1.4 - 50 SI. 52. 53. 54. 55. 56. 57. 58. l The Foundations:...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 50 SI. 52. 53. 54. 55. 56. 57. 58. l The Foundations: Logic and Proofs Show that VxPlx) v ‘9’): 9m and when) v thh are not logically equivalent. Show that E|.rP{x) A El.rQ(x) and ittPtx) A Q(x}) are not logically equivalent. As mentioned in the text. the notation Ellx P(.r) denotes “There exists a unique .r such that Plx] is true.” If the domain consists of all integers, what are the truth values of these statements? s) also > 1} b) anti? 2 l] c) El!x(x + 3 m 2):) d) 3!x{x = x +1) What are the truth values of these statements? a) El!.rP{.r'J —; Ele(.r) b) VxPLr) —r Elletx} c) 3!.Y—IP(X} —> —»V.tP[x) Write out E|!x Pfx ]. where the domain consists of the inte- gers l. 2. and 3. in terms of negations. conjunctions, and disjunctions. Given the Prolog facts in Example 28. what would Prolog return given these queries? 3) ?irtstructor l Chart . math273l b) ?instructortpatel,cs301l c) ?enrolledlx,cs301} d} ?enrolled(kiko.Yl e) ?teacheslgrossman,¥l Given the Prolog facts in Example 28. what would Prolog return when given these queries? 3) ?enrolledlkevin.ee2221 b) ?enrolledtkiko.math273l c} ?instructor lgrossmart . K} d} ?instructorlx,cs301) e) ?teachestx,kevinl Suppose that Prolog facts are used to define the predicates mother(M, Y) and farherl F . X ), which represent that M is the mother of Y and F is the father of X . respectively. Give a Prolog rule to define the predicate sibling{X. Y), which represents that X and l’ are siblings (that is, have the same mother and the same father). Suppose that Prolog facts are used to define the pred- icates mother(M_. Y) and fbthertF. X "J, which repre- sent that M is the mother of Y and F is the father of X , respectively. Give a Prolog rule to define the predicate gmndfarheflX , Y}, which represents that X is the grand- father of Y. [Hints You can write a disjunction in Prolog 1.4 Nested Quantifiers E Introduction i —,'i U either by using a semicolon to separate predicates or by putting these predicates on separate lines.] Exercises 59—62 are based on questions found in the book Symbolic Logic by Lewis Carroll. 59. 60. 61 62. Let PU}, le ), and Rtx) be the statements “.t is a profes- sor," “.r is ignorant," and “x is vain.” respectively. Express each of these statements using quantifiers: logical con— nectives; and PM}. le}. and Rlx). where the domain consists of all people. a) No professors are ignorant. b) All ignorant people are vain. c) No professors are vain. d) Does (cl follow from (a) and (bl? Let PU), Q{.r), and R(x} be the statements “.r is a clear explanation.“ “x is satisfactory.” and “.r is an excuse." respectively. Suppose that the domain for .r consists of all English text. Express each ofthese statements using quan— tifiers. logical connectives, and PU), th). and Rtxl. a) All clear explanations are satisfactory. b) Some excuses are unsatisfactory. e) Some excuses are not clear explanations. it‘d) Does (cl follow from (a) and {b}? . Let Plx), th}, Rtx}, and S(.r} be the statements “x is a baby," “.r is logical,“ "x is able to manage a crocodile." and “x is despised." respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and Pix], QM), RU}. and 50:). a) Babiesareillogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. ti) Babies cannot manage crocodiles. fie) Does (d) follow from (a). lb], and {c}? lfnot. is there a correct conclusion? Let P(.r). Q{.t). Rlxl, and 3(x} be the statements “.1' is a duck," “x is one of my poultry,” “x is an officer,” and “x is willing to waltz,” respectively. Express each of these statements using quantifiers; logical connectives; and Phi). Q(.r), Rtx}, and S(.r}. a) No ducks are willing to waltz. b) No officers ever decline to waltz. c) All my poultry are ducks. (1) My poultry are not officers. are) Does (d) follow from (a). (b), and [c]? If not. is there a correct conclusion? In Section 1.3 we defined the existential and universal quantifiers and showed how they can be used to represent mathematical statements. We also explained how they can be used to translate l-Fl’ EXAMPLE 1 Extra Examines " EXAMPLE 2 1.4 Nested Quantifiers 51 English sentences into logical expressions. In this section we will study nested quantifiers. Two quantifiers are nested if one is within the scope of the other, such as Vx3y(x +y = 0). Note that everything within the scope of a quantifier can be thought of as a propositional function. For example, VxEIy{x +y = 0) is the same thing as VxQ(x)_. where Q(x_) is Hyflx. y), where P(x. y) is x +y = O. Nested quantifiers commonly occur in mathematics and computer science. Although nested quantifiers can sometimes be difficult to understand, the rules we have already studied in Section 1.3 can help us use them. To understand these statements involving many quantifiers, we need to unravel what the quantifiers and predicates that appear mean. This is illustrated in Examples 1 and 2. Assume that the domain for the variables x and y consists of all real numbers. The statement VxVy(x + y = y + x) says that x + y = y + x for all real numbers x and y. This is the commutative law for addition of real numbers. Likewise, the statement Vxflflx + y = 0) says that for every real number .1: there is a real number y such that .r + y = 0. This states that every real number has an additive inverse. Similarly, the statement VxVsz{x + ("v + 2) = (x + y) + z) is the associative law for addition of real numbers. { Translate into English the statement VxVy((x s 0) A (y a. 0) —> (xy < 0)}. where the domain for both variables consists of all real numbers. Solution: This statement says that for every real number .1: and for every real number y, if x > O and y < 0. then xy < 0. That is, this statement says that for real numbers it and y, if .r is positive and y is negative, then xy is negative. This can be stated more succinctly as “The product of a positive real number and a negative real number is always a negative real number.” 