In 46–54, refer to the Tarski world given in

In 59–61, find the answers Prolog would give if the following questions were added to the program given in Example 3.3.11. a. \(\text { ?isabove }\left(w_{2}, b_{3}\right)\) b. ?color(X, gray) c. ?isabove(g, X) Text Transcription: ?isabove (w_2, b_3)

The following statement is true: “\(\forall\) real numbers 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. \(x=10^{10^{10}}\) Text Transcription: forall exists x=10^8 x=10^10^10

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. \(G\left(\frac{1}{2}, \frac{1}{2}\right)\) d. G(?2, 2) Text Transcription: x^2 >y G(1/2, 1/2)

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(Tokyo, Japan) b. P(Athens, Egypt) c. P(Paris, France) d. P(Miami, Brazil)

For all circles x there is a square y such that x and y have the same color.

For all squares x there is a circle y such that x and y have different colors and y is above x.

There is a triangle x such that for all squares y, x is above y.

Problem 9E Let D = E = {?2, ?1, 0, 1, 2}. Explain why the following statements are true. a. ?x in D, ?y in E such that x + y = 0. ________________ b. ?x in D such that ?y in E, x + y = y.

Problem 3E The following statement is true: “? nonzero numbers x, ? 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 = 3/4

There is a triangle x such that for all circles y, y is above x.

This exercise refers to Example 3.3.3. Determine whether each of the following statements is true or false. a. \(\forall\) students S, \(\exists\) a dessert D such that S chose D. b. \(\forall\) students S, \(\exists\) a salad T such that S chose T . c. \(\exists\) a dessert D such that \(\forall\) students S, S chose D. d. \(\exists\) a beverage B such that \(\forall\) students D, D chose B. e. \(\exists\) an item I such that \(\forall\) students S, S did not choose I . f. \(\exists\) a station Z such that \(\forall\) students S, \(\exists\) an item I such that S chose I from Z. Text Transcription: forall exists forall exists exists forall exists forall exists forall exists forall exists

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, Casablanca). b. \(\forall s \in S\), V(s, Star Wars). c. \(\forall s \in S, \exists m \in M\) such that V(s,m). d. \(\exists m \in M \) such that \(\forall s \in S\), V(s,m). e. \(\exists s \in S, \exists t \in S\), and \(\exists m \in M\) such that \(s \neq t\) and V(s,m) \(\wedge\) V(t,m). f. \(\exists s \in S \) and \(\exists t \in S\) such that \(s \neq t \) and \(\forall m \in M\), \(V(s, m) \rightarrow V(t, m)\). Text Transcription: exists s in S forall s in S forall s in S, exists m in M exists m in M forall s in S exists s in S, exists t in S exists m in M s neq t wedge exists s in S exists t in S s neq t forall m in M V(s, m) rightarrow V(t, m)

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. \(\forall\) odd integers n, \(\exists\) an integer k such that n = 2k + 1. Text Transcription: forall exists forall exists

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, \(\exists y\) in E 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 \(x y \geq y\). d. \(\exists x\) in D such that \(\forall y\) in E, \(x \leq y\). Text Transcription: forall x exists y exists x forall y forall x exists y x y geq y exists x forall y 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. \(\exists\) a real number u such that \(\forall\) real numbers v, uv = v. Text Transcription: forall exists exists forall

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. \(\forall\) colors C, \(\exists\) an animal A such that A is colored C. Text Transcription: forall exists forall exists

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. \(\forall r \in \mathbf{Q}\), \(\exists\) integers a and b such that r = a/b. Text Transcription: forall exists forall r in mathbf Q exists

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 on the next page 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 all squares y there is a triangle x such that x and y have different color. (2) There is a triangle x such that for all squares y, x and y have different colors. b. (1) For all circles y there is a square x such that x and y have the same color. (2) There is a square x such that for all circles y, x and y have the same color.

Problem 18E In each of 13–19, (a) rewrite the statement in English without using the symbol ? or ? or variables and expressing your answer as simply as possible, and (b) write a negation for the statement. ?x ? R, ? a real number y such that x + y = 0.

