Discrete Structures Homework #1

Due: Thursday, Sept. 5, 2002

Section 1.1 Exercises (pp. 14-19): 5, 8, 17, 18

5. Several forms of negation are given for each of the following statements. Which are correct?

a. The answer is either 2 or 3.

1. Neither 2 nor 3 is the answer.

2. The answer is not 2 or not 3.

3. The answer is not 2 and it is not 3.

b. Cucumbers are green and seedy.

1. Cucumbers are not green and not seedy.

2. Cucumbers are not green or not seedy.

3. Cucumbers are green and not seedy.

c. 2 < 7 and 3 is odd.

1. 2 > 7 and 3 is even.

2. 2 7 and 3 is even.

3. 2 7 or 3 is odd.

4. 2 7 or 3 is even.

8. Let A, B, and C be the following statements:

A Roses are red.

B Violets are blue.

C Sugar is sweet.

Translate the following compound statments into symbolic notation.

a. Roses are red and violets are blue.

b. Roses are red, and either violets are blue or sugar is sweet.

c. Whenever violets are blue, roses are red and sugar is sweet.

d. Roses are red only if violets aren't blue or sugar is sour.

e. Roses are red and, if sugar is sour, then violets aren't blue or sugar is sweet.

17. Verify by constructing truth tables that the following wffs are tautologies.

a.

b.

c.

d.

e. (DeMorgan's law)

f. (DeMorgan's law)

g.

18. Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side. You may use any of the equivalencies in the list on page 8 or in Exercise 17.

a.

b.

c.

d.

e.