Page 100 - Discrete Mathematics and Its Applications

1.6 Rules of Inference 79

10. For each of these sets of premises, what relevant conclu- b) “Each of five roommates, Melissa, Aaron, Ralph, Ve-
sion or conclusions can be drawn? Explain the rules of in- neesha, and Keeshawn, has taken a course in discrete
ference used to obtain each conclusion from the premises. mathematics. Every student who has taken a course in
a) “If I play hockey, then I am sore the next day.” “I discrete mathematics can take a course in algorithms.
use the whirlpool if I am sore.” “I did not use the Therefore, all five roommates can take a course in
whirlpool.” algorithms next year.”
c) “All movies produced by John Sayles are wonder-
b) “IfIwork,itiseithersunnyorpartlysunny.”“Iworked
lastMondayorIworkedlastFriday.”“Itwasnotsunny ful. John Sayles produced a movie about coal miners.
on Tuesday.” “It was not partly sunny on Friday.” Therefore, there is a wonderful movie about coal min-
ers.”
c) “All insects have six legs.” “Dragonflies are insects.”
d) “There is someone in this class who has been to
“Spiders do not have six legs.” “Spiders eat dragon-
France. Everyone who goes to France visits the
flies.”
Louvre. Therefore, someone in this class has visited
d) “Every student has an Internet account.” “Homer does the Louvre.”
not have an Internet account.” “Maggie has an Internet 15. For each of these arguments determine whether the argu-
account.” ment is correct or incorrect and explain why.
e) “All foods that are healthy to eat do not taste good.” a) All students in this class understand logic. Xavier is
“Tofu is healthy to eat.” “You only eat what tastes a student in this class. Therefore, Xavier understands
good.” “You do not eat tofu.” “Cheeseburgers are not logic.
healthy to eat.” b) Every computer science major takes discrete math-
f) “I am either dreaming or hallucinating.” “I am not ematics. Natasha is taking discrete mathematics.
dreaming.” “If I am hallucinating, I see elephants run- Therefore, Natasha is a computer science major.
ning down the road.” c) All parrots like fruit. My pet bird is not a parrot.There-
fore, my pet bird does not like fruit.
11. Show that the argument form with premises
p 1 ,p 2 ,...,p n and conclusion q → r is valid if the d) Everyone who eats granola every day is healthy. Linda
argument form with premises p 1 ,p 2 ,...,p n ,q, and is not healthy. Therefore, Linda does not eat granola
conclusion r is valid. every day.
16. For each of these arguments determine whether the argu-
12. Show that the argument form with premises (p ∧ t) → ment is correct or incorrect and explain why.
(r ∨ s), q → (u ∧ t), u → p, and ¬s and conclusion
q → r is valid by first using Exercise 11 and then us- a) Everyone enrolled in the university has lived in a dor-
ing rules of inference from Table 1. mitory. Mia has never lived in a dormitory. Therefore,
Mia is not enrolled in the university.
13. For each of these arguments, explain which rules of in- b) A convertible car is fun to drive. Isaac’s car is not a
ference are used for each step. convertible. Therefore, Isaac’s car is not fun to drive.
a) “Doug, a student in this class, knows how to write c) Quincy likes all action movies. Quincy likes the movie
programs in JAVA. Everyone who knows how to write Eight Men Out. Therefore, Eight Men Out is an action
programs in JAVA can get a high-paying job. There- movie.
fore, someone in this class can get a high-paying job.” d) All lobstermen set at least a dozen traps. Hamilton is a
b) “Somebody in this class enjoys whale watching. Ev- lobsterman. Therefore, Hamilton sets at least a dozen
ery person who enjoys whale watching cares about traps.
ocean pollution. Therefore, there is a person in this 17. What is wrong with this argument? Let H(x) be “x is
class who cares about ocean pollution.” happy.” Given the premise ∃xH(x), we conclude that
c) “Each of the 93 students in this class owns a personal H(Lola). Therefore, Lola is happy.
computer. Everyone who owns a personal computer 18. What is wrong with this argument? Let S(x, y) be “x is
can use a word processing program. Therefore, Zeke, shorter than y.” Given the premise ∃sS(s, Max), it follows
a student in this class, can use a word processing pro- that S(Max, Max). Then by existential generalization it
gram.” follows that ∃xS(x, x), so that someone is shorter than
himself.
d) “Everyone in New Jersey lives within 50 miles of the
ocean. Someone in New Jersey has never seen the 19. Determine whether each of these arguments is valid. If an
ocean. Therefore, someone who lives within 50 miles argument is correct, what rule of inference is being used?
of the ocean has never seen the ocean.” If it is not, what logical error occurs?
2
14. For each of these arguments, explain which rules of in- a) If n is a real number such that n> 1, then n > 1.
2
ference are used for each step. Suppose that n > 1. Then n> 1.
2
b) If n is a real number with n> 3, then n > 9.
a) “Linda, a student in this class, owns a red convertible. 2
Everyone who owns a red convertible has gotten at Suppose that n ≤ 9. Then n ≤ 3. 2
least one speeding ticket. Therefore, someone in this c) If n is a real number with n> 2, then n > 4.
2
class has gotten a speeding ticket.” Suppose that n ≤ 2. Then n ≤ 4.