10 1/The Foundations:Logic and Proofs EXAMPLE 10 Let p be the statement"You can take the flight,"and let g be the statement"You buy a ticket." Then pq is the statement "You can take the flight if and only if you buy a ticket." atedare ether both re o both false,that is ifyou buya ticket he flight or if you do not buy a tic et and you cannot take flight.It is fals h en yo (such as when the airline bumps you) IMPLICIT USE OF BICONDITIONALS You should be aware that biconditionals are not always explicit in natural language.In particular.the "if and only if"construction used in biconditionals is rarely used in common language.Instead,biconditionals are often expressed it,then"or an“only The other part of th "if and only if"is implicit. rse Is If you finis For examp cons r the stat you can hav This las ment in yo t is re ou can ha eal then u can have des only if you finish your meal.Because of this imprecision n natural langu we need to make an assumption whether a conditional statement in natural language implicitly includes its converse.Because precision is essential in mathematics and in logic,we will always distinguish between the conditional statement pq and the biconditional statement pq. Truth Tables of Compound Propositions We hay mportant logical connective nts and hic ell as nectives to build up complicated compound pro ositions involving variables.We can use truth tables to determine the truth values of these compound propositions as Example 11 illustrates.We use a separate column to find the truth value of each ion of t expression that occurs in the compound propositi ilt up.The truth valu the values of the propositiona EXAMPLE 11 Construct the truth table of the compound proposition (pVq)→(pAq). Solution Because this truth table involves two p nositional v riables oi this tuth table.one foreac of the paio uth auesIFand F The and g.ther re fou two columns are used for the truth values of p and g.respectively.In the third column we find the truth ly,the truth TABLE7 The Truth Table of(pv一q)→(pAg pv-q
10 1 / The Foundations: Logic and Proofs EXAMPLE 10 Let p be the statement “You can take the flight,” and let q be the statement “You buy a ticket.” Then p ↔ q is the statement “You can take the flight if and only if you buy a ticket.” This statement is true if p and q are either both true or both false, that is, if you buy a ticket and can take the flight or if you do not buy a ticket and you cannot take the flight. It is false when p and q have opposite truth values, that is, when you do not buy a ticket, but you can take the flight (such as when you get a free trip) and when you buy a ticket but you cannot take the flight (such as when the airline bumps you). ▲ IMPLICIT USE OF BICONDITIONALS You should be aware that biconditionals are not always explicit in natural language. In particular, the “if and only if” construction used in biconditionals is rarely used in common language. Instead, biconditionals are often expressed using an “if, then” or an “only if” construction. The other part of the “if and only if” is implicit. That is, the converse is implied, but not stated. For example, consider the statement in English “If you finish your meal, then you can have dessert.” What is really meant is “You can have dessert if and only if you finish your meal.” This last statement is logically equivalent to the two statements “If you finish your meal, then you can have dessert” and “You can have dessert only if you finish your meal.” Because of this imprecision in natural language, we need to make an assumption whether a conditional statement in natural language implicitly includes its converse. Because precision is essential in mathematics and in logic, we will always distinguish between the conditional statement p → q and the biconditional statement p ↔ q. Truth Tables of Compound Propositions We have now introduced four important logical connectives—conjunctions, disjunctions, conditional statements, and biconditional statements—as well as negations. We can use these connectives to build up complicated compound propositions involving any number of propositional variables. We can use truth tables to determine the truth values of these compound propositions, as Example 11 illustrates. We use a separate column to find the truth value of each compound expression that occurs in the compound proposition as it is built up. The truth values of the compound proposition for each combination of truth values of the propositional variables in it is found in the final column of the table. EXAMPLE 11 Construct the truth table of the compound proposition (p ∨ ¬q) → (p ∧ q). Solution: Because this truth table involves two propositional variables p and q, there are four rows in this truth table, one for each of the pairs of truth values TT, TF, FT, and FF. The first two columns are used for the truth values of p and q, respectively. In the third column we find the truth value of ¬q, needed to find the truth value of p ∨ ¬q, found in the fourth column. The fifth column gives the truth value of p ∧ q. Finally, the truth value of (p ∨ ¬q) → (p ∧ q) is found in the last column. The resulting truth table is shown in Table 7. ▲ TABLE 7 The Truth Table of (p ∨ ¬ q) → (p ∧ q). p q ¬q p ∨ ¬q p ∧ q (p ∨ ¬q) → (p ∧ q) T T F T T T T F T T F F F T F F F T F F T T F F
