This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: {3} 61. 62. 63. 64. Steve would like to determine the relative salaries ofthree
coworkers using two facts. First. he knows that if Fred
is not the highest paid of the three. then Janice is. Sec
ond. he knows that if Janice is not the lowest paid. then
Maggie is paid the most. is it possible to determine the
relative salaries of Fred, Maggie. and Janice from what
Steve knows? If so, who is paid the most and who the
least‘? Explain your reasoning. Five friends have access to a chat room. Is it possible to
determine who is chatting ifthe following information is
known? Either Kevin or Heather. or both. are chatting.
Either Randy or Vijay. but not both. are chatting. If Abby
is chatting. so is Randy. Vijay and Kevin are either both
chatting or neither is. If Heather is chatting, then so are
Abby and Kevin. Explain your reasoning. A detective has interviewed four witnesses to a crime.
Front the stories of the witnesses the detective has con
cluded that ifthe butler is telling the truth then so is the
cook; the cook and the gardener cannot both be telling the
truth; the gardener and the handyman are not both lying;
and if the handyman is telling the truth then the cook is
lying. For each of the four witnesses, can the detective de—
termine whether that person is telling the truth or lying?
Explain your reasoning. Four friends have been identiﬁed as suspects for an unau
thorized access into a computer system. They have made
statements to the investigating authorities. Alice said
“Carlos did it." John said “I did not do it.“ Carlos said 1.2 Propositional Equivalences 1.2 Propositional Equivalences 21 “Diana did it.” Diana said “Carlos lied when he said that I did it.” a) If the authorities also know that exactly one of the four
suspects is telling the truth. who did it? Explain your reasoning.
b) If the authorities also know that exactly one is lying. who did it? Explain your reasoning. *765. Solve this famous logic puzzle. attributed to Albert Einstein. and known as the zebra puzzle. Five men with
different nationalities and with different jobs live in con .g seculive houses on a street. These houses are painted different colors. The men have different pets and have dif
ferent favOrite drinks. Determine who owns a zebra and
whose favorite drink is mineral water (which is one of the
favorite drinks] given these clues: The Englishman lives
in the red house. The Spaniard owns a dog. The Japanese
man is a painter. The Italian drinks tea. The Norwegian
lives in the ﬁrst house on the left. The green house is
immediately to the right of the white one. The photogra
pher breeds snails. The diplomat lives in the yellow house.
Milk is drunk in the middle house. The owner ofthe green
house drinks coffee. The Norwegian’s house is next to the
blue one. The violinist drinks orange juice. The fox is in
a house next to that of the physician. The horse is in a
house next to that of the diplomat. [Hints Make a table
where the rows represent the men and columns represent
the color oftheir houses, rlreirjobs, their pets. and their
favorite drinks and use logical reasoning to determine the
correct entries in the table] D Introduction An important type of step used in a mathematical argument is the replacement of a statement
with another Statement with the same truth value. Because of this. methods that produce propo
sitions with the same truth value as a given compound proposition are used extensively in the
construction of mathematical arguments. Note that we will use the term “compound propo
sition” to refer to an expression formed from propositional variables using logical operators, such as p A q. We begin our discussion with a classiﬁcation of compound propositions according to their possible truth values. EFINITION 1 A compound proposition that is always true, no matter what the truth values of the propositions that occur in it. is called a morning A compound proposition that is always false is called
a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency. Tautologies and contradictions are often important in mathematical reasoning. Example 1 illus
trates these types of compound propositions. 22 l I The Feundations: Logic and Proofs . TABLE 2 De Morgan’s
Laws. etpAqEvpch TABLE 1 Examples of a Tautology and a
Contradiction. "(PVQ)E“P*’\“1 EXAMPLE 1 We can construct examples of tautologies and contradictions using just one propositional vari
able. Consider the truth tables ofp v —p and p A p. shown in Table 1. Because p v p is
always true, it is a tautology. Because p A mp is always false. it is a contradiction. { Logical Equivalences Compound propositions that have the same truth values in all possible cases are called logically
"an"! equivalent. We can also deﬁne this notion as follows. DEFINITION 2 The compound propositions p and q are called logically equivalent if p H q is a tautology.
