# Introduction to Logic

## Exercises for Section 2: Logical Equivalence, Tautologies, and Contradictions

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Construct the truth tables for the following expressions.

 1. p(~q) 2. p(~q) 3. ~(~p)p 4. p(~p) 5. (~p)(~q) 6. (~p)(~q) 7. (pq)r 8. p(qr) 9. p(qr) 10. (pq)(pr)

Use truth tables to verify the following logical equivalences:

11. pp p
12. pp p
13. pq qp. . . the Commutative Law for disjunction.
14. pq qp. . . the Commutative Law for conjunction.
15. ~(pq) (~p)(~q)
16. ~(p(~q)) (~p)q
17. (pq)r p(qr) . . . the Associative Law for conjunction.
18. (pq)r p(qr). . . the Associative Law for disjunction.
19. p(q(~q)) p
20. p(~p) q(~q)

Use truth tables to check whether the following are tautologies, contradictions or neither.

 21. p(~p) 22. pp 23. p~(pq) 24. p~(pq) 25. p~(pq) 26. q~(p(~p)).

Apply the stated logical equivalence to each of the following statements:

27. p(~p); the Commutative law
28. p(~q); the Commutative law
29. ~(p(~q)); De Morgan's Law
30. ~(q(~q)); De Morgan's Law
31. p~(pq); De Morgan's Law
32. q~(p(~p)); De Morgan's Law
33. p((~p)q); the Distributive Law
34. (~q)((~p)q); the Distributive Law.

Use logical equivalences to rewrite each of the following sentences. If possible, rewrite more simply.

35. It is not true that both I am Julius Caesar and you are a fool.

36. It is not true that either I am Julius Caesar or you are a fool.

37. Either it's raining and I have forgotten my umbrella, or it's raining and I have forgotten my hat.

38. I forgot my hat or my umbrella, and I forgot my hat or my glasses.

39. My computer crashes when it has been on a long time, and when it's not the case that either the air is dry or the moon is not full.

40. The study determined that the market crashed because interest rates rose, or because it was not the case that both earnings rose and the moon was not full.

41. The warning light will come on if the pressure drops while the temperature is high, or if the pressure drops while not both the emergency override and the manual controls are activated.

42. The alarm will sound if the door is opened and the override button is not pushed while the alarm is activated, or if there is motion and it is not the case that either the override button is pushed or the alarm is not activated.

Communication and Reasoning Exercises

43. . If two popositions are logically equivalent, what can be said about their truth tables?

44. If a proposition neither a tautology nor a contradiction, what can be said about its truth table?

45. . Can an atomic statement be a tautology or a contradiction? Explain.

46. Can a statement with a single variable p be a tautology or a contradiction? Explain.

47. .If A and B are two (possibly compound statements) such that AB is a contradiction, what can you say about A and B?

48. If A and B are two (possibly compound statements) such that AB is a tautology, what can you say about A and B?

49. Your friend has been telling everyone that all tautologies are logically equivalent to each other. Is he correct? Explain.

50. Another friend has been going around saying that, if two statements are logically equivalent to each other, then they must either be tautologies or contradictions. Is she correct? Explain.

Last Updated: September, 2001