4 THINKING OF QUANTIFICATION AS LOOPS In working with quantifications of more than one variable, it is sometimes helpful to think in terms of nested loops. (Of course, if there are infinitely many elements in the domain of some variable, we cannot actually loop through all values. Nevertheless, this way of thinking is helpful in understanding nested quantifiers.) For example, to see whether VxVyP(x. y) is true. we loop through the values for x, and for each x we loop through the values for ,v. If we find that Ptx, y) is true for all values for x and y, we have determined that V_rV_vP(.r. y) is true. If we ever hit a value x for which we hit a value y for which P(x. y) is false, we have shown that VxVyP(x. y) is false. Similarly, to determine whether VxEIyP(x, y) is true, we loop through the values for x. For each x we loop through the values for __v until we find a y for which P(x. y) is true. If for 52 l t The Foundations: Logic and Proofs EXAMPLE 3 [Film Ettamntes " EXAMPLE 4 1—53 every I we hit such a y, then VxEIyP(x, y) is true; if for some .r we never hit such a y, then Wily P (x. y} is false. To see whether 3xVyP(_x, y) is true, we loop through the values for 1: until we find an x for which P{'.\'. y) is always true when we loop through all values for y. Once we find such an x, we know that Eley P (x. y) is true. If we never hit such an x, then we know that itVyPLr. y) is false. Finally, to see whether SxElyPLr. y) is true. we loop through the values for x, where for each .r we loop through the values for y until we hit an x for which we hit a y for which P(x, y) is true. The statement 3x3yP(x, y) is false only if we never hit an x for which we hit a y such that P{.r_. y} is true. The Order of Quantifiers Many mathematical statements involve multiple quantifications of propositional functions in» volving more than one variable. [t is important to note that the order of the quantifiers is important. unless all the quantifiers are universal quantifiers or all are existential quantifiers. These remarks are illustrated by Examples 3—5. Let P(:c. y) be the statement “at + y = y + .1: What are the truth values of the quantifications VxVyPix. y) and V‘vi P{x. y) where the domain for all variables consists of all real numbers? Sohttfon: The quantification VxV_t:P(.r. y) denotes the proposition “For all real numbers x, for all real numbers y. x + y = y + x.” Because P(.r, y) is true for all real numbers 3: and y (it is the commutative law for addition, which is an axiom for the real numbers—see Appendix 1), the proposition VxVItrPlx. y) is true. Note that the statement V’ny P(.r. Iv) says “For all real numbers y, for all real numbers x. x + y = y + x.” This has the same meaning as the statement as “For all real numbers x. for all realnumbersyu +y = y + x.”Thatis,VxVyP(x. y} and VyV.rP(x, y) have the same meaning, and both are true. This illustrates the principle that the order of nested universal quantifiers in a statement without other quantifiers can be changed without changing the meaning of the quantified statement. 4 Let Q(_r, y) denote “x + y = 0.“ What are the truth values of the quantifications 5|ny Q(.r. y) and V.r3yQ{x. y}, where the domain for all variables consists of all real numbers? Solution: The quantification Elny th. __t-'] denotes the proposition “There is a real number y such that for every real number x, Q(x. y).” No matter what value ofy is chosen, there is only one value ofx for which .1: + y = 0. Because there is no real number y such that x + y = 0 for all real numbers 1', the statement 1vi Q(x. y) is false. )‘ —53 EXAMPLE 5 1.4 Nested Quantifiers 53 TABLE 1 Quantifieations of Two Variables. VxVyPLt. y) P(x. y} is true for every pairx. y. There is a pair x. y for which P(x. y) is false. There is an .t such that P(x. y} is false for every _v. For every x there is a y for which PLr. y} is false. Ptx. y) is false for every pair I. ,v. VJ-'V.t P (x . y) For every .r there is a ,v for VxElyPtx. y} which P(.r. y) is true. EleyP(x. y) There is an x for which Ptx. y] is true for every y. HxElyP{x. y) EIyEIx P(x. y) The quantification There is a pair x. y for which P(x. Iv) is true. Vx El y Q {.1' , y) denotes the proposition “For every real number x there is a real number y such that Q(x. y)“ Given a real number x, there is a real number y such that x + y = 0; namely, y = —x. Hence, the statement VxElyth. y) is true. { Example 4 illustrates that the order in which quantifiers appear makes a difference. The state- ments 5|ny P (x. y) and Vx By P (x. y) are not logically equivalent. The statement 3’ny P (I. y) is true if and only if there is a y that makes P(x. y) true for every 3:. So, for this statement to be true, there must be a particular value of y for which P (x, y) is true regardless of the choice of x. On the other hand, VinyPUc. y) is true