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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 mid-nineteenth 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 PMEdP-rnql alpfiqlfipfivq cinematic-61 {p—rqlMp—rrEp—rtqmi [p-rr‘letq—rrlEIchil—H' tp-rqlvtp—H'Ep-qul (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 {fit-p) 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 1-3.7 [.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)E-1[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 computer-generated 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 l-28 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) [patio-r 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) [qulVr-EpVMVr} 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. easier-to-understand 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‘V-ts,-p\fq V-Is,q Vr‘V-Is,q V-v‘V-us. ~pv—tgv-vs, pvr Vs. ander‘ v-o‘ canbe made 1.3 Predicates and Quantifiers E Introduction I-jti 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\(pv-|q V-I5)A {-tp v -r V ms) A (p v q v w-] ntp v -v‘ v his) c) (qu w-Mlpv—q V—'5}f\(q v—q- V's-tn (-‘pvr'wtntepvq Vnthpch v-v-l 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. ...
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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.

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1.2 - {-3} 61. 62. 63. 64. Steve would like to determine...

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