bits to make them easier to read.) Solurion: The bitwise OR, bitwise AND, and bitwise XOR of these strings are obteinnely the OR, AND, and XOR of the corresponding bits, respectively. This gives us 0110110110 1100011101 1110111111 bitwise OR 0100010100 bitwise AND 1010101011 bitwise XOR Exercises 1. Which of these sentences are propositions? What are the d) 4+x=5. truth values of those that are propositions? e) The moon is made of green cheese. a) Boston is the capital of Massachusetts. f) 2n100. b) Miami is the capital of Florida. c) 2+3=5. d) 5+7=10. 3. What is the negation of each of these proposiinsey e) x+2=11. a) Mei has an MP3 player. b) There is no pollution in New Jersey. f) Answer this question. c) 2+1=3. 2. Which of these are propositions? What are the truth values d) The summer in Maine is hot and sunny. of those that are propositions? 4. What is the negation of each of these propositime a) Do not pass go. a) Jennifer and Teja are friends. b) What time is it? b) There are 13 items in a baker's dozen. c) There are no black flies in Maine. c) Abby sent more than 100 text messages enen d) 121 is a perfect square. 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 a) Experience with C++ or Java is required. do every exercise in this book or you get an A on the b) Lunch includes soup or salad. final. c) To enter the country you need a passport or a votei 15. Let p,4, and r be the propositions registration card. d) Publish or perish. p : Grizly bears have been seen in the area. q : Hiking is safe on the trail. 21. For each of these sentences, state what the sentence means r : Berries are ripe along the trail. if the logical connective or is an inclusive or (that is, a dis. Write these propositions using p,q, and r and logical junction) versus an exclusive or. a) Berries are ripe along the trail, but grizely bears have a) To take discrete mathematics, you must have taken calculus or a course in computer science. b) Grizzly bears have not been seen in the area and hik- b) When you buy a new car from Acme Motor Company, ing on the trail is safe, but berries are ripe along the you get $2000 back in cash or a 2% car loan. c) If berries are ripe along the trail, hiking is safe if and c) Dinner for two includes two items from column A or three items from column B. only if grizaly bears have not been seen in the area. d) It is not safe to hike on the trail, but grizzly bears have d) School is closed if more than 2 feet of snow falls or if not been seen in the area and the berries along the trail the wind chill is below 100. are ripe. e) For hiking on the trail to be safe, it is necessary but not 22. Write each of these statements in the form "if p, then q in English. [Hint: Refer to the list of common ways to exsufficient that berries not be ripe along the trail and press conditional statements provided in this section.] for grizzly bears not to have been seen in the area. f) Hiking is not safe on the trail whenever grizzly bears a) It is necessary to wash the boss's car to get promoted, have been seen in the area and berries are ripe along b) Winds from the south imply a spring thaw. the trail. c) A sufficient condition for the warranty to be good is false. a) 2+2=4 if and only if I+1=2. b) 1+1=2 if and only if 2+3=4. e) You can access the website only if you pay a subscrip. tion fee. f) Getting elected follows from knowing the right peois true or false. a) If 1+1=2, then 2+2=5. b) If I+1=3, then 2+2=4. c) If I+1=3, then 2+2=5. d) If monkeys can fly, then 1+1=3. Determine whether each of these conditional statements 5 true or false. If I+1=3, then unicoms exist. If 1+1=3, then dogs can fly. If 1+1=2, then dogs can fiy. If 2+2=4, then 1+2=3. 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 worldeach of these sentences, determine whether an inive or, or an exclusive or, is intended. Explain your famous. 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. E.S 40. Explain, withoent using a truth table, why (g)vy)A. because Fred is happy most of the time, and the truth valie (q,vr) a (r,vp) is mue whee p,q. and r have the same truth value and it is falve otherwise. 0.4 can be assigned to the statement "John is happy," becaule Solon is happy slightly less than half the time. Use these truith F. 41. Fixplain, without using a truth able, why (pqr) a values to solve Exercises 4547. (p,v) is true when at Jeast cee of p,4, and r. 45. The truth value of the negation of a proposition in fugry. is true and ar least one is false. beit is false when alt three logic is I minus the truth value of the proposition. What variables bave the same truth value are the truth values of the statements "Fred is not happr" 42. What is the value of x after each of these statements is and "John is not happy?" encounered in a compoter poogram, if x=1 before the 46. The truth value of the conjunction of two propositions in statement is reachsd? a) if x+2=3 then x:x+1 furzy logic is the muinimum of the truth salues of the taxe b) if (x+1=3) OR (2x+2=3) then x=x+1 propositions. What are the truth values of the stateroents c) if (2x+3=5)AND(3x+4=7) then x:=x+1 "Fred and John are happy" and "Neither Fred nor John is d) if (x+1=2)XOR(x+2=3) then x:=x+1 happy?" e) if x