ifand only iffor every value ofx there is a value ofy for which P(x. y) is true. So, for this statement to be true, no matter which x you choose, there must be a value of y (possibly depending on the x you choose) for which P(x. y) is true. In other words, in the second case, y can depend on x, whereas in the first case, y is a constant independent of it. From these observations, it follows that if ELvi P(x. y) is true, then VxEIyP(x. y) must also be true. However, if VxEIyP(x, y) is true, it is not necessary for EnyPtx, y) to be true. (See Supplementary Exercises 24 and 25 at the end of this chapter.) Table 1 summarizes the meanings of the different possible quantifications involving two variables. Quantifications of more than two variables are also common, as Example 5 illustrates. Let Q(x. y. 2) be the statement “x + y = 2.” What are the truth values of the statements VxVyEIzQ(x.y, z) and ElexVyQ(_x._v.z), where the domain of all variables consists of all real numbers? Solution .' Suppose that x and y are assigned values. Then, there exists a real number 2 such that x + y = 2. Consequently, the quantification VxVyEIzQ{x,y.z). 54 l I The Foundations: Logic and Proofs EXAMPLE 6 mm _\ minutes *' EXAMPLE 7 which is the statement “For all real numbers it and for all real numbers y there is a real number 2 such that x + y = z,” is true. The order of the quantification here is important, because the quantification ElexVyQ(x, y, 2), which is the statement “There is a real number 2 such that for all real numbers .1: and for all real numbers y it is true that x + y = z,” is false, because there is no value of 2 that satisfies the equation x + y = z for all values of x and y. 4 Translating Mathematical Statements into Statements Involving Nested Quantifiers Mathematical statements expressed in English can be translated into logical expressions as Examples 6~8 show. Translate the statement “The sum of two positive integers is always positive" into a logical expression. Sciatica: To tranSIate this statement into a logical expression, we first rewrite it so that the implied quantifiers and a domain are shown: “For every two integers, if these integers are both positive, then the sum of these integers is positive.“ Next, we introduce the variables x and '1' to obtain “For all positive integers x and y, x + y is positive." Consequently, we can express this statement as VxVy((x > 0) A (y r; 0) —> {x + y > 0)). where the domain for both variables consists of all integers. Note that we could also translate this using the positive integers as the domain. Then the statement “The sum of two positive integers is always positive” becomes “For every two positive integers, the sum of these integers is positive. We can express this as VxVy(x + y > 0), where the domain for both variables consists of all positive integers. { Translate the statement “Every real number except zero has a multiplicative inverse.” (A mul— tiplicative inverse of a real number x is a real number y such that xy = 1.) Solution: We first rewrite this as “For every real number x except zero, x has a multiplicative inverse.” We can rewrite this as “For every real number x, if x 75 0, then there exists a real number y such that xy = 1.” This can be rewritten as Vxflx 5e 0) —+ Elytxy = 1)). <l One example that you may be familiar with is the concept of limit, which is important in calculus. EXAMPLE 8 EXAMPLE 9 EXAMPLE 10 1.4 Nested Quantifiers 55 (Requires caicuius) Express the definition of a limit using quantifiers. Solution: Recall that the definition of the statement lim f(x) = L Nat? is: For every real number 6 > 0 there exists a real number 6 > 0 such that |f{.t) — L| c 6 whenever 0 < Ix — at < 8. This definition ofa limit can be phrased in terms of quantifiers by VEEIdew < Ix ~ al < 5 —+ [f(.r) — L| 4 E). where the damain for the variables 8 and 6 consists of all positive real numbers and for it consists of all real numbers. This definition can also be expressed as V€>036>0V.r(0 < Ix—al <5—+ |f{x)—L| <6) when the domain for the variables 6 and 6 consists of all real numbers, rather than just the positive real numbers. [Here restricted quantifiers have been used. Recall that Vx; {i P(x) means that for all x with x> 0, Put) is true] { Translating from Nested Quantifiers into English Expressions with nested quantifiers expressing statements in English can be quite complicated. The first step in translating such an expression is to write out what the quantifiers and predicates in the expression mean. The next step is to express this meaning in a simpler sentence. This process is illustrated in Examples 9 and [0. Translate the statement Vx(C{x) V El_v{_C(y) A F(x. y))) into English, where C (x) is “x has a computer," F (x. y) is “x and y are friends,“ and the domain for both x and y consists of all students in your school. Solution: The statement says that for every student x in your school, it has a computer or there is a student y such that y has a computer and x and y are friends. In other words, every student in your school has a computer or has a friend who has a computer. { Translate the statement BxVsz({F(x.y) A For. 2) A (y as 2}} —> fiFU‘ZD into English, where F ((3.27) means a and b are friends and the domain for x, y, and 2 consists of all students in your school. Solution: We first examine the expression (Rx. y) A F(x, 2} A 0/ ¢ 2]) —> fiFLy. z). This expression says that if students it and y are friends, and students it and z are friends, and further- more, if y and z are not the same student, then y and z are not friends. It follows that the original statement. which is triply quantified, says that there is a student at such that for all students y and all students 2 other than y, if x and y are friends and x and z are friends, then y and z are not friends. In other words, there is a student none of whose friends are also friends with each other. 