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. \(\exists x \in \mathbf{R}\) such that for all real numbers y, x + y = 0. Text Transcription: forall exists exists x in mathbf R

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. \(\exists\) a book b such that \(\forall\) people p, p has read b. Text Transcription: forall exists exists forall

For each of the following equations, determine which of the following statements are true: (1) For all real numbers x, there exists a real number y such that the equation is true. (2) There exists a real number x, such that for all real numbers 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. \(x^{2}-2 x y+y^{2}=0\) d. (x ? 5)(y ? 1) = 0 e. \(x^{2}+y^{2}=-1\) Text Transcription: x^2 -2x y+y^2 =0 x^2 +y^2 =-1

Problem 22E Rewrite each statement without using variables or the symbol ? or ?. Indicate whether the statement is true or false. Exercise a. ? real numbers x, ? a real number y such that x + y = 0. b. ? a real number y such that ? real numbers x, x + y = 0.

Problem 23E Rewrite each statement without using variables or the symbol ? or ?. Indicate whether the statement is true or false. Exercise a. ? nonzero real numbers r, ? a real number s such that rs = 1. ________________ b. ? a real number r such that ? nonzero real numbers s, rs = 1.

Use the laws for negating universal and existential statements to derive the following rules: a. \(\sim(\forall x \in D(\forall y \in E(P(x, y)))) \equiv \exists x \in D(\exists y \in E(\sim P(x, y)))\) b. \(\sim(\exists x \in D(\exists y \in E(P(x, y)))) \equiv \forall x \in D(\forall y \in E(\sim P(x, y)))\) Text Transcription: sim(forall x in D(forall y in E(P(x, y)))) equiv exists x in D(exists y in E(sim P(x, y))) sim(exists x in D(exists y in E(P(x, y)))) equiv forall x in D(forall y in E(sim 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, (b) write a negation for the statement (referring, if you wish, to the result in exercise 24). \(\forall\) circles x and \(\forall\) squares y, x is above y. Text Transcription: forall forall

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, (b) write a negation for the statement (referring, if you wish, to the result in exercise 24). \(\forall\) circles x and \(\forall\) triangles y, x is above y. Text Transcription: forall forall

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, (b) write a negation for the statement (referring, if you wish, to the result in exercise 24). \(\exists\) a triangle x and \(\exists\) a square y such that x is above y and x and y have the same color. Text transcription: exists exists

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. \(\forall x \in \mathbf{R}\), \(\exists y \in \mathbf{R}\) such that x < y. Text Transcription: forall exists forall x in mathbf R exists y in mathbf R

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. \(\exists x \in \mathbf{R}\) such that \(\forall y \in \mathbf{R}^{-}\) (the set of negative real numbers), x > y. Text Transcription: forall exists exists x in mathbf R forall y in mathbf R^-

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, (b) write a negation for the statement (referring, if you wish, to the result in exercise 24). \(\exists\) a circle x and \(\exists\) a square y such that x is above y and x and y have different colors. Text Transcription: exists exists

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Everybody loves somebody.

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Somebody loves everybody.

Problem 31E Consider the statement “Everybody is older than some body.” Rewrite this statement in the form “? people x, ? _____.”

Problem 32E Consider the statement “Somebody is older than every body.” Rewrite this statement in the form “? a person x such that ? _____.”

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Everybody trusts somebody.

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Somebody trusts everybody.

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Any even integer equals twice some integer.

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. Every action has an equal and opposite reaction.

In 33–39, (a) rewrite the statement formally using quantifiers and variables, and (b) write a negation for the statement. There is a program that gives the correct answer to every question that is posed to it.

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. Text Transcription: forall exists

The following is the definition for \(\lim _{x \rightarrow a} f(x)=L\): For all real numbers \(\varepsilon>0\), there exists a real number \(\delta>0\) such that for all real numbers x, if \(a-\delta<x<a+\delta\) and \(x \neq a\) then \(L-\varepsilon<f(x)<L+\varepsilon\). Write what it means for \(\lim _{x \rightarrow a} f(x) \neq L\). In other words, write the negation of the definition. Text Transcription: lim_x \rightarrow a f(x)=L varepsilon>0 delta>0 a-delta < x < a+delta x neq a L-varepsilon < f(x) < L+varepsilon lim_x rightarrow a f(x) neq L

