Hints, Answers, and Solutions to Selected Exercises

1.1 Formal Logic

1. (a) (*q* ∧ *p*) → ¬*r*

(b) If the car will start, then there isn't a blown head gasket or water in the cylinders.

3. (a) (*q* ∨ *r*) → ¬*p*

(b) ¬¬*p* → ¬(*q* ∨ *r*)

(C) If you can vote, then you aren't under 18 or from Mars.

5. (a) If the ground is wet, then it is raining.

(b) If the ground is not wet, then it is not raining.

7. (a) Yes, the first components of equal ordered pairs must be equal.

(b) If *a* = *c*, then (*a*, *b*) = (*c*, *d*).

(c) No. For example, 1 = 1, but (1, 2) ≠ (1, 5).

9.

11. The statements ¬(*p* → *q*) and ¬*p* → ¬*q* are not logically equivalent:

13.

15. (a)

(b) There are only two ...

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