4 56 | The Foundations: Logic and Proofs EXAMPLE 11 EXAMPLE 12 EXAMPLE 13 Translating English Sentences into Logical Expressions In Section 1.3 we showed how quantifiers can be used to translate sentences into logical expres- sions. However. we avoided sentences whose translation into logical expressions required the use of nested quantifiers. We now address the translation of such sentences. Express the statement “If a person is female and is a parent, then this person is someone’s mother” as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives. Solution: The statement “If a person is female and is a parent, then this person is someone’s mother” can he expressed as “For every person .r. if person .r is female and person .1: is a parent. then there exists a person Av such that person 1' is the mother of person y." We introduce the propositional functions F (x) to represent "3: is female,” P (x) to represent “.1: is a parent,” and Mtr. y) to represent “x is the mother of); The original statement can be represented as V.\‘((F(x) A P[x)_} —» ElyM(x.y)). Using the null quantification rule in part (b) of Exercise 4? in SectiOn 1.3, we can move Ely to the left so that it appears just after Vx, because y does not appear in F (x) A P(x). We obtain the logically equivalent expression V.rEly({F(x) A P{.r)) —> M(x. '12)). ‘ Express the statement “Everyone has exactly one best friend" as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives. Solution: The statement “Everyone has exactly one best friend” can be expressed as “For every person .r, person .1' has exactly one best friend.” Introducing the universal quantifier. we see that this statement is the same as “W: (person it has exactly one best fi'iend).” where the domain consists of all people. To say that x has exactly one best friend means that there is a person y who is the best friend of x, and furthermore, that for every person 2, if person 2 is not person in then 2 is not the best friend of x. When we introduce the predicate B(.r, y) to be the statement “y is the best friend of x," the statement that x has exactly one best friend can be represented as 3y(B(X« .v) A Vzttz as y) -> —-B(x. 2D)- Consequently, our original statement can be expressed as VxEI,v(B(x- r) A Vzttz s9 y) 4+ nBU. 2D)- [Note that we can write this statement as VxHIyB (x. y), where El! is the “uniqueness quantifier" defined on page 31] 4 Use quantifiers to express the statement “There is a woman who has taken a flight on every airline in the world." RSSGSSITIEIII EXAMPLE 14 Extra Examples EXAMPLE 15 EXAMPLE 16 I .4 Nested Quantifiers 57 Solution: Let P(w, f) be “w has taken f" and Q(f. o) be “f is a flight on a.” We can express the statement as EleoElf(P(w. f) A QU; 0)). where the domains of discourse for w. f, and o consist of all the women in the world, all airplane flights, and all airlines, respectively. The statement could also be expressed as 3ththR(w. f. a). where R(w. f. a) is “w has taken f on a Although this is more compact, it somewhat obscures the relationships among the variables. Consequently, the first solutiOn is usually preferable. { Negating Nested Quantifiers Statements involving nested quantifiers can be negated by successively applying the rules for negating statements involving a single quantifier. This is illustrated in Examples 14—16. Express the negation of the statement V): Ely{xy = 1) so that no negation precedes a quantifier. Solution: By successively applying De Morgan ’s laws for quantifiers in Table 2 of Section 1.3, we can move the negation in -Vx Ely(x_v = l ) inside all the quantifiers. We find that —-Vx E] y(x y = l) is equivalent to Elx-Ely{xy : 1), which is equivalent to Eley-{xy = 1}. Because -1(xy = l)can be expressed more simply as xy 73 l, we conclude that our negated statement can be expressed as 3xVy(xy a 1). < Use quantifiers to express the statement that “There does not exist a woman who has taken a flight on every airline in the world." Solution: This statement is the negation of the statement “There is a woman who has taken a flight on every airline in the world” from Example 13. By Example 13, our statement can be expressed as -:E|onEIf(P{w. f) A Q{f, 0)), where P(w. f) is “w has taken f” and Q(f. a) is “f is a flight on a." By successively applying De Morgan’s laws for quantifiers in Table 2 of Section 1.3 to move the negation inside successive quantifiers and by applying De Morgan’s law for negating a conjunction in the last step, we find that our statement is equivalent to each of this sequence of statements: Vw—VaEI.f(P(w, f) A 902 an E Vw-Baeaflfiw. f) A Qtf. aJ) E Vwaan—(th, f) A Qtf. 4)) E VwEIanf-IPW. f) v —Q(f, an. This last statement states “For every woman there is an airline such that for all flights, this woman has not taken that flight or that flight is not on this airline.” ‘ (Requires coicultts} Use quantifiers and predicates to express the fact that him—m flx} does not exist. Solution: To say that limxua fix) does not exist means that for all real numbers L, limxua f{x) 7E L. By using Example 8. the statement limp.