $n > x$ true. + +a. $x = 15.83$ + +b. $x = 10^8$ + +c. $x = 10^{10^{10}}$ + +The statements in exercises 5-8 refer to the Tarski world given in Figure 3.3.1. +Explain why each is true. + +5. For every circle $x$ there is a square $y$ such that $x$ and $y$ have the + same color. + +6. For every square $x$ there is a circle $y$ such that $x$ and $y$ have + different colors and $y$ is above $x$. + +7. There is a triangle $x$ such that for every square $y$, $x$ is above $y$. + +8. There is a triangle $x$ such that for every circle $y$, $y$ is above $x$. + +9. Let $D = E = \{-2, -1, 0, 1, 2\}$. Explain why the following statements are + true. + +a. $\forall x$ in $D$, $\exists y$ such that $x + y = 0$. + +b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = y$. + +10. This exercise refers to Example 3.3.3. Determine whether each of the + following statements is true or false. + +a. $\forall$ student $S$, $\exists$ a dessert $D$ such that $S$ chose $D$. + +b. $\forall$ student $S$, $\exists$ a salad $T$ such that $S$ chose $T$. + +c. $\exists$ a dessert $D$ such that $\forall$ student $S$, $S$ chose $D$. + +d. $\exists$ a beverage $B$ such that $\forall$ student $D$, $D$ chose $B$. + +e. $\exists$ an item $I$ such that $\forall$ student $S$, $S$ did not choose +$I$. + +f. $\exists$ a station $Z$ such that $\forall$ student $S$, $\exists$ an item +$I$ such that $S$ chose $I$ from $Z$. + +11. Let $S$ be the set of students at your school, let $M$ be the set of movies + that have ever been released, and let $V(s, m)$ be "student $s$ has seen + movie $m$." Rewrite each of the following statements without using the + symbol $\forall$, the symbol $\exists$, or variables. + +a. $\exists s \in S$ such that $V(s, \text{Casablanca})$. + +b. $\forall s \in S, V(s, \text{Star Wars})$. + +c. $\forall s \in S, \exists m \in M \text{ such that } V(s, m)$. + +d. $\exists m \in M \text{ such that } \forall s \in S, V(s, m)$. + +e. +$\exists s \in S, \exists t \in S, \text{ and } \exists m \in M \text{ such that } s \neq t \text{ and } V(s, m) \wedge V(t, m)$. + +f. +$\exists s \in S \text{ and } \exists t \in S \text{ such that } s \neq t \text{ and } \forall m \in M, V(s, m) \to V(t, m)$. + +12. Let $D = E = \{-2, -1, 0, 1, 2\}$. Write negations for each of the following + statements and determine which is true, the given statement or its negation. + +a. $\forall x$ in $D$, $\exists y$ such that $x + y = 1$. + +b. $\exists x$ in $D$ such that $\forall y$ in $E$, $x + y = -y$. + +c. $\forall x$ in $D$, $\exists y$ in $E$ such that $xy \geq y$. + +d. $\exists x$ in $D$ such that $\forall y$ in $E$, $x \leq y$. + +In each of 13-19, (a) rewrite the statement in English without using the symbol +$\forall$ or $\exists$ or variables and expressing your answer as simply as +possible, and (b) write a negation for the statement. + +13. $\forall$ color $C$, $\exists$ an animal $A$ such that $A$ is colored $C$. + +14. $\exists$ a book $b$ such that $\forall$ person $p$, $p$ has read $b$. + +15. $\forall$ odd integer $n$, $\exists$ an integer $k$ such that $n = 2k + 1$. + +16. $\exists$ a real number $u$ such that $\forall$ real number $v$, $uv = v$. + +17. $\forall r \in \mathbb{Q}$, $\exists$ integers $a$ and $b$ such that + $r = \dfrac{a}{b}$. + +18. $\forall x \in \mathbb{R}$, $\exists$ a real number $y$ such that + $x + y = 0$. + +19. $\exists x \in \mathbb{R}$ such that for every real number $y$, $x + y = 0$. + +20. Recall that reversing the order of the quantifiers in a statement with two + different quantifiers may change the truth value of the statement - but it + does not necessarily do so. All the statements in the pairs below refer to + the Tarski world of Figure 3.3.1. In each pair, the order of the quantifiers + is reversed but everything else is the same. For each pair, determine + whether the statements have the same or opposite truth values. Justify your + answers. + +a. + +(1) For every square $y$ there is a triangle $x$ such that $x$ and $y$ have +different colors. + +(2) There is a triangle $x$ such that for every square $y$, $x$, and $y$ have +different colors. + +b. + +(1) For every circle $y$ there is a square $x$ such that $x$ and $y$ have the +same color. + +(2) There is a square $x$ such that for every circle $y$, $x$ and $y$ have the +same color. + +21. For each of the following equations, determine which of the following + statements are true: + +(1) For every real number $x$, there exists a real number $y$ such that the +equation is true. + +(2) There exists a real number $x$, such that for every real number $y$, the +equation is true. + +Note that it is possible for both statements to be true or for both to be false. + +a. $2x + y = 7$ + +b. $y + x = x + y$ + +c. $x^2 - 2xy + y^2 = 0$ + +d. $(x - 5)(y - 1) = 0$ + +e. $x^2 + y^2 = -1$ + +In 22 and 23, rewrite each statement without using variables or the symbol +$\forall$ or $\exists$. Indicate whether the statement is true or false. + +22. + +a. $\forall$ real number $x$, $\exists$ a real number $y$ such that $x + y = 0$. + +b. $\exists$ a real number $y$ such that $\forall$ real number $x$, $x + y = 0$. + +23. + +a. $\forall$ nonzero real number $r$, $\exists$ a real number $s$ such that +$rs = 1$. + +b. $\exists$ a real number $r$ such that $\forall$ nonzero real number $s$, +$rs = 1$. + +24. Use the laws for negating universal and existential statements to derive the + following rules: + +a. +$\neg(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\exists y \in E(\neg P(x, y)))$ + +b. +$\neg(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D(\forall y \in E(\neg P(x, y)))$ + +Each statement in 25-28 refers to the Tarski world of Figure 3.3.1. For each, +(a) determine whether the statement is true or false and justify your answer, +and (b) write a negation for the statement (referring, if you wish, to the +result in exercise 24). + +25. $\forall$ circle $x$ and $\forall$ square $y$, $x$ is above $y$. + +26. $\forall$ circle $x$ and $\forall$ triangle $y$, $x$ is above $y$. + +27. $\exists$ a circle $x$ and $\exists$ a square $y$ such that $x$ is above $y$ + and $x$ and $y$ have different colors. + +28. $\exists$ a triangle $x$ and $\exists$ a square $y$ such that $x$ is above + $y$ and $x$ and $y$ have the same color. + +For each of the statements in 29 and 30, (a) write a new statement by +interchanging the symbols $\forall$ and $\exists$, and (b) state which is true: +the given statement, the version with interchanged quantifiers, neither or both. + +29. $\forall x \in \mathbb{R}, \exists y \in \mathhbb{R}$ such that $x < y$. + +30. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}^{-}$ (the set + of negative real numbers), $x > y$. + +31. Consider the statement "Everybody is older than somebody." Rewrite this + statement in the form "$\forall$ people $x$, $\exists$ ______." + +32. Consider the statement "Somebody is older than everybody." Rewrite this + statement in the form "$\exists$ a person $x$ such that $\forall$ ______." + +In 33-39, (a) rewrite the statement formally using quantifiers and variables, +and (b) write a negation for the statement. + +33. Everybody loves somebody. + +34. Somebody loves everybody. + +35. Everybody trusts somebody. + +36. Somebody trusts everybody. + +37. Any even integer equals twice some integer. + +38. Every action has an equal and opposite reaction. + +39. There is a program that gives the correct answer to every question that is + posed to it. + +40. In informal speech most sentences of the form "There is ______ every ______" + are intended to be understood as meaning "$\forall$ ______ $\exists$ + ______," even though the existential quantifier _there is_ comes before the + universal quantifier _every_. Note that this interpretation applies to the + following well-known sentences. Rewrite them using quantifiers and + variables. + +a. There is a sucker born every minute. + +b. There is a time for every purpose under heaven. + +41. Indicate which of the following statements are true and which are false. + Justify your answers as best you can. + +a. $\forall x \in \mathbb{Z}^{+}, \exists y \in \mathbb{Z}^{+}$ such that +$x = y + 1$. + +b. $\forall x \in \mathbb{Z}, \exists y \in \mathbb{Z}$ such that $x = y + 1$. + +c. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x = y + 1$ . + +d. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that +$xy = 1$. + +e. $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ such that $xy = 1$. + +f. $\exists x \in \mathbb{R}$ such that $\forall y \in \mathbb{R}, x + y = y$. + +g. $\forall x \in \mathbb{R}^{+}, \exists y \in \mathbb{R}^{+}$ such that +$y < x$. + +h. $\exists x \in \mathbb{R}^{+}$ such that +$\forall y \in \mathbb{R}^{+}, x \leq y$. + +42. Write the negation of the definition of limit of a sequence given in Example + 3.3.7. + +43. The following is the definition for $\lim\limits_{x \to a}f(x) = L$: + +For every real number $\varepsilon > 0$, there exists a real number $\delta > 0$ +such that for every real number $x$, if $a - \delta < x < a + \delta$ and +$x \neq a$ then + +$$ L - \varepsilon < f(x) < L + \varepsilon $$ + +Write what it means for $\lim\limits_{x \to a}f(x) \neq L$. In other words, +write the negation of the definition. + +44. The notation $\exists !