# Introduction to Logic

## Exercises for Section 4: Tautological Implications and Tautological Equivalences

Answers To see an answer to any odd-numbered exercise, just click on the exercise number.

Use truth tables to check the following tautologies from the lists at the end of the section.

 1. Modus Tollens 2. Simplification: pqp 3. Addition 4. Disjunctive Syllogism: [(pq)(~p)]q 5. Transitivity 6. Double Negative 7. Commutative Law: (pq)(qp) 8. Commutative Law: (pq)(qp) 9. Switcheroo 10. Contrapositive.

Show that the following are not tautologies by giving examples of statements p and q for which these implications are false.

 11. (pq)p 12. p(pq) 13. (pq)(qp) 14. ~(pq)[(~p)(~q)] 15. (pq)(~p~q) 16. ((pq)(pr))(qr)

Write each of the following in symbolic form, and then decide whether it is a tautology or not.

17. If I am hungry and thirsty, then I am hungry.

18. If I am hungry or thirsty, then I am hungry.

19. If it's not true that roses are red and violets are blue, then roses are not red and violets are not blue.

20. If roses are not red or violets are not blue, then it's not true that roses are red and violets are blue.

21. For me to bring my umbrella it's necessary that it rain; therefore if it does not rain I will not bring my umbrella.

22. For me to bring my umbrella it's sufficient that it rain; therefore if it does not rain I will not bring my umbrella.

23. For me to bring my umbrella it's necessary and sufficient that it rain; therefore if it does not rain I will not bring my umbrella.

24. For me to bring my umbrella it's necessary and sufficient that it rain; therefore if I do not bring my umbrella it will not rain.

25. For me to pass math it is sufficient that I have a good teacher; therefore, I will either have a good teacher or I will not pass math.

26. For me to pass math it is necessary that I have a good teacher; therefore, I will either have a good teacher or I will not pass math.

27. I am either tired or hungry, but I am not tired, so I must be hungry.

28. I am either smart or athletic, and I am athletic, so I must not be smart.

29. To get good grades it is necessary to study, and if you get good grades you will get a good job; therefore, it is sufficient to study to get a good job.

30. To get good grades it is sufficient to study, and to get a good job it is necessary to get good grades; therefore, if you study you will get a good job.

31. To get good grades it is necessary to study, but John did not get good grades; therefore John did not study.

32. To get good grades it is necessary and sufficient to study, but Jill did not study; therefore Jill will not get good grades.

Communication and Reasoning Exercises

33. How would you convert a tautology of the form AB into a tautological implication?

34. How would you convert a tautology of the form (AB)(CD) into two tautological implications?

35. Complete the following sentence. A tautological equivalence can be expressed as tautological implications.

36. Complete the following sentence. If A(BC) is a tautological implication, then, givenand, we can always deduce .

Last Updated: October, 2001