“ fur) are L can be expressed as —-Ve>0§l§>0Vx(0 < I): —al 4 6 ——> lf(x)— Ll <: e). i 0'8 Successiver applying the rules for negating quantified expressions, we construct this sequence E? EIE}O"'35>UVX(0<II —al{6 —> |f{_t'}— Llee} E 36>0V5DU—IVX(O~<II -a|<6 —> [fix] # L|<EJ EEE>OV5>03x m(0€|x —a[<8 —) |f(x)—L|<€) In the last step we used the equivalence —{p —+ q} E p A -vq, which follows from the fifth Because the statement “limp”, f (x) does not exist" means for all real numbers L, This last statement says that for every real number L there is a real number 6 > 0 such that for every real number 6 > 0, there exists a real number .r such that 0 < |.r — al < 5 and 4 58 l I The Foundations: Logic and Proofs of equivalent statements —-V£>O 36>0Vx(0<]x —a| <5 —+ |f(x) H Ll <6} 2 3€)0V6}0 Elx(0< |x — a| <5 A |f(x} — Ll 26). equivalence in Table 7 of Section 1.2. limxaa flx) 7e L. this can be expressed as V1.36 >0V6‘90 3x(0 < lx — al «C 5 A |f(x) -— LI 3 6). If (X) - LI 2 e. Exercises _————_—————_— . Translate these statements into English, where the domain for each variable consists of all real numbers. a} Virille < y) b) VxVytux a 0) A {y 3 0)) —> {.ry 2 0)} c) VxV_vE!z(_t-y = z) . Translate these statements into English, where the domain for each variable consists of all real numbers. 8) itVflxy : .vl b) V.\'V_v(([x 2 0) A {y < 0)} —a- (x - y 3' 0)} c) VxVyEler 2 y + 2) Let (30:. y) be the statement “x has sent an e-mail mesA sage to y." where the domain for both .1: and y consists of all students in your class. Express each of these quantifi- cations in English. a) 3x3yQ(x._t-'} c) V.tE|yQ(x.)-‘l e) VnyQ(x.y) b) E|.rV_yQ{x.y) d) Elny Q(.t.y) f) VxVyQ(x_.y} . Let Pony} be the statement “student x has taken class y,” where the domain forx consists of all students in your class and for y consists of all computer science courses at your school. Express each of these quantificatiOns in English. a) 3x3yP(x,y) c) Vx3yP{x.y) e) Vyflx PU. y) b) Elx‘v’yPLLy} d) Ely V): P(x, y) f] V.\"V_vP(x.y) . Let WU. y) mean that student I has visited website y, where the domain for .1: consists of all students in your school and the domain for y consists of all websites. Express each of these statements by a simple English sentence. a) W(Sarah Smith, www.att.corn) b) 3xW(x, www.imdb.org) e) ElleJose Orez, y) d) 3,v(W(Ashok Puri, y) A WtCindy Yoon, _1-‘]) e) EIsz(y ¢ (David Belcher) A (W(David Belcher. z) —* Wty.2})) f) Elx3sz{(.t aé y) A (ll/(x. 2} H WU). z})) . Let C(x. y) mean that student I is enrolled in class y, where the domain for 1' consists of all students in your school and the domain for y consists of all classes being given at your school. Express each of these statements by a simple English sentence. a) C (Randy Goldberg, CS 252) b) 3xC(x, Math 695} c) HyC (Carol Sitea, y) d) 3x(C[.r, Math 222) A C(x, CS 252)} 9) 3x3}-'V—’{(x as y) A (Clix-7) -> CU" zll} f) irElszflx 7E y) A {C(x, 2] H C(y. 2D} . Let T(x. y) mean that student x likes cuisine y. where the domain for it consists of all students at your school and the domain for y consists of all cuisines. Express each of these statements by a simple English sentence. a) -=T[Abdallah Hussein, Japanese) b) irTLr, Korean} A V.tT{.t-. Mexican) 3-59 II]. I]. c) 3y(T(Moniquc Arsenault, y) V TtJay Johnson. y)) d) VszHyUx ¢ 2) —) -|(T(x. y) A T(z._v})} e) EIxEIsztTtx. v) <—> th. y)_] f) VIVZEIytTtx. y) H T(z. y}) Let Q(x._v} be the statement “student x has been a contestant on quiz show y." Express each of these sentences in terms of Q (x. y), quantifiers, and logical con- nectives, where the damain for it consists of all students at your school and for y consists of all quiz shows on television. 3) There is a student at your school who has been a con- testant on a television quiz show. [1) No student at your school has ever been a contestant on a television quiz show. c) There is a student at your school who Itas been a con- testant on Jeopardy and on Wheel ofFor-nme. d) Every television quiz show has had a student from your school as a contestant. e) At least two students from your school have been con- testants on Jeopardy. . Let L(.\'. y} be the statement “I loves ,v.“ where the do— main for both .1; and y consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody loves Jerry. [1) Everybody loves somebody. c) There is somebody whom everybody loves. d) Nobody loves everybody. e) There is somebody whom Lydia does not love. f) There is somebody whom no one loves. g) There is exactly one person whom everybody loves. h) There are exactly two people whom Lynn loves. E) Everyone loves himself or herself. j) There is someone who loves no one besides himself or herself. Let F(.t. y) be the statement “x can fool y." where the do- main consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody can fool Fred. b) Evelyn can fool everybody. c) Everybody can fool somebody. (1) There is no one who can fool everybody. e) Everyone can be fooled by somebody. i) No one can fool both Fred and Jerry. g) Nancy can fool exactly two people. b) There is exactly one person whom everybody can fool. i) No one can fool himself or herself. j) There is someone who can fool exactly one person besides himself or herself. Let S (x) be the predicate “x is a student." F (x) the pred— icate “x is a faculty member,” and A(x. y) the predicate “x has asked y a question,” where the domain consists of all people associated with your school. Use quantifiers to express each ofthese statements. a) Lois has asked Professor Michaels a question. b) Every student has asked Professor Gross a question. 