Discrete Mathematical Structures by Pearson

4-8 Discrete Mathematical Structures

Example 4 Determine the truth values of each of the following statements:

(a) 3 + 2 = 6 and 5 + 8 = 13, (b) 8 + 2 = 10 and 2 + 5 = 6

(c) 2 + 2 = 4 and 3 + 5 = 8, (d) 2 + 2 = 5 and 3 + 1 = 5

Solution: The statement ‘p and q’ is true only when both statements are true. Thus, (a) False

(b) False (c) True (d) False.

Example 5 Let a, b and c be any three real numbers and let p: a < b, q: b < c and r: a < c.

Represent the following sentences symbolically:

1. (a ≥ b) or (b < c)

2. (b ≥ c) or (a < c)

3. (a < b) and [(b ≥ c) or (c > a)]

4. (a ≥ b) or [(b < c) and (c > a)]

Solution:

1. ∼p ∨ q

2. ∼q ∨ r

3. p ∧ (∼q ∨ r)

4. ∼p ∨ (q ∧ r)

4.4.5 Conditional Statements

Many mathematical statements are implications: that ...