1.1 Propositional Logic 11 Precedence of Logical Operators TABLE8 Logical Operators. in a compound pr of pand.However,to reduce the number of parentheses.we specify that the negation Operator Precedence operator is applied before all other logical operators.This means that-pAq is the conjunction 1 of-p andq.namely,(-p)A q,not the negation of the conjunction of p andg,namely-(p A q) nother general rule of precedence is that the conjunction operator takes precedence over the disiun ion and coniunction o rators is cloa Finally,it is an accepted rule that the conditional and biconditional oper rators and have lower precedence than the conjunction and disjunction operators.A and v.Consequently. pv-r is the same as (pv)r.We will use parentheses when the order of the con Logic and Bit Operations Computers represent information using bits.A bit is a symbol with two possible values,namely Truth Value Bit 0(zero)and 1 (one).This meaning of the word bit comes from binary digi,because zeros and s are the digits used in binary repre darRresetamhitciamohn ns of numbers.The well-know T 0 a 1 hit t ent true and a 0 bit to repre nt false That is ntsT(true)or variable is called a Boolean variable if its value is either true or false.Cons sequently.a Boolean variable can be represented using a bit. and abe bytthe conthe pading tue by on e opera roUniversity.where he studied mathe er's deg ree in chemist ch to his lik ontro nd im he Statistics Depa ade significant contributions to many areas o rom a single Re the te s bit and
1.1 Propositional Logic 11 Precedence of Logical Operators We can construct compound propositions using the negation operator and the logical operators defined so far. We will generally use parentheses to specify the order in which logical operators in a compound proposition are to be applied. For instance, (p ∨ q) ∧ (¬r) is the conjunction of p ∨ q and ¬r. However, to reduce the number of parentheses, we specify that the negation operator is applied before all other logical operators. This means that ¬p ∧ q is the conjunction of ¬p and q, namely,(¬p) ∧ q, not the negation of the conjunction ofp and q, namely ¬(p ∧ q). Another general rule of precedence is that the conjunction operator takes precedence over the disjunction operator, so that p ∧ q ∨ r means (p ∧ q) ∨ r rather than p ∧ (q ∨ r). Because this rule may be difficult to remember, we will continue to use parentheses so that the order of the disjunction and conjunction operators is clear. TABLE 8 Precedence of Logical Operators. Operator Precedence ¬ 1 ∧ 2 ∨ 3 → 4 ↔ 5 Finally, it is an accepted rule that the conditional and biconditional operators → and ↔ have lower precedence than the conjunction and disjunction operators, ∧ and ∨. Consequently, p ∨ q → r is the same as (p ∨ q) → r. We will use parentheses when the order of the conditional operator and biconditional operator is at issue, although the conditional operator has precedence over the biconditional operator. Table 8 displays the precedence levels of the logical operators, ¬, ∧, ∨, →, and ↔. Logic and Bit Operations Computers represent information using bits. A bit is a symbol with two possible values, namely, 0 (zero) and 1 (one). This meaning of the word bit comes from binary digit, because zeros and ones are the digits used in binary representations of numbers. The well-known statistician John Tukey introduced this terminology in 1946. A bit can be used to represent a truth value, because there are two truth values, namely, true and false. As is customarily done, we will use a 1 bit to represent true and a 0 bit to represent false. That is, 1 represents T (true), 0 represents F (false). A variable is called a Boolean variable if its value is either true or false. Consequently, a Boolean variable can be represented using a bit. Truth Value Bit T 1 F 0 Computer bit operations correspond to the logical connectives. By replacing true by a one and false by a zero in the truth tables for the operators ∧, ∨, and ⊕, the tables shown in Table 9 for the corresponding bit operations are obtained. We will also use the notation OR, AND, and XOR for the operators ∨, ∧, and ⊕, as is done in various programming languages. JOHN WILDER TUKEY (1915–2000) Tukey, born in New Bedford, Massachusetts, was an only child. His parents, both teachers, decided home schooling would best develop his potential. His formal education began at Brown University, where he studied mathematics and chemistry. He received a master’s degree in chemistry from Brown and continued his studies at Princeton University, changing his field of study from chemistry to mathematics. He received his Ph.D. from Princeton in 1939 for work in topology, when he was appointed an instructor in mathematics at Princeton. With the start of World War II, he joined the Fire Control Research Office, where he began working in statistics. Tukey found statistical research to his liking and impressed several leading statisticians with his skills. In 1945, at the conclusion of the war, Tukey returned to the mathematics department at Princeton as a professor of statistics, and he also took a position at AT&T Bell Laboratories. Tukey founded the Statistics Department at Princeton in 1966 and was its first chairman. Tukey made significant contributions to many areas of statistics, including the analysis of variance, the estimation of spectra of time series, inferences about the values of a set of parameters from a single experiment, and the philosophy of statistics. However, he is best known for his invention, with J. W. Cooley, of the fast Fourier transform. In addition to his contributions to statistics, Tukey was noted as a skilled wordsmith; he is credited with coining the terms bit and software. Tukey contributed his insight and expertise by serving on the President’s Science Advisory Committee. He chaired several important committees dealing with the environment, education, and chemicals and health. He also served on committees working on nuclear disarmament. Tukey received many awards, including the National Medal of Science. HISTORICAL NOTE There were several other suggested words for a binary digit, including binit and bigit, that never were widely accepted. The adoption of the word bit may be due to its meaning as a common English word. For an account of Tukey’s coining of the word bit, see the April 1984 issue of Annals of the History of Computing
12 1/The Foundations:Logic and Proofs 0 Information is often represented using bit strings,which are lists of zeros and ones.When this is done,operations on the bit strings can be used to manipulate this information. DEFINITION 7 a sequene ofromore bits.The engh of thisstring is the numbero EXAMPLE 12 101010011 is a bit string of length nine We can extend bit operations to bit strings.We define the bitwise OR,bitwise AND.and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR AND,anc XOR of the co e use the symb se XORoperations,respectively EXAMPLE13 and bits to make then asier to ad ho strings Soltion:The bitwise OR.bitwise AND.and bitwise YORof these strings are obtained by taking the OR.AND.and XOR of the corresponding bits,respectively.This gives us 1110111111 bitwise OR 01 0001 0100 bitwise 4.VD 1010101011 bitwise XOR Exercises 1.Which of these positions?What are the d4+x=5. truth values of those that are e)The moon is made of green cheese. a)Boston is the capital of Massachusetts. f)2"≥100. b)Miami is the capital of Florida. 3.What is the negation of each of these propositions? +2=11 f)Answer this question d)The summer in Maine is hot and sunmv. 2.Whichof these are pro ositions?What are the truth values 4.What is the negation of each of these propositions? of those that are propositions? a)lennifer and Teia are friends a)Do not pass go. b)There are 13 items in a baker's dozen. There are no blac k flies in Maine
12 1 / The Foundations: Logic and Proofs TABLE 9 Table for the Bit Operators OR, AND, and XOR. x y x ∨ y x ∧ y x ⊕ y 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 Information is often represented using bit strings, which are lists of zeros and ones. When this is done, operations on the bit strings can be used to manipulate this information. DEFINITION 7 A bit string is a sequence of zero or more bits. The length of this string is the number of bits in the string. EXAMPLE 12 101010011 is a bit string of length nine. ▲ We can extend bit operations to bit strings. We define the bitwise OR, bitwise AND, and bitwise XOR of two strings of the same length to be the strings that have as their bits the OR, AND, and XOR of the corresponding bits in the two strings, respectively. We use the symbols ∨, ∧, and ⊕ to represent the bitwise OR, bitwise AND, and bitwise XOR operations, respectively. We illustrate bitwise operations on bit strings with Example 13. EXAMPLE 13 Find the bitwise OR, bitwise AND, and bitwise XOR of the bit strings 01 1011 0110 and 11 0001 1101. (Here, and throughout this book, bit strings will be split into blocks of four bits to make them easier to read.) Solution: The bitwise OR, bitwise AND, and bitwise XOR of these strings are obtained by taking the OR, AND, and XOR of the corresponding bits, respectively. This gives us 01 1011 0110 11 0001 1101 11 1011 1111 bitwise OR 01 0001 0100 bitwise AND 10 1010 1011 bitwise XOR ▲ Exercises 1. Which of these sentences are propositions? What are the truth values of those that are propositions? a) Boston is the capital of Massachusetts. b) Miami is the capital of Florida. c) 2 + 3 = 5. d) 5 + 7 = 10. e) x + 2 = 11. f ) Answer this question. 