The notation p E q denotes that p and q are logically equivalent. Remark: The symbol a is not a logical connective and p E q is not a compound proposition
but rather is the statement that p H q is a tautology. The symbol 4: is sometimes used instead
of 2 to denote logical equivalence. One way to determine whether two compound propositions are equivalent is to use a truth
table. In particular, the compound propositions p and q are equivalent if and only if the columns
giving their truth values agree. Example 2 illustrates this method to establish an extremely
important and useful logical equivalence, namely, that of (p v q) of p A —q. This logical
equivalence is one of the two De Morgan laws, shown in Table 2, named after the English
mathematician Augustus De Morgan, of the midnineteenth century. Extra 9 Examples EXAMPLE 2 Show that '(p v q) and mp A q are logically equivalent. Solution: The truth tables for these compound propositions are displayed in Table 3. Because
the truth values of the compound propositions (p v q) and —p A q agree for all possible
combinations ofthe truth values ofp and q, it follows that ﬁ(p v q) H (—p A Hg) is a tautology
and that these compound propositions are logically equivalent. 4 TABLE 3 Truth Tables for —:{p V q) and 1p /\ :q. m «M F
F
F
T EXAMPLE 3 EXAMPLE 4 [.2 Propositional Equivalences 23 TABLE 4 Thu]: Tables for Ip V q and
P “‘t 9 p q 1 Show that p —> g and mp V q are logically equivalent. Solution: We construct the truth table for these compound propositions in Table 4. Because the
truth values of Ip v q and p —> g agree, they are logically equivalent. { We will now establish a logical equivalence of two compound propositions involving three
different propositional variables p, q, and r. To use a truth table to establish such a logical
equivalence, we need eight rows. one for each possible combination of truth values of these three variables. We symbolically represent these combinations by listing the truth values of p,
q, and r, respectively. These eight combinations of truth values are TTT, TTF, TFT, TFF, FTT,
FTF, FFT, and F FF; we use this order when we display the rows of the truth table. Note that we
need to double the number of rows in the truth tables we use to show that compound propositions
are equivalent for each additional propositional variable, so that 16 rows are needed to establish
the logical equivalence of two compound propositions involving four propositional variables,
and so on. In general, 2" rows are required if a compound proposition involves a propositional
variables. Show that p v (q A r) and (p v q} A (p v r] are logically equivalent. This is the distributive
law of disjunction over conjunction. Solution: We construct the truth table for these compound propositions in Table 5. Because
the truth values ofp v (q A r) and (p v q) A (p v r) agree, these compound propositions are
logically equivalent. ‘ TABLE 5 A Demonstration That p V (q A r) and (p V q} A [p V r) Are Logically
Equivalent. _ M
T T T T T T T T
T T F F T T T T
T F T F T T T T
T F F F T T T T
F T T T T T T T
F T F F F T F F
F F T F F F T F
F F F F F F F F 24 l r The Foundations: Logic and Proofs I 24 TABLE 6 Logical Equivalences. Identity laws Domination laws idempotent laws "l—uﬂl E p Double negation law Commutative laws [p v q) v r E p v [g V r) Associative laws
(pAqlAr' EpAtq Ar) p v (q A r) E (p v q) A (p v r) Distributive laws
pA(_q vr) E (p A11] v(pAr} De Morgan’s laws P V (P A q) E [7 Absorption laws
.0 A (P V q} E P Negation laws Table 6 contains some important equivalences.* In these equivalences, T denotes the com
pound proposition that is always true and F denotes the compound proposition that is al
ways false. We also display some useful equivalences for compound propositions involving
conditional statements and biconditional statements in Tables 7' and 8, respectively. The reader
is asked to verify the equivalences in Tables 6e8 in the exercises at the end of the section. The associative law for disjunction shows that the expression p v q V r is well deﬁned,
in the sense that it does not matter whether we ﬁrst take the disjunction of p with q and then
the disjunction of p v q with r. or if we ﬁrst take the disjunction of q and r and then take the
disjunction ofp withq v r. Similarly, the expression p A q A r is well deﬁned. By extending this
reasoning, it follows that p] v p; V .   v p" and p1 A p; A    A p,1 are well deﬁned whenever
p1. p2. . . . . p” are propositions. Furthermore, note that De Morgan’s laws extend to nlpi szV“VpnlEl‘PtA‘P2A"‘/\*Pn} and
—'(Pl APEA'” APII)E(_'pl VTPE V'” Vb‘pn)‘ {Methods for proving these identities will be given in Section 4.1.) 'Readers familiar with the concept of a Boolean algebra will notice that these identities are a special case of identities that
hold for any Boolean algebra. Compare them with set identities in Table l in Section 2.2 and with Boolean identities in Table Sin Section 1.l. 1.2 Propositional Equivalences 25 TABLE 8 Logical Equivalences Involving
Biconditionals. TABLE 7 Logical Equivalences
Involving Conditional Statements.
