Logic as a Tool

Chapter 6Answers and Solutions to Selected Exercises

Section 1.1

1. 1.1.1 (a) Yes. (b) Yes. (c) No. (d) No. (e) Yes. (f) No. (Imperative sentence.) (g) No. (“her” is not defined in this sentence, so truth value cannot be assigned to it.) (h) Yes. (i) No. (Question.) (i) Yes. (k) Yes. (True.) (l) No. (Either truth value can be assigned.)
2. 1.1.2 (a) true. (c) true. (e) false. (g) true.
3. 1.1.3
   1. (a)  Antecedent: “John talks.” Consequent: “Everyone else listens.”
   2. (c)  Antecedent: “John talks.” Consequent: “Everyone else listens.”
   3. (e)  Antecedent: “The cube of the integer n is positive.”

      Consequent: “An integer n is positive.”

   4. (g)  Antecedent: “A function is differentiable.”

      Consequent: “The function is continuous.”

   5. (i)  Antecedent: “A function is continuous.”

      Consequent: “The function is differentiable.”

4. 1.1.4 (a) Neither. (c) Sufficient. (e) Necessary and sufficient. (g) Necessary. (i) Necessary and sufficient.
5. 1.1.5 Denote:
   1. “The Earth rotates around itself” by A (true),
   2. “The Moon rotates around the Earth” by B (true),
   3. “The Sun rotates around the Moon” by C (false),
   4. “The Sun rotates around the Earth” by D (false),
   5. “The Earth rotates around the Moon” by E (false).

      Then we have:

      (a) : false. (c) : false. (e) : true.

6. 1.1.6
   1. (a)  By the truth definition of the implication, must be ...