$ stands for the words "there exists a unique." Thus, + for instance, "$\exists ! x$ such that $x$ is prime and $x$ is even" means + that there is one and only one even prime number. Which of the following + statements are true and which are false? + +a. $\exists !$ real number $x$ such that $\forall$ real number $y$, $xy = y$. + +b. $\exists !$ integer $x$ such that $\dfrac{1}{x}$ is an integer. + +c. $\forall$ real number $x$, $\exists !$ real number $y$ such that $x + y = 0$. + +45. Suppose that $P(x)$ is a predicate and $D$ is the domain of $x$. Rewrite the + statement "$\exists ! x \in D \text{ such that } P(x)$" without using the + symbol $\exists !$. (See exercise 44 for the meaning of $\exists !$.) + +In 46-54, refer to the Tarski world given in Figure 3.1.1, which is shown again +here for reference. The domains of all variables consist of all the objects in +the Tarski world. For each statement, (a) indicate whether the statement is true +or false and justify your answer, (b) write the given statement using the formal +logical notation illustrated in Example 3.3.10, and (c) write a negation for the +given statement using the formal logical notation of Example 3.3.10. + +46. There is a triangle $x$ such that for every square $y$, $x$ is above $y$. + +47. There is a triangle $x$ such that for every circle $y$, $x$ is above $y$. + +48. For every circle $x$, there is a square $y$ such that $y$ is to the right of + $x$. + +49. For every object $x$, if $x$ is a circle then there is a square $y$ such + that $y$ has the same color as $x$. + +50. For every object $x$, if $x$ is a triangle then there is a square $y$ such + that $y$ is below $x$. + +51. There is a square $x$ such that for every triangle $y$, if $y$ is above $x$ + then $y$ has the same color as $x$. + +52. For every circle $x$ and for every triangle $y$, $x$ is to the right of $y$. + +53. There is a circle $x$ and there is a square $y$ such that $x$ and $y$ have + the same color. + +54. There is a circle $x$ and there is a triangle $y$ such that $x$ has the same + color as $y$. + +Let $P(x)$ and $Q(x)$ be predicates and suppose $D$ is the domain of $x$. In +55-58, for the statement forms in each pair, determine whether (a) they have the +same truth value for every choice of $P(x)$, $Q(x)$ and $D$, or (b) there is a +choice of $P(x)$, $Q(x)$, and $D$ for which they have opposite truth values. + +55. $\forall x \in D, (P(x) \wedge Q(x)) \text{ and } (\forall x \in D, P(x)) \wedge (\forall x \in D, Q(x))$ + +56. $\exists x \in D, (P(x) \wedge Q(x)) \text{ and } (\exists x \in D, P(x)) \wedge (\exists x \in D, Q(x))$ + +57. $\forall x \in D, (P(x) \vee Q(x)) \text{ and } (\forall x \in D, P(x)) \vee (\forall x \in D, Q(x))$ + +58. $\exists x \in D, (P(x) \vee Q(x)) \text{ and } (\exists x \in D, P(x)) \vee (\exists x \in D, Q(x))$ + +In 59-61, find the answers Prolog would give if the following questions were +added to the program given in Example 3.3.11. + +59. + +a. $?\text{isabove}(b_1, w_1)$ + +b. $?\text{color}(X, white)$ + +c. $?\text{isabove}(X, b_3)$ + +60. + +a. $?\text{isabove}(w_1, g)$ + +b. $?\text{color}(w_2, blue)$ + +c. $?\text{isabove}(X, b_1)$ + +61. + +a. $?\text{isabove}(w_2, b_3)$ + +b. $?\text{color}(X, gray)$ + +c. $?\text{isabove}(g, X)$ diff --git a/chapter_3/test_yourself.md b/chapter_3/test_yourself.md index df3f045..77de8ea 100644 --- a/chapter_3/test_yourself.md +++ b/chapter_3/test_yourself.md @@ -63,3 +63,41 @@ property $P$." "For every $x$, if $x$ does not have property $P$, then $x$ does not have property $Q$." + +--- + +**Test Yourself** + +Page 165 + +1. To establish the truth of a statement of the form + "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)$," + you imagine that someone has given you an element $x$ from $D$ but that you + have no control over what that element is. Then you need to find _______ with + the property that the $x$ the person gave you together with the _______ you + subsequently found satisfy _______. + +2. To establish the truth of a statement of the form + "$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$," + you need to find _______ so that no matter what _______ a person might + subsequently give you, _______ will be true. + +3. Consider the statement + "$\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$." + A negation for this statement is "_______." + +4. Consider the statement + "$\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}$." + A negation for this statement is "_______." + +5. Suppose $P(x, y)$ is some property involving $x$ and $y$, and suppose the + statement + "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$" + is true. Then the statement + "$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$" + +a. is true. + +b. is false. + +c. may be true or may be false.