Introduction to Logic
by
Stefan Waner and Steven R. Costenoble

Exercises
for
Section 2: Logical Equivalence, Tautologies and Contradictions

1. Statements and Logical Operators 2. Logical Equivalence, Tautologies ... 3. The Conditional and the Biconditional Main Logic Page "Real World" Page
Answers to Odd-Numbered Exercises

Construct the truth tables for the following expressions.

Use truth tables to verify the following logical equivalences:

11. ppp
12. ppp
13. pqqp. . . the Commutative Law for disjunction.
14. pqqp. . . the Commutative Law for conjunction.
15. ~(pq)(~p)(~q)
16. ~(p(~q))(~p)q
17. (pq)rp(qr) . . . the Associative Law for conjunction.
18. (pq)rp(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.

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.

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.

1. Statements and Logical Operators 2. Logical Equivalence, Tautologies ... 3. The Conditional and the Biconditional Main Logic Page "Real World" Page
Answers to Odd-Numbered Exercises

We would welcome comments and suggestions for improving this resource. Mail us at:
Stefan Waner (matszw@hofstra.edu) Steven R. Costenoble (matsrc@hofstra.edu)
Last Updated: July, 1996
Copyright © 1996 StefanWaner and Steven R. Costenoble