Teall TB Chapter 05 - Fordham University



Chapter 5

Setting a Table—Truth Tables

in Propositional Logic

TRANSLATION

Translate the following English-language sentences into logical notation.

1. Ginger and the Professor are stranded on a desert island.

Answer: G • P

2. Neither Gilligan nor the Skipper went on a three-hour cruise.

Answer: ~(G v S)

3. If Jan and Marcia had hair of gold, then Cindy was in curls or the boys were all alone.

Answer: (J • M) ( (C v B)

4. Either if Scooby-Doo hangs out with Shaggy, then Daphne hangs with Fred, or Shaggy does not hang out with Thelma if Scooby-Doo eats Scooby snacks.

Answer: (S ( D) v (E ( ~T)

5. The Beav does not like Eddie Haskell only if both Wally plays baseball and both June and Ward work full time.

Answer: ~B ( [W • (J • F)]

6. Either I am going to twenty meetings in five days and will go insane or if I remain sane, then I will have to escape to Puerto Rico.

Answer: (T • I) v ( S ( E)

7. The President has a high popularity rating and is not running the risk of impeachment only if it is not the case that Hillary Clinton and Monica Lewinsky are members of the same country club.

Answer: (P • ~I) ( ~(H • M)

8. Either you will graduate or you will not graduate and still get a good job.

Answer: G v (~G • J)

9. If a professor and student are involved in a study project, then either they will get a paper published or the student will graduate with honors and the professor will get tenure.

Answer: (P • S) ( [T v (G • W)]

10. Either both if the President dies in a plane crash, then the Vice President will become the new President and Newt will be the Speaker of the House forever or Ed will become president only if Chuck is Valedictorian.

Answer: [P ( V) • N] v (E ( C)

11. If a professor and student are involved in a study project, then either they will get a paper published or the student will graduate with honors and the professor will get tenure.

Answer: (P • S) ( [G v (W • T)]

12. Either I am going to New York City and to Boston during Memorial Day weekend or my wife is staying home.

Answer: (N • B) v W

13. It is not the case that either Mel Gibson or Bruce Springsteen won Oscars and Kevin Spacey did not.

Answer: N [(M v B) • ~K]

14. If Bill Clinton and Ted Kennedy do not go to Ft. Lauderdale for spring break, then Hillary Clinton will go home to Arkansas only if Chelsea Clinton or Socks has the run of the White House.

Answer: (~B • ~T) ( [H ( (C v S)]

15. All students are busy with exams and are not thinking about going home early.

Answer: S • ~T

16. If Marcia Brady will be homecoming queen or will get the upstairs bedroom, then she will break her nose and not become the most popular person in high school.

Answer: (M v B) ( (N • ~P)

17. Anyone who likes Jimmy Buffett is a Parrothead only if they will listen to his music and they are not passed out and have sprung a few leaks.

Answer: P ( [(L • ~P) • S]

18. Greg Brady will be a hippie and go out with Florence Henderson.

Answer: G • F

19. MSMC will go undefeated if and only if NJIT plays MSMC in soccer and either SUNY-New Paltz does not play its next game or MSMC recruits five seven foot centers.

Answer: M ( [N • (~S v R)]

20. I am getting a lot of presents, and I am not getting a lump of coal for Christmas.

Answer: P • ~L

21. Brad Pitt and Matthew McConaughey are not Academy Award winners.

Answer: ~B • ~M

22. Adam Sandler sings Lunch Lady Land if he and Chris Farley are in Billy Madison.

Answer: (S • F) ( A

23. It is not that case that either if Donald Sutherland is Keifer’s father, then Keifer dated Julia Roberts or Robert DeNiro is the taxi driver only if Samuel L. Jackson is the hit man.

Answer: ~[(D ( K) v (R ( S)]

24. Either Danny and Keith play guitar or Mrs. Partridge and Mr. Kincade will ride on top of the bus.

Answer: (D • K) v (P • K)

25. Scooby is scared of ghosts if and only if both Thelma has lost her glasses and either Fred or Shaggy eat Scooby snacks.

Answer: S ( [T • (F v E)]

26. If you pass this class, then if you will want to be a logician, then you will need to go to graduate school.

Answer: P ( (L ( G)

27. If either Ren has a dime in his ear or Stimpy is really Killer Kadoogan, then Superman is from the planet Krypton if Spiderman and Batman are humans.

Answer: (R v S) ( [(P • B) ( K]

28. Gilligan was stranded on a desert island only if Mr. Magoo was not on the same island.

Answer: G ( ~M

29. The Skipper was a mighty sailing man and either the Professor or Mary Ann did not have their own shack.

Answer: S • (~P v ~M)

30. Either both Trapper and Hawkeye had their own still, or if Henry was commander, then Frank was not commander.

Answer: (T • H) v (C ( ~F)

31. It is not the case that if the Partridge Family or the Brady Bunch did not own a bus, then Eddie's father was not married.

