Why Read This Book? - Index of

34 Chapter 1 Language and Mathematics
Solution Applying DeMorgan’s laws, we have
1. ¬(p ∧ q ∧ r) ⇔¬p∨¬q∨¬r 2. ¬[(p ∨ q) ∧ (r ∨ s)] ⇔¬(p ∨ q) ∨¬(r ∨ s) ⇔ (¬p ∧¬q) ∨ (¬r ∧¬s) �
Example 1.4.2 Use DeMorgan’s laws to express in words a negation of the
following statements.
1. Jacob has brown hair and blue eyes.
2. Either I’m crazy or there is a pink elephant floating overhead.
3. Meghan is at least 25 years old, she has a valid driver’s license, and she
either has her own insurance or has purchased coverage from the car rental
company.
Solution Don’t hesitate to word the statements in a way that makes them easy
to understand.
1. Either Jacob does not have brown hair, or he does not have blue eyes.
2. I am not crazy, and there is no pink elephant floating overhead.
3. Either Meghan is under 25, or she doesn’t have a valid driver’s license, or she
has no insurance of her own and has not purchased coverage from the car rental
company. �
EXERCISE 1.4.3 State in words a negation for each of the following statements.
(a) Either Melanie is naturally blond, or she bleaches her hair.
(b) Eric has a boarding pass and either a driver’s license or passport.
(c) Either f is not continuous or it crosses the x-axis at some point. (f is a given
function.)
(d) Been there; done that.
1.4.2 Negations of If-Then Statements
In Section 1.2, we defined p → q to be logically equivalent to ¬p ∨ q. To construct
a negation of p → q, we can use this fact with DeMorgan’s law.
¬(p → q) ⇔¬(¬p ∨ q) ⇔ (¬¬p) ∧¬q ⇔ p ∧¬q
This might be a little confusing at first, but later you will want to think of it in
the following way. If someone makes a claim that p implies q, then he or she is
claiming that the truth of p is by necessity accompanied by the truth of q. If you