p a q E *9 V a Prqu—‘q—t‘l’ Poqatpeqlotq—ipl H Eﬁ)ﬁ_l
qua—rp—rq P ‘1’ l' q PﬁqEIPquf‘po‘qi PMEdPrnql
alpﬁqlﬁpﬁvq cinematic61
{p—rqlMp—rrEp—rtqmi
[prr‘letq—rrlEIchil—H'
tprqlvtp—H'Epqul
(per‘lvtrteriﬁwom—H' Using De Morgan’s Laws The two logical equivalences known as De Morgan’s laws are particularly important. They tell
us how to negate conjunctions and how to negate disjunctions. In particular, the equivalence
—(p v q) E ﬁp A —Iq tells us that the negation of a disjunction is formed by taking the con
junction of the negations of the component propositions. Similarly, the equivalence (p A q) E
—p V He; tells us that the negation of a conjunction is formed by taking the disjunction of the
negations of the component propositions. Example 5 illustrates the use of De Morgan’s laws. EXAMPLE 5 Use De Morgan‘s laws to express the negations of “Miguel has a cellphone and he has a laptop assessment a computer" and “Heather will go to the concert or Steve will go to the concert.“ Solution: Let p be “Miguel has a cellphone" and q be “Miguel has a laptop computer.“ Then
“Miguel has a cellphone and he has a laptop computer” can be represented by p A q. By the
ﬁrst of De Morgan’s laws, (p A q) is equivalent to vp V —q. Consequently. we can express AUGUSTUS DE MORGAN “Silo—1871) Augustus De Morgan was born in India, where his father was a
colonel in the Indian army. De Morgan’s family moved to England when he was 7 months old. He attended
private schools. where he developed a strong interest in mathematics in his early teens. Dc Morgan studied at
Trinity C ollegc, Cambridge, graduating in I 82?. Although he considered entering medicine or law, he decided
on a career in mathematics. He won a position at University College. London. in l828. but resigned when the
college dismissed a fellow professor without giving reasons. However. he resumed this position in l836 when
his successor died staying there until 1866. De Morgan was a noted teacher who stressed principles over techniques. His students included many famous
mathematicians, including Augusta Ada, Countess of Lovelace, who was Charles Babbage’s collaborator in his work on computing machines (see page 27 for biographical notes on Augusta Ada}. (De Morgan cautioned the countess against
studying too much mathematics. hocause it might interfere with her childbearing abilities!) De Morgan was an exo‘emely proliﬁc writer. He wrote more than I000 articles for more than 15 periodicals. Dc Morgan also
wrote textbooks on many subjects, including logic. probability, calculus. and algebra. In 1838 he presented what was perhaps the ﬁrst
clear explanation of an important proof technique known as mathematical induction [discussed in Section 4,] of this text). a term
he coined. ln the 18405 Dc Morgan made fundamental contributions to the development of symbolic logic. He invented notations
that helped him prove propositional equivalences, such as the laws that are named after him. in [842 De Morgan presented what
was perhaps the ﬁrst precise deﬁnition of a limit and developed some tests for convergence of inﬁnite series. De Morgan was also
interested in the history of mathematics and wrote biographies of Newton and Halley. ln i837 De Morgan married Sophia Frend, who wrote his biography in ISSZ. De Morgan’s research. writing, and teaching left
little time for his family or social life. Nevertheless. he was noted for his kindness. humor. and wide range of knowledge. 26  t The Foundations: Logic and Proofs EXAMPLE 6 Extra __
Estimates ‘" EXAMPLE 7 {—26 the negation of our original statement as “Miguel does not have a cellphone or he does not have
a laptop computer.” Let r be “Heather will go to the concert” and s be “Steve will go to the concert.“ Then
“Heather will go to the concert or Steve will go to the concert" can be represented by r V 5.