2. Which of these are propositions?What are the truth values of those that are propositions? a) Do not pass go. b) What time is it? c) There are no black flies in Maine. d) 4 + x = 5. e) The moon is made of green cheese. f ) 2n ≥ 100. 3. What is the negation of each of these propositions? a) Mei has an MP3 player. b) There is no pollution in New Jersey. c) 2 + 1 = 3. d) The summer in Maine is hot and sunny. 4. What is the negation of each of these propositions? a) Jennifer and Teja are friends. b) There are 13 items in a baker’s dozen. c) Abby sent more than 100 text messages every day. d) 121 is a perfect square
1.1Propositional Logic 13 sitions? 10 Let Stev ap each of these compound propositions asan English sen nails and texts from Jenifer. b) d)Diane rode her bicycle 100 mileson Sunday. pvq 6.Suppose that SmartphoneA has256 MB RAMand 32GB P qv(-PAq) of its camera is4MP:and Smartphone C has 11. Letpand be the propositions 128 MB RAM 132 GB ROM Write the tions usingpand and logical co nectives (including negations). a)It is below freezing and snowing ROM o a higher resolution g c)Smartphone B has more RAM.more ROM,and a d)It is either snowing or below freezing (or both). dfserTesolutio eezing.it is also s s snowing.but it is Smartphone C.then it also has a higher resolution g)That it is below freezing is necessary and sufficient and only if Smartphone B has more RAMthan Smart- phone A 7.Suppose that during the most recent fisca al year.the an- p You final examination r:You pass the course Express each of these propositions as an English sen- tence. nd its net a pq 3 billion dollars.Determine the truth value of each of these propositions for the most recent fiscal year. 8pr)vg→-r) (pAa)v(Ar) 13.Let p andbe the propositions Computer had the la c)Acme snet protruote gar a or ho if Acme Co d only You drveiper hour.but you do not get anuter had the largest annual revenue 8.Let p andbe the propositions p:I bought a lot ry ticket this week I won the million dollar jackpot. d)If you do not drive over 65 miles per hour,then you Express each of these propositions as an English sen- speeding ticket. b)pvq pAg )-pv(pA) g)Whenever you get a speeding ticket.you are driving 9.Let p and be the propositions"Swimming at the Nev over 65 miles per hour. “Sharks ha chofthepotee 14.Let p..andr be the propositions pound propositions as an English sentence. P You get an A on the final a-g b)pAg c)-pvg 0p→g Write thes sitions using
1.1 Propositional Logic 13 5. What is the negation of each of these propositions? a) Steve has more than 100 GB free disk space on his laptop. b) Zach blocks e-mails and texts from Jennifer. c) 7 · 11 · 13 = 999. d) Diane rode her bicycle 100 miles on Sunday. 6. Suppose that SmartphoneA has 256 MB RAM and 32 GB ROM, and the resolution of its camera is 8 MP; Smartphone B has 288 MB RAM and 64 GB ROM, and the resolution of its camera is 4 MP; and Smartphone C has 128 MB RAM and 32 GB ROM, and the resolution of its camera is 5 MP. Determine the truth value of each of these propositions. a) Smartphone B has the most RAM of these three smartphones. b) Smartphone C has more ROM or a higher resolution camera than Smartphone B. c) Smartphone B has more RAM, more ROM, and a higher resolution camera than Smartphone A. d) If Smartphone B has more RAM and more ROM than Smartphone C, then it also has a higher resolution camera. e) Smartphone A has more RAM than Smartphone B if and only if Smartphone B has more RAM than Smartphone A. 7. Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was 138 billion dollars and its net profit was 8 billion dollars, the annual revenue of Nadir Software was 87 billion dollars and its net profit was 5 billion dollars, and the annual revenue of Quixote Media was 111 billion dollars and its net profit was 13 billion dollars. Determine the truth value of each of these propositions for the most recent fiscal year. a) Quixote Media had the largest annual revenue. b) Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue. c) Acme Computer had the largest net profit or Quixote Media had the largest net profit. d) If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue. e) Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue. 8. Let p and q be the propositions p : I bought a lottery ticket this week. q : I won the million dollar jackpot. Express each of these propositions as an English sentence. a) ¬p b) p ∨ q c) p → q d) p ∧ q e) p ↔ q f ) ¬p → ¬q g) ¬p ∧ ¬q h) ¬p ∨ (p ∧ q) 9. Let p and q be the propositions “Swimming at the New Jersey shore is allowed” and “Sharks have been spotted near the shore,” respectively. Express each of these compound propositions as an English sentence. a) ¬q b) p ∧ q c) ¬p ∨ q d) p → ¬q e) ¬q → p f ) ¬p → ¬q g) p ↔ ¬q h) ¬p ∧ (p ∨ ¬q) 10. Let p and q be the propositions “The election is decided” and “The votes have been counted,” respectively. Express each of these compound propositions as an English sentence. a) ¬p b) p ∨ q c) ¬p ∧ q d) q → p e) ¬q → ¬p f ) ¬p → ¬q g) p ↔ q h) ¬q ∨ (¬p ∧ q) 11. Let p and q be the propositions p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical connectives (including negations). a) It is below freezing and snowing. b) It is below freezing but not snowing. c) It is not below freezing and it is not snowing. d) It is either snowing or below freezing (or both). e) If it is below freezing, it is also snowing. f ) Either it is below freezing or it is snowing, but it is not snowing if it is below freezing. g) That it is below freezing is necessary and sufficient for it to be snowing. 12. Let p, q, and r be the propositions p : You have the flu. q : You miss the final examination. r : You pass the course. Express each of these propositions as an English sentence. a) p → q b) ¬q ↔ r c) q → ¬r d) p ∨ q ∨ r e) (p → ¬r) ∨ (q → ¬r) f ) (p ∧ q) ∨ (¬q ∧ r) 13. Let p and q be the propositions p : You drive over 65 miles per hour. q : You get a speeding ticket. Write these propositions using p and q and logical connectives (including negations). a) You do not drive over 65 miles per hour. b) You drive over 65 miles per hour, but you do not get a speeding ticket. c) You will get a speeding ticket if you drive over 65 miles per hour. d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket. e) Driving over 65 miles per hour is sufficient for getting a speeding ticket. f ) You get a speeding ticket, but you do not drive over 65 miles per hour. g) Whenever you get a speeding ticket, you are driving over 65 miles per hour. 14. Let p, q, and r be the propositions p : You get an A on the final exam. q : You do every exercise in this book. r : You get an A in this class. Write these propositions using p, q, and r and logical connectives (including negations)
14 1/The Foundations:Logic and Proofs a)You get an A in this class.but you do not do every a)Coffee or tea comes with dinner. exercise in this bo book and you ong c)To get annh a course in number is class,it is necessary for you to get d)You get an on the final.but you don't do every ex d)You can pay using U.S.dollars or euros. ercise in this book:nevertheless.you get an A in this 20.For each of these sentences.determine whether an in- clusive or.or an exclusive or.is intended.Explain your n a on the final and doing e in Java is required 目You will get anA ifand c)To enter the country you need a passport or a voter egistration carc 15.Let p.g.and r be the pre ositions d)Publish or perish Grizzly bears have been seen in the area. 21.Fo ch of thes sentences,state vhat the sentence me Write and r and logical of or do you think is intended a)To take discrete mathematics.you must have taken a)Berries are ripe alo the trail.but grizzly bears have b) ourse in compute not been seen in the area b u get $2000 back in cash or a 2%car loan. mpany the trail is safe.but be c)Dinner for two includes two items from column A or column are ripe along the trail,hiking is safe if and d the wind chill is 2 feet of snow falls or if d)It isnot safe to hike on thetrail.but griy bears hav 22.Writ h of thes in the form"if p.then not been seen in the area and the berries along the trail Refer to the list of co on ways to ex- )on the trail tobe safe.it is nec press conditional statements provided in this section. ry but no sh the boss's car to get promoted. sufficient that berries not be ripe along the trail and mt出 ave seen in area. od is have been seen in the area and berries are ripe along that you bought the computer less than a vear ago. the trai d)Willy gets caught whenever he cheats. oteyou py if1⊥ elected follows from knowing the right peo- nle ly if monkeys can fly. g)Carol gets seasick whenever she is on a boat. 11. 00 r 23.Write each of these statements in the form"if p.then hether each of these conditional statements in English. Refer to the list of common ways to nd blows from then h)The c)That the Pis onship implies that 1=3 they beat the Lakers 18 De nditional statements d)It is ary to walk 8 miles to get to the top of is true or false I's Poak as a professor.it is sufficient to be world- famous. 1= 3.then dogs can fly df2+2=4hem12=3 )rermyoledoby 19.For each of these sentences,deter rmine whether an in. clusive or,or an exclusive or,is intended.Explain your olaver less than 90 days ago. od only if you bought your CD answer h)Jan will go swimming unless the water is too cold