Problem 41E Indicate which of the following statements are true and which are false. Justify your answers as best you can. a. ?x ? Z+, ?y ? Z+ such that x = y + 1. ________________ b. ?x ? Z, ?y ? Z such that x = y + 1. ________________ c. ?x ? R such that ?y ? R, x = y + 1. ________________ d. ?x ? R+, ?y ? R+ such that xy = 1. ________________ e. ?x ? R, ?y ? R such that xy = 1. ________________ f. ?x ? Z+ and ?y ? Z+, ?z ? Z+ such that z = x ? y. ________________ g. ?x ? Z and ?y ? Z, ?z ? Z such that z = x ? y. h. ?u ? R+ such that ?v ? R+, uv

Problem 42E Write the negation of the definition of limit of a sequence given in Example. Example The Definition of Limit of a Sequence The definition of limit of a sequence, studied in calculus, uses both quantifiers ? and ? and also if-then. We say that the limit of the sequence an as n goes to infinity equals L and write if, and only if, the values of an become arbitrarily close to L as n gets larger and larger without bound. More precisely, this means that given any positive number ?, we can find an integer N such that whenever n is larger than N, the number an sits between L ? ? and L + ? on the number line. Symbolically: ?? > 0, ? an integer N such that ? integers n, if n > N then L ? ? + ?. Considering the logical complexity of this definition, it is no wonder that many students find it hard to understand.

Suppose that P(x) is a predicate and D is the domain of x. Rewrite the statement “\(\exists ! x \in D\) such that P(x)” without using the symbol \(\exists !\). (See exercise 44 for the meaning of \(\exists !\).) Text transcription: exists ! x in D exists !

Problem 44E The notation ?! stands for the words “there exists a unique.” Thus, for instance, “?! 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? Explain. a. ?! real number x such that ? real numbers y, xy = y. ________________ b. ?! integer x such that 1/x is an integer. ________________ c. ? real numbers x, ?! real number y such that x + y = 0.

Problem 46E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. There is a triangle x such that for all squares y, x is above y.

Problem 47E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. There is a triangle x such that for all circles y, x is above y.

Problem 48E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. For all circles x, there is a square y such that y is to the right of x.

Problem 49E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. For every object x, there is an object y such that x ? y and x and y have different colors.

In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. For every object x, there is an object y such that if x \(\neq\) y then x and y have different colors. Text Transcription: neq

Problem 51E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. There is an object y such that for all objects x, if x ? y then x and y have different colors.

In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. For all circles x and for all triangles y, x is to the right of y.

Problem 53E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. There is a circle x and there is a square y such that x and y have the same color.

Problem 54E In 46–54, refer to the Tarski world given in Figure 3.1.1, which is printed 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 the negation of the given statement using the formal logical notation of Example 3.3.10. There is a circle x and there is a triangle y such that x and y have the same color.

Problem 55E 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. ?x ? D, (P(x) ? Q(x)), and (?x ? D, P(x)) ? (?x ? D, Q(x))

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. \(\exists x \in D,(P(x) \wedge Q(x))\), and \((\exists x \in D, P(x)) \wedge(\exists x \in D, Q(x))\) Text Transcription: exists x in D,(P(x) wedge Q(x)) (exists x in D, P(x)) wedge (exists x in D, Q(x))

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. \(\forall x \in D,(P(x) \vee Q(x))\), and \(\forall x \in D, P(x)) \vee(\forall x \in D, Q(x))\) Text Transcription: forall x in D,(P(x) vee Q(x)) forall x in D, P(x)) vee (forall x in D, Q(x))

Problem 58E 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. ?x ? D, (P(x) ? Q(x)), and (?x ? D, P(x)) ? (?x ? 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. a. \(\text { ?isabove }\left(b_{1}, w_{1}\right)\) b. ?color(X, white) c. \(\text { ?isabove }\left(X, b_{3}\right)\) Text Transcription: ?isabove (b_1, w_1) ?isabove (X, b_3)

In 59–61, find the answers Prolog would give if the following questions were added to the program given in Example 3.3.11. a. \(\text { ?isabove }\left(w_{1}, g\right)\) b. \(\text { ?color( } \left.w_{2}, \text { blue }\right)\) c. \(\text { ?isabove }\left(X, b_{1}\right)\) Text Transcription: ?isabove (w_1, g) ?color(w_2, blue) ?isabove (X, b_1)