By the second of De Morgan’s laws, —~(r v s) is equivalent to or A ts. Consequently. we can
express the negation of our original statement as “Heather will not go to the concert and Steve
will not go to the concert." ‘ Constructing New Logical Equivalences The logical equivalences in Table 6, as well as any others that have been established (such as
those shown in Tables 7 and 8), can be used to construct additional logical equivalences. The
reason for this is that a proposition in a compound proposition can be replaced by a compound
proposition that is logically equivalent to it without changing the truth value of the original
compound proposition. This technique is illustrated in Examples 6—8, where we also use the
fact that if p and q are logically equivalent and q and r are logically equivalent, then p and r
are logically equivalent (see Exercise 56). Show that —I{'p —> q) and p A q are logically equivalent. Solution: We could use a truth table to show that these compound propositions are equivalent
(similar to what we did in Example 4). Indeed, it would not be hard to do so. However, we want
to illustrate how to use logical identities that we already know to establish new logical identities,
something that is of practical importance for establishing equ ivalences of compound propositions
with a large number of variables. So, we will establish this equivalence by developing a series of
logical equivalences. using one of the equivalences in Table 6 at a time, starting with —I( p —~> q)
and ending with p A ‘q. We have the following equivalences. —'lP ‘9 q) E _'(_'p V g) by Example 3
E *(ﬁp) A m; by the second Dc Morgan law
E p A —Iq by the double negation law ‘ Show that I(p v (—Ip A q)) and  p A q are logically equivalent by developing a series of
logical equivalences. Solution: We will uSe one of the equivalences in Table 6 at a time. starting with (p V (p A q))
and ending with ‘ p A ﬁg. (Note: we could also easily establish this equivalence using a truth table.) We have the following equivalences. by the second De Morgan law ntp V (“P A qt) 2 pp A ntcp A q) E p A {ﬁtp) V mg] by the ﬁrst De Morgan law E 1 p A (p V mg) by the double negation law E {p A p) V (—p A mg) by the second distributive law E F v (—p A q) because inp E F E (p A ﬁq) V F by the commutative law for disjunction
E —p A q by the identity law for F Consequently ﬁt p v (1p A 11)) and I p A q are logically equivalent. 4 13.7 [.2 Propositional Equivalences 27 EXAMPLE 8 Show that (p A q) —r (p v q) is a tautology. Solution: To show that this statement is a tautology, we will use logical equivalences to demon
strate that it is logically equivalent to T. (Note: This could also be done using a truth table.) (pAq)—=r(pvq)E1[pf\q)v(pvq) hyExttmple3
E (—p V q) V (p V q) by the ﬁrst Dc Morgan law
E (p V ,0) V (“‘q V g} by the associative and commutative
laws for disjunction
E T V T by Example 1 and the commutative
law for disjunction E T by the domination law { A truth table can be used to determine whether a compound proposition is a tautology.