14 1 / The Foundations: Logic and Proofs a) You get an A in this class, but you do not do every exercise in this book. b) You get an A on the final, you do every exercise in this book, and you get an A in this class. c) To get an A in this class, it is necessary for you to get an A on the final. d) You get an A on the final, but you don’t do every exercise in this book; nevertheless, you get an A in this class. e) Getting an A on the final and doing every exercise in this book is sufficient for getting an A in this class. f ) You will get an A in this class if and only if you either do every exercise in this book or you get an A on the final. 15. Let p, q, and r be the propositions p : Grizzly bears have been seen in the area. q : Hiking is safe on the trail. r : Berries are ripe along the trail. Write these propositions using p, q, and r and logical connectives (including negations). a) Berries are ripe along the trail, but grizzly bears have not been seen in the area. b) Grizzly bears have not been seen in the area and hiking on the trail is safe, but berries are ripe along the trail. c) If berries are ripe along the trail, hiking is safe if and only if grizzly bears have not been seen in the area. d) It is not safe to hike on the trail, but grizzly bears have not been seen in the area and the berries along the trail are ripe. e) For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area. f ) Hiking is not safe on the trail whenever grizzly bears have been seen in the area and berries are ripe along the trail. 16. Determine whether these biconditionals are true or false. a) 2 + 2 = 4 if and only if 1 + 1 = 2. b) 1 + 1 = 2 if and only if 2 + 3 = 4. c) 1 + 1 = 3 if and only if monkeys can fly. d) 0 > 1 if and only if 2 > 1. 17. Determine whether each of these conditional statements is true or false. a) If 1 + 1 = 2, then 2 + 2 = 5. b) If 1 + 1 = 3, then 2 + 2 = 4. c) If 1 + 1 = 3, then 2 + 2 = 5. d) If monkeys can fly, then 1 + 1 = 3. 18. Determine whether each of these conditional statements is true or false. a) If 1 + 1 = 3, then unicorns exist. b) If 1 + 1 = 3, then dogs can fly. c) If 1 + 1 = 2, then dogs can fly. d) If 2 + 2 = 4, then 1 + 2 = 3. 19. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer. a) Coffee or tea comes with dinner. b) A password must have at least three digits or be at least eight characters long. c) The prerequisite for the course is a course in number theory or a course in cryptography. d) You can pay using U.S. dollars or euros. 20. For each of these sentences, determine whether an inclusive or, or an exclusive or, is intended. Explain your answer. a) Experience with C++ or Java is required. b) Lunch includes soup or salad. c) To enter the country you need a passport or a voter registration card. d) Publish or perish. 21. For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended? a) To take discrete mathematics, you must have taken calculus or a course in computer science. b) When you buy a new car from Acme Motor Company, you get $2000 back in cash or a 2% car loan. c) Dinner for two includes two items from column A or three items from column B. d) School is closed if more than 2 feet of snow falls or if the wind chill is below −100. 22. Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements provided in this section.] a) It is necessary to wash the boss’s car to get promoted. b) Winds from the south imply a spring thaw. c) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago. d) Willy gets caught whenever he cheats. e) You can access the website only if you pay a subscription fee. f ) Getting elected follows from knowing the right people. g) Carol gets seasick whenever she is on a boat. 23. Write each of these statements in the form “if p, then q” in English. [Hint: Refer to the list of common ways to express conditional statements.] a) It snows whenever the wind blows from the northeast. b) The apple trees will bloom if it stays warm for a week. c) That the Pistons win the championship implies that they beat the Lakers. d) It is necessary to walk 8 miles to get to the top of Long’s Peak. e) To get tenure as a professor, it is sufficient to be worldfamous. f ) If you drive more than 400 miles, you will need to buy gasoline. g) Your guarantee is good only if you bought your CD player less than 90 days ago. h) Jan will go swimming unless the water is too cold