Section 3 3 Truth Tables for Negation Conjunction

Section 3. 3 Truth Tables for Negation, Conjunction, and Disjunction Objectives 1. Use the definitions of negation, conjunction, and disjunction. 2. Construct truth tables. 3. Determine the truth value of a compound statement for a specific case. 12/17/2021 Section 3. 3 1

Truth Tables • Negation (not): Opposite truth value from the statement. • Conjunction (and): Only true when both statements are true. • Disjunction (or): Only false when both statements are false. Negation p ~p T F F T Disjunction Conjunction p q p q T T T T F F F T T F F F F 12/17/2021 Section 3. 3 2

Example 1 Using the Definitions of Negation, Conjunction, and Disjunction Let p and q represent the following statements: p: 10 > 4 This is a true statement q: 3 < 5 This is a true statement Determine the truth value for each statement: a. p q Since both are true, the conjunction is true. b. ~ p q Since p is true then, ~p is false, the conjunction is false. c. p q Since both are true and a disjunction is only false when both components are false, then this is true. c. ~p ~q Since both are false, the disjunction is false. 12/17/2021 Section 3. 3 3

Example 2 Constructing Truth Tables Construct a truth table for ~(p q) • Step 1: First list the simple statements on top and show all the possible truth values. • Step 2: Make a column for p q and fill in the truth values. 12/17/2021 p T T F q T F T p q T F F F Section 3. 3 4

Example 2 continued • Step 3: Construct one more column for ~(p q). The final column tells us that the statement is false only when both p and q are true. For example: p q p q ~(p q) p: Harvard is a college. (true) q: Yale is a college. (true) T T T F ~(p q): It is not true that T F F T Harvard and Yale are colleges. A compound statement that is F T always true is called a tautology. F F F T Is this a tautology? NO 12/17/2021 Section 3. 3 5

Example 3 Constructing a Truth Table Construct a truth table for (~p q) ~q. 12/17/2021 p q ~p ~p q ~q (~p q) ~q T T F F T T T F F T T Section 3. 3 6

Constructing a Truth Table with Eight Cases • There are eight different true-false combinations for compound statements consisting of three simple statements. 12/17/2021 Section 3. 3 7

Example 4 Constructing a Truth Table with Eight Cases Construct a truth table for the following statement: I study hard and ace the final, or I fail the course. Suppose that you study hard, you do not ace the final and you fail the course. Under these conditions, is this compound statement true or false? • Solution: We represent our statements as follows: p: I study hard. q: I ace the final. r: I fail the course. • 12/17/2021 Section 3. 3 8

Example 4 continued • Writing the given statement in symbolic form: • The completed table is: • The statement is True. 12/17/2021 Section 3. 3 9

Example 5 Determining Truth Values for Specific Cases • We can determine the truth value of a compound statement for a specific case in which the truth values of the simple statements are known, without constructing an entire truth table. • Substitute the truth values of the simple statements into the symbolic form of the compound statement and use the appropriate definitions to determining the truth value of the compound statement. 12/17/2021 Section 3. 3 10

Example 5 continued • Use the information in the circle graphs to determine the truth value of the following statement: It is not true that freshmen make up 24% of the undergraduate college population and account for more than one-third of the undergraduate deaths, or seniors do not account for 30% of the undergraduate deaths. 12/17/2021 Section 3. 3 11

Example 5 continued • Substitute the truth values for p, q, and r that we obtained from the circle graphs to determine the truth value for the given compound statement. ~(p q) ~r This is the given compound statement in symbolic form. ~(T T) ~F Substitute the truth value obtained from the graph. F T Replace T T with T. Conjunction is true when both parts are true. T Replace F T with T. Disjunction is true when at least one part is true. We conclude that the given statement is true. 12/17/2021 Section 3. 3 12