Unks� JOHN WILDER TUKEY (1915-2000) Tukey, Born in New Bedford, Massachusetts, Was an Only Child
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16 1 / The Foundations: Logic and Proofs 1-16 Solution: The bitwise OR, bitwise AND, and bitwise XOR ofthese 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 ofthese sentences are propositions? What are the b) There is no pollution in New Jersey. truth values of those that are propositions? c) 2 + 1 = 3. a) Boston is the capital of Massachusetts. d) The summerin Maine is hot and sunny. b) Miami is the capital of Florida. 4. Let p and q be the propositions c) p : I bought a lottery ticket this week. 2 + 3 = 5. d) 5 + 7 = 10. Answer this question. q : I won the million dollar jackpot on Friday. e) x + 2 = II. 1) 2. Which ofthese are propositions? What are the truth values Express each ofthese propositions as an English sentence. of those that are propositions? a) --'p b) p v q c) p --+ q d) p q e) p - q 1) --'p --+ --.q a) Do not pass go. /\ g) --'p --.q --'p (p q) b) What time is it? /\ b) v /\ 5. Let p and be the propositions "Swimming at the New c) There are no black flies in Maine. q Jersey shore is allowed" and "Sharks have been spotted d) 4 +x 5. = near the shore," respectively. Express each of these com The moon is made of green cheese. e) pound propositions as an English sentence. 1) 2n ::: 100. a) --.q b) p /\ q c) --'p v q 3. What is the negation of each ofthese propositions? d) p --+ --.q e) --.q --+ p 1) --'p --+ --.q p --.q --'p (p --.q) a) Today is Thursday. g) _ b) /\ V unkS� 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 fo rmal 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 ofthe war, Tukey returnedto 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 ofvariance, 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 1. 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. 1-1 7 1.1 Propositional Logic 17 6. Let p and q be the propositions "The election is de Write these propositions using p, q, and r and logical cided" and "The votes have been counted," respectively. connectives. Express each of these compound propositions as an a) Yo u get an A in this class, but you do not do every English sentence. exercise in this book. a) ""'p b) p vq b) You get an A on the final,you do every exercise in this c) ""'p/\ q d) q --+ P book, and you get an A in this class. e) ""'q --+ ""'p I) ""'p --+ ""'q c) To get an A in this class, it is necessary for you to get an A on the final. g) p ++ q b) ....,qv(....,P /\ q) d) You get an A on the final, but you don't do every ex Let p and q be the propositions 7. ercise in this book; nevertheless, you get an A in this p : It is below freezing. class. q : It is snowing. e) Getting an A on the final and doingevery exercise in Write these propositions using p and q and logical this book is sufficientfor getting an A in this class. connectives. I) Yo u 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 a) It is below freezing and snowing. final. b) It is below freezingbut not snowing. 11. Let p, q, and r be the propositions c) It is not below freezingand it is not snowing. d) It is either snowing or below freezing (or both). p : Grizzly bears have been seen in the area. e) If it is below freezing, it is also snowing. q : Hiking is safe on the trail. I) It is either below freezingor it is snowing, but it is not r : Berries are ripe along the trail. snowing if it is below freezing. Write these propositions using p, q, and r and logical g) That it is below freezing is necessary and sufficient connectives. for it to be snowing. a) Berries are ripe along the trail, but grizzly bears have 8. Let p, q, and r be the propositions not been seen in the area. p : Yo u have the flu. b) Grizzly bears have not been seen in the area and hiking q : Yo u miss the final examination. 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 r : Yo u pass the course. ) only if grizzly bears have not been seen in the area. Express each ofthese propositions as an English sentence. d) It is not safe to hike on the trail, but grizzly bears have a) p --+ q b) ""'q ++ r not been seen in the area and the berries along the trail c) q --+ ....,r d) p v q V r are ripe. e) (p --+ ....,r) v (q --+ ....,r) e) For hiking on the trail to be safe, it is necessary but I) (p /\q)v(""'q /\r) not sufficient that berries not be ripe along the trail 9. Let p and q be the propositions and for grizzly bears not to have been seen in the area. p Yo u drive over 65 miles per hour. : I) Hiking is not safe on the trail whenever grizzly bears q Yo u get a speeding ticket. : have been seen in the area and berries are ripe along Write these propositions using p and q and logical the trail. connectives. 12. Determine whether these biconditionals are true or false. a Yo u do not drive over 65 miles per hour. ) a 2 + 2 = 4 if and only if + 1 = 2. b) Yo u drive over 65 miles per hour, but you do not get a ) I b) 1 + 1 2 if and only if2 + 3 4. speeding ticket. = = c 1 + = 3 if and only if monkeys can fly. c) Yo u will get a speeding ticket if you drive over ) I d) 0> if and only if2 > l. 65 miles per hour. I 13. Determine whether each of these conditional statements d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket. is true or false. e) Driving over 65 miles per hour is sufficientfor getting a) If I + I = 2, then 2 + 2 = 5. a speeding ticket. b) If 1 + 1 = 3, then 2 + 2 = 4. c I) Yo u get a speeding ticket, but you do not drive over 65 ) If I + I = 3, then 2 + 2 = 5. miles per hour. d) If monkeys can fly, then 1 + I = 3. g) Whenever you get a speeding ticket, you are driving 14. Determine whether each of these conditional statements over 65 miles per hour. is true or false. 10. Let p, q, and r be the propositions a) If I + I = 3, then unicorns exists. p : Yo u get an A on the final exam. b) If 1 + 1 = 3, then dogs can fly. c q : Yo u do every exercise in this book. ) If I + I = 2, then dogs can fly. A 4, r : Yo u get an in this class. d) If2 + 2 = then I + 2 = 3. 18 I / The Foundations: Logic and Proofs I-Iii " 15. For each of these sentences, determine whether an inclu 20. Write each ofthese statements in the form "if p, then q sive or or an exclusive or is intended.