This can be done by hand for a compound proposition with a small number of variables, but
when the number of variables grows, this becomes impractical. For instance, there are 220 =
1,048,576 rows in the truth table for a compound proposition with 20 variables. Clearly, you
need a computer to help you determine, in this way, whether a compound proposition in 20
variables is a tautology. But when there are 1000 variables, can even a computer determine in
a reasonable amount of time whether a compound proposition is a tautology? Checking every
one of the 2luau {a number with more than 300 decimal digits) possible combinations of truth
values simply cannot be done by a computer in even trillions of years. Furthermore, no other
procedures are known that a computer can follow to determine in a reasonable amount of time
whether a compound proposition in such a large number of variables is a tautoiogy. We will
study questions such as this in Chapter 3. when we study the complexity of algorithms. linlts AUGUSTA ADA. L‘UUN'I‘ESS OF LOVELAL‘E {lb’lS—ISSE] Augusta Ada was the only child ﬁ'om the
marriage of the famous poet Lord Byron and Lady Byron, Annabella Millbanke, who separated when Ada
was I month old, because of Lord Byron‘s scandalous affair with his half sister. The Lord Byron had quite a
reputation, being described by one of his lovers as “mad. bad, and dangerous to know." Lady Byron was noted for
her intellect and had a passion for mathematics; she was cal led by Lord Byron “The Princess of Parallelograms.“
Augusta was raised by her mother, who encouraged her intellectual talents especially in music and mathematics.
to counter what Lady Byron considered dangerous poetic tendencies. At this time, women were not allowed to
attend universities and could not join learned societies. Nevertheless, Augusta pursued her mathematical studies
independently and with mathematicians, including William F rend. She was also encouraged by another female
mathematician. Mary Somerville, and in 1834 at a dinner party hosted by Mary Somerville, she learned about Charles Babbage’s
ideas for a calculating machine, called the Analytic Engine. In 1838 Augusta Ada married Lord King, later elevated to Earl of Lovelace. Together they had three children. Augusta Ada continued her mathematical studies after her marriage. Charles Babbage had continued work on his Analytic
Engine and lectured on this in Europe. In [842 Babbage asked Augusta Ada to translate an article in French describing Babbage‘s
invention. When Babbage saw her translation. he suggested she add her own notes, and the resulting work was three times the
length of the original. The most complete accounts of the Analytic Engine are found in Augusta Ada’s notes. In her notes, she
compared the working of the Analytic Engine to that of the Jacquard loom. with Babbage’s punch cards analogous to the cards used
to create patterns on the loom. Furthermore, she recognized the promise of the machine as a general purpose computer much better
than Babbage did. She stated that the “engine is the material expression of any indeﬁnite function of any degree of generalin and
complexity.“ Her notes on the Analytic Engine anticipate many future developments, including computergenerated music. Augusta
Ada published her writings under her initials A.A.L. concealing her identity as a women as did many women did at a time when
women were not considered to be the intellectual equals of men. After 1845 she and Babbage worked toward the development of a
system to predict horse races. Unfortunately, their system did not work well, leaving Augusta Ada heavily in debt at the time of her
death at an unfortunately young age from uterine cancer. In 1953 Augusta Ada’s notes on the Analytic Engine were republished more than 100 years after they were written, and after
they had been long forgotten. in his work in the 9505 on the capacity of computers to think (and his famous Turing Test}, Alan
Turing responded to Augusta Ada ‘s statement that “The Analytic Engine has no pretensions whatever to originate anything. It can do
whatever we know how to order it to perform.” This “dialogue” between Turing and Augusta Ada is still the subject of controversy.
Because of her fundamental contributions to computing. the programming language Ada is named in honor of the Countess of Lovelace. 28  .t The Foundations: Logic and Proofs l28 Exercises
________________—_——— 9. Show that each of these conditional statements is a tau
tology by using truth tables. Use truth tables to verify these equivalences. a) pATEp b}vaEp
c) pAFEF d)pVTET altpaql—ip bJpetpWr)
e)prEp DpApEp amoeba—re} dltpoqlatp+ql e)n(p—)q}—rp t} nip—*qlenq
Show that each of these conditional statements is a tau
tology by using truth tables. a) [anpvqller a b) [(P —* qqu —r Fl] * (p e t“) 9) [patior all—>6! d) [(qu}A{p—> r}A(q —» l‘}] —> : Show that [p) and p are logically equivalent.
