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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 lFl’ 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 5ny 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 5ny 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 ElxEly{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 :EonEIf(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 VwBaeaﬂﬁw. f) A Qtf. aJ)
E Vwaan—(th, f) A Qtf. 4))
E VwEIanfIPW. 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(_ty = 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 email 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.tEyQ(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 (Clix7) > 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) 359 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 ofFornme. 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 emaii 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 email message to Koko.
b) Arlene has never sent an email message to or tele phoned Sarah.
c) Jose has never received an email message from Deborah.
(1) Every student in your class has sent an email 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 email message. g) There is a student in your class who has sent everyone
else in your class an email message. b) There is someone in your class who has either sent an
email message or telephoned everyone else in your class. i) There are two different students in your class who
have sent each other emaii messages. j) There is a student who has sent himself or herself an
email 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 email and the second
student has telephoned the ﬁrst student. it} There are two different students in your ciass who
between them have sent an email 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. lr’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 ViaX 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) Exit1° (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) EyEl.r Ptx. y} b) waﬁyPtx, y) c) —Ely{Q(y} A VxﬁRtx. y)) d) ‘Ey(ElxR(x. y} V VxSix. y)} e) Ey(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) want3 —.\' «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 QtYI 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 QtyD
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. ...
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