CK-12-Pre-Calculus Concepts

16.3. Negative Statements http://www.ck12.org

Start by building up to theoriginalstatement. Then, add a column that completely negates the original statement.
Notice that there is only one row where the final negated statement is true. That is whenPis false andQis true.
Therefore, the negation of the original sentence is true when “I am not cold” and “it is snowing”.

Vocabulary

De Morgan’s Lawtransforms a conjunction into a disjunction using negation.
Atautologyis a logical statement that is always true. A tautology is a type of basic theorem.

Guided Practice

1. Show that the following statements are equivalent in a truth table. The symbol≡means equivalent. To be
   equivalent in this case means to be true at the same time and false at the same time.
   P∧Q≡∼(∼P∨∼Q)


2. Write a sentence in two different ways illustrating the mathematical statement in Guided Practice #1.


3. Demonstrate the following tautology in a truth table.
   B→(A∨∼A)
   Answers:


4. While a truth table is not a proof, it can help you recognize when two statements have the same truth values.

TABLE16.12:

P Q ∼P ∼Q ∼P∨∼Q ∼ (∼ P∨ ∼

Q)

P∧Q

T T F F F T T

T F F T T F F

F T T F T F F

F F T T T F F

Notice that the final two columns are identical.
2.P∧Q: I like movies and I like TV.∼(∼P∨∼Q): It is not the case that either I don’t like movies or I don’t like
TV.

3. A tautology is a logical statement that is always true.

TABLE16.13:

A ∼A B A∨∼A B→(A∨∼A)

T F F T T

T F T T T

F T T T T

F T F T T

Practice

I’m either going to go skiing or snowboarding next weekend.

1. Identify the atomic statements in the above sentence and use logical connectives to rewrite the sentence with