Use truth tables to verify the commutative laws alpvqaqvp Miami;an
Use truth tables to verify the associative laws :1) [qulVrEpVMVr} 10. b) [p A q) A r E p A (q A r} ll. Show that each conditional statement in Exercise 9 is a
. Use a truth table to verify the distributive law “ll“0103)” Without “Sing mull tableS
p A ((1 V r) E (p A (1') V (p A m 12. Show that each conditional statement in Exercise 10 is a
Use a truth table to verify the ﬁrst De Morgan law laumlogy Will[om “3mg tmth tables'
_{ p A (A E _p v ﬁg [3. Use truth tables to verify the absorption laws.
V E b V E
Use De Morgan’s laws to ﬁnd the negation of each ofthe a) p .[p A q) p ) p A {p q} p .
following statements‘ [4. Determine whether ("wp A (p —» q}) —> q is a
. . tautology.
z) gulls “Galas? halppy' 15. Determine whether {—q A (p F» q}) —) —p is a
) at 05 w: icyc e or run tomorrow. mummy. c) Mei walks or takes the bus to class.
d) Ibrahim is smart and hard working. Use De Morgan's laws to ﬁnd the negation of each of the
following statements. a) Kwame will take a job in industry or go to graduate Each of Exercises 16—28 asks you to show that two compound
propositions are logically equivalent. To do this either show
that both sides are true, or that both sides are false. for exactly
the same combinations of truth values of the propositional
variables in these expressions ( whichever is easier). school. 16. Show that p H q and (p Aq) vt—tp A “IqJ are
b) Yoshiko knows Java and calculus. equivalent.
c) James is young and strong. 11'. Show that ﬁtp H q} and p H —q are logically
:1) Rita will move to Oregon or Washington. equivalent. _—___—_—_————————~ HENRY MAURICE SHEFFER (1883 964i Henry Maurice Shetfer. born to Jewish parents in the western
Ukraine, emigrated to the United States in [892 with his parents and six siblings. He studied at the Boston Latin
School before entering Harvard where he completed his undergraduate degree in 1905. his master’s in 1907‘
and his PhD. in philosophy in [908. After holding a postdoctoral position at Harvard. Henry traveled to Europe
' ' on a fellowship. Upon returning to the United States, he became an academic nomad. spending one year each
at the University of Washington, Cornell, the University of Minnesota. the University of Missouri. and City
College in New York. In I916 he returned to Harvard as a faculty member in the philosophy department. He  . remained at Harvard until his retirement in 1952.
Shelter introduced what is now known as the Sheﬁ'er stroke in 1913; it became well known only after its use in the I925 edition of Whitehead and Russell’s Pri'ncipia Maikemair‘ca. [n this same edition Russell wrote that Sheffer had invented
a powerful method that could be used to simplify the Principt’u. Because of this comment. Sheﬂ‘er was something of a mystery man
to logicians. especially because Sheﬁ'er. who published little in his career, never published the details of this method only describing it in mimeographed notes and in a brief published abstract.
Sheﬁ‘er was a dedicated teacher of mathematical logic. He liked his classes to be small and did not like auditors. When strangers appeared in his classroom. Shaffer would order them to leave, even his colleagues or distinguished guests visiting Harvard. Sheffer
was barely ﬁve feet tall: he was noted for his wit and vigor, as well as for his nervousness and irritability. Although widely liked, he
was quite lonely. He is noted for a quip he spoke at his retirement: “Old professors never die. they just become emeriti.“ Sheffer is
also credited with coining the term “Boolean algebra" (the subject of Chapter I I of this text). Sheffer was briefly married and lived
most of his later life in small rooms at a hotel packed with his logic books and vast ﬁles of slips of paper he used to jot down his
ideas. Unfortunately. Sheffer suffered from severe depression during the last two decades of his life. i—JS‘ 18. Show that p —r q and "q —+ p are logically equivalent. [9. Show that p <—> q and p «a —q are logically equivalent. 20. Show that {p 63 q} and p <—> q are logically equivalent. 21. Show that —[p H q) and —p H q are logically
equivalent. 22. Show that {p —> qlA (p —> Handp a (q Ar]arelog
ieaily equivalent. 23. Show that {p —> r)A (q —> r)and (p Vq) —r r are log
ically equivalent. 24. Show that (p —r q} V [p n» r) and p —> it; V r] are loge
ically equivalent. 25. Show that {p —» r) v (q —> r} and (p Ag) —> r are log
ically equivalent. 26. Show that Ip —> ((1 —> rJandq + (p v r) are logically
equivalent. 2?. Show that p <—> q and (p —+ q) A {q —r p} are logically
equivalent.