Answer: ~[(~P v ~B) ~ E]

32. Either we were all good this year and we get the presents we want, or if we do not get the presents we want, then we were naughty and Santa knows it.

Answer: (G • P) v [~P ( (N • S)]

33. The weather outside is frightful and inside is so delightful if and only if Rudolph has a red nose and he could not play in the reindeer games.

Answer: (F • D) ( (R · ~P)

34. All students are busy with exams or thinking about going home early.

Answer: S v T

35. Either this test will be easy for everyone, or if everyone studies, then there will not be anyone who fails.

Answer: T v (E ( ~F)

36. Neither my heat works nor will my oven work if the toaster is broken.

Answer: ~[H v (T ( O)]

37. Gilligan was stranded on a desert island and Mr. Magoo was not on the same island.

Answer: G • ~M

38. The Skipper was a mighty sailing man only if either the Professor or Mary Ann had their own shack.

Answer: S ( (P v M)

39. Both either Trapper or Hawkeye had their own still, and if Henry was commander, then Frank was not commander.

Answer: (T v H) • (C ( ~F)

40. It is not the case that if the Partridge Family or the Brady Bunch did not own a bus, then Eddie's father was married.

Answer: ~[(~P v ~B) ( E]

Truth Tables

Determine the type of proposition (using truth tables) for the following.

1. [(A ( B) • C] ( {[~A v (A v B) • C}

Answer: Contingent

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2. (A ( ~C) ( (C • B)

Answer: Contingent

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3. [A v (B • C)] ( [(A v B) • (A v B)]

Answer: Contingent

[pic]

4. ~{(E • F) v [~G ( (E v G)]}

Answer: Contingent

[pic]

5. ~(A v C) ( (~B • A)

Answer: Contingent

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6. ~(A ( C) v (B • A)

Answer: Contingent

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Using the method of truth tables, determine what the relationship is between the following statements.

7. ~[(M ( N) • (N ( M)] (~~M • ~N) v ~(N ( M)

Answer: Equivalent

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8. ~[(A ( B) v D] ~D • ~(~A v B)

Answer: Equivalent

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9. (X v Y) ( Z [(X v Y) ( Z] • [(Y v X) • Z]

Answer: Consistent

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10. [(X • Z) ( (Y v Z)] ~ [X ( (Y ( ~X)]

Answer: Consistent

[pic]

11. ~[~(A • C)] ~~A ( ~C

Answer: Contradictory

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Use truth tables to determine whether the following arguments are valid or invalid.

12. ~(D v G) ( (G • D)

~D ( (~G ( D)

G v (D ( ~D)

( ~[(G • D) v ~D]

Answer: Invalid

[pic]

13. M • (N v R)

~N ( (M ( ~R)

( (R v M) v (N • R)

Answer: Valid

[pic]

14. [(A v C) v B]

[(A v C) v B] ( ~(A ( C)

( ~(A ( C)

Answer: Valid

[pic]

15. ~X v (Z • W)

Z ( (Y v Z)

~~X • Y

( (Z • W)

Answer: Valid

[pic]

16. ~M ( N

(~M ( M) • N

( N v M

Answer: Valid

[pic]

17. (A • B) ( (D v C)

D

( (C v B)

Answer: Valid

[pic]

18. (A v B) ( ~(B • A)

~A ( B

( ~B v ~A

Answer: Valid

[pic]

19. ~(X ( Z)

Y v ~[Z v (Y ( X)]

X • ~Y

( [(Z • Y) v X] • ~X

Answer: Valid

[pic]

20. B v A

~(C • B)

~A ( ~(B v C)

( (A • B) • (~A • ~B)

Answer: Invalid

[pic]

21. Y ( Z

( (Z ( Y) • (Y ( Z)

Answer: Valid

[pic]

22. M v (L v M)

~[M • (N • L)]

N • ~L

( ~L v (M • ~M)

Answer: Valid

[pic]

23. (A ( B) v ~B

~{[~~A • (~B v A)] ( ~~B}

~[(A • B) ( (~A v ~B)]

( (A v ~B)

Answer: Valid

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24. Z • (~Z v Y)

X ( (Z ( X)

~Y

( ~[(Z ( Y) v ~X]

Answer: Valid

[pic]

25. ~(C • D) ( E

B v (D v ~E)

C • ~B

~[(C ( ~D) v C]

D ( ~B

( E • ~E

Answer: Valid

[pic]

26. (A ( C)

[(A v C) v B] ( (A ( C)

( ~[(A v C) v B]

Answer: Valid

[pic]

27. ~X v (Z • Y)

W ( (W v Z)

~~X • Y

( (Z • Y)

Answer: Valid

[pic]

28. ~(A • ~B) v C

C ( B

( A ( (B ( C)

Answer: Valid

[pic]

29. ~[X • (~Z v X)]

X

Z • ~X

( ~Z v X

Answer: Valid

[pic]

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