Stefan Waner and Steven R. Costenoble

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.

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

**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.

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Communication and Reasoning Exercises
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**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 .