28. Show that p H q and —p H 1q are logically equivalent.
29. Showthattp —> g) A (q —> r) —r (p + r}isatautology.
30. Show that (p v q) A (—p Vr} —» [g V r) is a tautology.
3]. Show that (p —> q) —) r and p —> {q —> r) are not
logically equivalent.
32. Show that (p Aq) —> r and {p —> HA (q —> r) are not
logically equivalent.
33. Showthat(p —> g) —) (r —) s}and{p —» r) —> (q —> s)
are not logically equivalent.
The dual of a compound proposition that contains only the
logical operators v, A, and  is the compound proposition
obtained by replacing each v by A. each A by v, each T by
F. and each F by T. The dual of s is denoted by s".
34. Find the dual of each of these compound propositions.
alpvnq b)pA(qvtrAT})
c) {pA—q)vtq AF)
35. Find the dual of each of these compound propositions.
a) pA—‘qA—‘l‘ b) (pAqrxr)\/s
c) (vaMtq vT)
36. When does 5* = s, where s is a compound proposition?
3?. Show that (5* )" = s when s is a compound proposition. 38. Show that the logical equivalences in Table 6, except for
the double negation law. come in pairs. where each pair
contains compound propositions that are duals of each other. “‘39. Why are the duals of two equivalent compound proposi—
tions also equivalent. where these compound propositions
contain only the operators A. V, and m? 40. Find a compound proposition involving the propositional
variables p. q, and r that is true when p and q are true and
r is false. but is false otherwise. [Hint Use a conjunction
of each propositional variable or its negation] 41. Find a compound proposition involving the propositional
variables ,5. q. and r that is true when exactly two ofp. q,
and r are true and is false otherwise. [Hints Form a dis—
junction of conjunctions. Include a conjunction for each [.2 Propositional Equivalenccs 29 combination of values for which the compound proposi
tion is true. Each conjunction should include each of the
three propositional variables or its negationsj F312. Suppose that a truth table in n propositional variables is speciﬁed. Show that a compound propositiOn with this
truth table can be formed by taking the disjunction of
conjunctions of the variables or their negations, with one
conjunction included for each combination of values for
which the compound proposition is true. The resulting
compound proposition is said to be in disjunctive normal
form. A collection of logical operators is called functionally com
plete if every compound proposition is logically equiva
lent to a compound proposition involving only these logical
operators. 43. Show that —u, A, and v form a functionally complete col
lection of logical operators. [Hints Use the fact that every
compound proposition is logically equivalent to one in
disjunctive normal form. as shown in Exercise 42.] *44. Show that u and A form a functionally complete collec
tion of logical operators. [Hint First use a De Morgan law
toshow that p v q islogically equivalent to {p A —g).] *45. Show that —1 and v form a functionally complete collec
tion of logical operators. The following exercises involve the logical operators NAND
and NOR. The proposition p NAND q is true when either p or
q. or both. are false: and it is false when both p and q are true.
The proposition p NOR q is true when both p and q are false,
and it is false otherwise. The propositions p NAND q and p NOR q are denoted by p l q and p .1, q, respectively. (The op
erators  and J, are called the Sheffer stroke and the Peirce arrow after H. M. Sheﬁ’er and C. S. Peirce. respectively.) 46. Construct a truth table for the logical operator NAND.