12. 13. 1.4 Nested Quantifiers 59 c) Every faculty member has either asked Professor Miller a question or been asked a question by Profes- sor Miller. d) Some student has not asked any faculty member a question. c) There is a faculty member who has never been asked a question by a student. f) Some student has asked every faculty member a question. g) There is a faculty member who has asked every other faculty member a question. h) Some student has never been asked a question by a faculty member. Let I (x) be the statement “x has an Internet connection" and C {.r. y) be the statement “x and y have chatted over the Internet." where the domain for the variables x and y consists of all students in your class. Use quantifiers to express each of these statements. 9) Jerry does not have an Internet connection. 11) Rachel has not chatted over the Internet with Chelsea. c) Jan and Sharon have never chatted over the Internet. (1) No one in the class has chatted with Bob. e) Sanjay has chatted with everyone except Joseph. 1) Someone in your class dOes not have an Internet connection. g) Not everyone in your class has an Internet connection. h) Exactly one student in your class has an Internet connection. i) Everyone except one student in your class has an Internet connection. j) Everyone in your class with an Internet connection has chatted over the Internet with at least one other student in your class. It) Someone in your class has an Internet connection but has not chatted with anyone else in your class. I) There are two students in your class who have not chatted with each other over the Internet. In) There is a student in your class who has chatted with everyone in your class over the Internet. 11) There are at least two students in your class who have not chatted with the same person in your class. 0) There are two students in the class who between them have chatted with everyone else in the class. Let Mix. y) be “.r has sent y an e-maii message" and T(x,y) be “x has telephoned y,“ where the domain consists of all students in your class. Use quantifiers to express each of these statements. {Assume that all e~mail messages that were sent are received. which is not the way things often work.) a) Chou has never sent an e-mail message to Koko. b) Arlene has never sent an e-mail message to or tele- phoned Sarah. c) Jose has never received an e-mail message from Deborah. (1) Every student in your class has sent an e-mail mes- sage to Ken. 60 14. 15. l The Foundations: Logic and Proofs e) No one in your class has telephoned Nina. t] Everyone in your class has either telephoned Avi or sent him an e-mail message. g) There is a student in your class who has sent everyone else in your class an e-mail message. b) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class. i) There are two different students in your class who have sent each other e-maii messages. j) There is a student who has sent himself or herself an e-mail message. k) There is a student in your class who has not received an e—mail message from anyone else in the class and who has not been called by any other student in the class. I) Every student in the ciass has either received an e~mail message or received a telephone call from another student in the class. n1) There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student. it} There are two different students in your ciass who between them have sent an e-mail message to or telephoned everyone else in the class. Use quantifiers and predicates with more than one variable to express these statements. at) There is a student in this ciass who can speak Hindi. b) Every student in this class plays some sport. c) Some student in this class has visited Alaska but has not visited Hawaii. d) All students in this ciass have learned at least one programming language. e) There is a student in this class who has taken ev- ery course offered by one of the departments in this school. i] Some student in this class grew up in the same town as exactly one other student in this ciass. g) Every student in this class has chatted with at ieast one other student in at least one chat group. Use quantifiers and predicates with more than one variable to express these statements. a} Every computer science student needs a course in discrete mathematics. b) There is a student in this ciass who owns a personal computer. c) Every student in this class has taken at least one computer science course. d) There is a student in this class who has taken at least one course in computer science. e) Every student in this class has been in every building on CEII'DPUS. ft There is a student in this class who has been in every room of at least one building on campus. g) Every student in this 13135:; has been in at least one room ofevery building on campus. 16. 