47. Show that p I q is logically equivalent to —'[p A q).
48. Construct a truth table for the logical operator NOR.
49. Show that p l, q is logically equivalent to —‘(p V q}.
50. In this exercise we will show that it} is a functionally
complete collection of logical operators.
a) Show that p J, p is logically equivalent to mp.
b) Show that {p J, q) i (p l, qlis logically equivalent to
v .
c) gonglude from parts (a) and (b), and Exercise 49, that
{t} is a functionally complete collection of logical
operators.
*51. Find a compound proposition logically equivalent to
p —> q using only the logical operator 4,.
52. Show that ll] is a functionally complete collection oflog
ical operators.
53. Show that p  q and q  p are equivalent.
54. Show thatp  [q Ir) andtp  q) l r are not equivalent, so
that the logical operator  is not associative.
*55. How many different truth tables of compound proposi—
tions are there that involve the propositional variables p
and q? 30 56. 57. 58. 59. l i" The Foundations: Logic and Proofs Show that if p. q. and r are compound propositions
such that p and q are logically equivalent and q and r
are logically equivalent, then p and r are logically
equivalent. The following sentence is taken from the speciﬁcation of
a telephone system: “If the directory database is opened
then the monitor is put in a closed state. if the system is
not in its initial state.“ This speciﬁcation is hard to underw
stand because it involves two conditional statements. Find
an equivalent. easiertounderstand speciﬁcation that in
volves disjunctions and negations but not conditional
statements How many of the disjunctions p v q. mp v q, (1 v r.
q v —v'. and ﬁg V Hr can be made simultaneously true
by an assignment of truth values to p, q, and r? How many ofthe disjunctions p v q v s. —p v r v s,
mpV—v‘Vts,p\fq VIs,q Vr‘VIs,q Vv‘Vus.
~pv—tgvvs, pvr Vs. ander‘ vo‘ canbe made 1.3 Predicates and Quantiﬁers E Introduction Ijti simultaneously true by an assignment of truth values to
p. q. r. and s? A compound proposition is satisﬁable if there is an assign
ment of truth values to the variables in the compound propo
sition that makes the compound proposition true 60. Which ofthese compound propositions are satisﬁablc? 61. a) (qu v—«ﬂnlpv—q V—‘slAlp v—wV—MA
(Ip v—q V s}r\ [p Vq V—IS) b) {Ip V—Iq Vr) A (hp Vq V rs)r\(pvq VI5)A
{tp v r V ms) A (p v q v w] ntp v v‘ v his) c) (qu wMlpv—q V—'5}f\(q v—q V'stn
(‘pvr'wtntepvq Vnthpch vvl A
(«p v q v s) A (—Ip v ﬁr V 5} Explain how an algorithm for determining whether a compound proposition is satisﬁable can be used to de— termine whether a compound proposition is a tautology. [Hint Look at mp, where p is the compound proposition that is being examined] Propositional logic, studied in Sections 1.1 and 1.2, cannot adequately express the meaning of
statements in mathematics and in natural language. For example, suppose that we know that “Ever com uter connected to the university network is functionin ro erl ."
Y P gP P Y No rules of propositional logic allow us to conclude the truth of the statement “MATH3 is functioning properly," where MATH3 is one of the computers connected to the university network. Likewise, we cannot
use the rules of propositional logic to conclude from the statement “C52 is under attack by an intruder,” where C82 is a computer on the university network, to conclude the truth of “There is a computer on the university network that is under attack by an intruder." In this section we will introduce a more powerful type of logic called predicate logic. We
will see how predicate logic can be used to express the meaning of a wide range of statements
in mathematics and computer science in ways that permit us to reason and explore relationships
between objects. To understand predicate logic, we ﬁrst need to introduce the concept of a
predicate. Afterward, we will introduce the notion of quantiﬁers, which enable us reason with
statements that assert that a certain property holds for all objects of a certain type and with
statements that assert the existence of an object with a particular property. ...
View
Full
Document
This note was uploaded on 12/16/2009 for the course MATH 1014 taught by Professor Ganong during the Spring '09 term at York University.
 Spring '09
 ganong

Click to edit the document details