17. l' —6() A discrete mathematics class contains 1 mathematics major who is a freshman. l2 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors. 2 computer science majors who arejuniors. and 1 computer science major who is a senior. Express each of these state- ments in terms of quantifiers and then determine its truth value. a} There is a student in the class who is a junior. h) Every student in the class is a computer science major. c) There is a student in the class who is neither a math— ematics major not a junior. d) Every student in the class is either a sophomore or a computer science major. c) There is a major such that there is a student in the class in every year of study with that major. Express each of these system specifications using predi- cates, quantifiers, and logical connectives. if necessary. a) Every user has access to exactly one mailbox. b) There is a process that continues to run during all error conditions only if the kernel is working correctly. c} All users on the campus network can access all web- sites whose url has a .edu extension. *d) There are exactiy two systems that monitor every 18. 19. 20. remote server. Express each of these system specifications using predi- cates, quantifiers, and logical connectives, if necessary. a) At least one console must be accessible during every fault condition. b) The c~mail address of every user can be retrieved whenever the archive contains at least one message sent by every user on the system. c) For every security breach there is at least one mecha- nism that can detect that breach if and only if there is a process that has not been compromised. C!) There are at least two paths connecting every two distinct endpoints on the network. e) No one knows the password of every user on the sys- tem except for the system administrator, who knows all passwords. Express each of these statements using mathematical and logical operators, predicates, attd quantifiers. where the domain consists of all integers. a) The sum of two negative integers is negative. b) The difference of two positive integers is not neces- sarily positive. c) The sum of the squares of two integers is greater than or equal to the square of their sum. d) The absolute value of the product of two integers is the product of their absolute values. Express each of these statements using predicates. quan- tifiers, logical connectives, and mathematical operators where the domain consists of all integers. a) The product of two negative integers is positive. b) The average of two positive integers is positive. l-r’il 21. 22. 23. 24. 25. 26. 27. 28. e) The difference of two negative integers is not neces- sarily negative. :1) The absolute value of the sum of two integers does not exceed the sum of the absolute values of these integers. Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that every positive integer is the sum of the squares of four integers. Use predicates. quantifiers, logical connectives, and math ematical operators to express the statement that there is a positive integer that is not the sum of three squares. Express each of these mathematical statements using predicates. quantifiers. logical connectives. and mathe- matical operators. 3} The product of two negative real numbers is positive. it) The difference of a real number and itself is zero. e} Every positive real number has exactly two square roots. (1) A negative real number does not have a square root that is a real number. Translate each of these nested quantifications into an En- glish statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a} 3_rVy(.t' + y = y} b) V.tV}-'({(x z 0} A {y < 0}) —) (x — y > 0)) c} Elelyqu g 0) A (y 5 0}) A (.t —_v > 0)) d) V.I.'V_v({.r 9e 0) A U‘ 95 0) 4-) {Iy 59$ 0}) Translate each of these nested quantifications into an En- glish statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) Eleth'y = y} b) VxVyirultx < 0) A (y < 0)} —> (xy > 0)} c) El.rEly((x2 > y) A(.t' < y)) d) VxVyElz(x + y = 2') Let Q(.r.__v} be the statement “x +y = x — y.” If the domain for both variables consists of all integers, what are the truth values? 3!) Qt I. 1} c) V_vQ[l._i:) e) 3x3_vQ(x. y) g) Ely‘v‘x Q(x. y} i) Vx‘r‘yth. y) Determine the truth value of each of these statements if the domain for all variables consists of all integers. b) Etthin 4: m2) '3) le. 0} d) 5LT le. 2) f) VIElyQiX. y) M Via-X le - J’) a) Vn3m(n3 < m) c) VnElmt’n + m = 0] d) Eian(nm = m} e) Eli'iElrttlrt2 + in2 z 5} f) Elnilmtn2 + m2 = 6) g) Eln3m{n +m :4 An — m = l_) h) Elrt3min + m = 4 A n — m = 2) i) VanBptp = [m + my?) Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a} V.\'3_l’(.\‘2 = y') b) VxEly(x = y2) 29. 30. 31. 32. 33. 34. 35. 36. L4 Nested Quantifiers 61 c} Eley{xy = 0} (1) Elelny —— y 79 y + x) e) Vx(x 7E 0 —+ 3'1r(x_r-' :2 U} D EleyIly 99E 0 —+ .ry = l} g) VxEIyLr —— y = l} h) El.\'Ely(x + 2y 2 2 A 21' + 4y = 5} i) V_t'Ely{x +y = 2 A 2): — y =1} j) V_rV_vElz{z = (.1' + ylf2) Suppose the domain of the propositional function P(.r. y) consists of pairs x and y. where x is l, 2. or 3 and y is l. 2. or 3. Write out these propositions using disjunctittns and conjunctions. a) V,rVyP{x.y) h) Exit-1° (x. y} c) 3xVyP(x.y] d) VyElx P{.r.y) Rewrite each of these statements so that negations ap- pear only within predicates {that is. so that no negation is outside a quantifier or an expression involving logical connectives). a) -E|yEl.r Ptx. y} b) wafiyPtx, y) c) —-Ely{Q(y} A VxfiRtx. y)) d) -‘E|y(ElxR(x. y} V VxSix. y)} e) -E|y(Vx32T(x. y. 2) v 3.szU(x, y. 2)} Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) VxEIvi-‘Vthx. y. z) b) VxElpHx. y] v VxHyQLr. y) c) anytPtx. y) A athx. y. 2)) d] VxEl_v(P{.r. y) “4 le. yil Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) EszVthx. y. z) b) irElyPtx, y) A VxV_vQ{x. y) c} Bxflyith.y} H W. x)) d] VyElelzt’T{_x,y. 2) v Q(x.y)) Rewrite each of these statements so that negations ap— pear only within predicates {that is, so that no negation is outside a quantifier or an expression involving logical connectives). a) HVxVyPLLy) b) fiVyEleU'JJ c) —-Vny{P{x.y) V Q[x.y)) d) -(3xEly—rP(x. y} A V.rVyQ{x. y)} e) -vVJr(3szP{x. y. z) A Elz‘t’y P{x. y. 2)} Find a common domain for the variables x. y, and z for which the statement VxVyux ¢ y} —) Vzitz = .r) v (z = y)_}) is true and another domain for which it is false. Find a common domain for the variables .r. y. z. and w for which the statement VxVszflwttw # x] A {w # y) A (w ¢ 2}) is true and another common domain for these variables For which it is false. Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the lefi of a quantifier. Next. express the negation in 62 37. 38. 39. 40. 41. 42. 43. 44. l - The Foundations: Logic and Proofs simple English. [Do not simply use the words “it is not the case that") a) No one has lost more than one thousand dollars play- ing the lottery. b) There is a student in this class who has chatted with exactly one other student. e) No student in this class has sent email to exactly two other students in this class. (1) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book. Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the words “It is not the case that") a) Every student in this class has taken exactly two mathematics classes at this school. it) Someone has visited every country in the world except Libya. c) No one has climbed every mountain in the Himalayas. d) Every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon. Express the negations of these propositions using quan- tifiers, and in English. a) Every student in this class likes mathematics. b} There is a student in this class who has never seen a computer. it) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus. Find a counterexample. if possible, to these universally quantified statements. where the domain for all variables consists of all integers. a) ‘r’x‘o’ytx2 : y: —> .t' = y) 1)) way-()9 = .1‘ i c) V.rV_t-‘(xy 2 )6) Find a counterexample, if possible, to these universally quantified statements. where the domain for all variables consists ofall integers. a) V.\'3_t‘(x = My) b) want-3 —.\' «r. IOU) c) Vx'Vi-‘Lt': # _t-'3) Use quantifiers to express the associative law for multi— plication of real numbers. Use quantifiers to express the distributive laws of multi- plication over addition for real numbers. Use quantifiers and logical connectives to express the fact that every linear polynomial {that is, polynomial of degree 1) with real coefiicients and where the coefficient of .t is nonzero. has exactly one real root. Use quantifiers and logical connectives to express the fact #6.? that a quadratic polynomial with real number coefficients has at most two real roots. 45. Determine the truth value of the statement Vx 3_v(.ty = I) if the domain for the variables consists of a} the nonzero real numbers. b) the nonzero integers. c) the positive real numbers. 46. Determine the truth value of the statement SxV‘er 5 3:2) if the domain for the variables consists of a) the positive real numbers. b) the integers. c) the nonzero real numbers. 47. Show that the two statements -EI.rV,vP{.r. y) and Vx 3y—-P(x, y), where both quantifiers over the first vari- able in P(.r. yihave the same domain, andboth quantifiers over the second variable in Ptx. y) have the same domain. are logically equivalent. *48. Show that V.rP(x) v V.\‘Q(x) and V.1'V_‘i-‘(Pl.\'} v Qty”, where all quantificrs have the same nonempty domain, are logically equivalent. (The new variable y is used to combine the quantifications correctly.) *49. a) Show that V.1‘P(.\‘}x\ EIxQ(x) is logically equivalent to Vx 3y ( Ptx) A Qua), where all quantifiers have the same nonempty domain. 11) Show that Vx Plx] v Elx th) is equivalent to VxElytPtx) v Qtyj). where all quantifiers have the same nonempty domain. A statement is in prenex normal form {PNF} if and only if it is of the form Qt-YI sze ' ' - QtIt Piv‘fi - x2. . . . . xi). where each Q;. i = l. 2, . . . . k. is either the existentialquanti~ fier or the universal quantifier. and Pm . . . . . xi.) is a predicate involving no quantifiers. For example. 3xVy{ Pix. y) A Qty-D is in prenex normal form, whereas it P{.r) V VI Q(x) is not (because the quantifiers do not all occur first). Every statement formed from propositional variables, predicates, T, and F using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 5| asks for a proof of this fact. *50. Put these statements in prenex normal form. [Hirit.' Use logical equivalence From Tables 6 and 'r' in Section 1.2. Table 2 in Section 1.3. Example 19 in Section 1.3, Ex- ercises 45 and 46 in Section 1.3, and Exercises 48 and 49 in this section] a) 3x P{.r) v 3x Q(.r) v A, where A is a proposition not involving any quantifiers. b) -|(Vx P().') v VI th1) c) 3.\'P(x) —> Elelx) “'51. Show how to transform an arbitrary statement to a state- ment in prenex normal form that is equivalent to the given statement. *52. Express the quantification Six P(_x). introduced on page 37, using universal quantifications. existential quantifica- tions. and logical operators. ...
View Full Document

Page1 / 13

1.4 - 50 SI. 52. 53. 54. 55. 56. 57. 58. l The Foundations:...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online