Statements of Symbolic Logic Name Symbolic Form Negation ~p Conjunction p  q Disjunction p  q Conditional p  q Biconditional p  q Negation ~(p) "It is false that,..." " Is not true that,.." " It is not the case that.."

3.2: Truth Tables: Negation, Conjunction, and Disjunction p q r T F A truth table is used to determine when a compound statement is true or false. p q T T T F F T F F p T F 2

Negation (not): Opposite truth value from the statement. Truth Tables Negation p ~p T F Negation (not): Opposite truth value from the statement. Conjunction p q p  q T T T F F T F F Conjunction (and): Only true when both statements are true. 3

Truth Tables Disjunction (or): Only false when both statements are false. Disjunction p q p  q T T T F F T F F Disjunction p q p  q T T T T F F T F F F 4

Construct a Truth Table Construct a truth table for ~p ⋀ ~q. p q ~p ~q ~p ⋀ ~q T T T F F T F F 5

Constructing a Truth Table Construct a truth table for (~p  q)  ~q. p q ~p  q ~q (~p  q)  ~q T T T F F T F F p q ~p ~p  q ~q (~p  q)  ~q T T T F F T F F 6

Truth Table with a Negation Construct a truth table for ~(~q ⋁ p). p q T T T F F T F F 7

Truth Table with a 8 Cases Construct a truth table for (p ^ ~q) v r. p q r T T T T T F T F T T F F F T T F T F F F T F F F 8

Determine the Truth Value of a Compound Statement Determine the truth value for each simple statement. Then, using these truth values, determine the truth value of the compound statement. 15 is less than or equal to 9. Let p: 15 is less than 9. q: 15 is equal to 9. 9

Truth Tables for Conditional Statements 3.3: Truth Tables: Conditional, biconditional Truth Tables for Conditional Statements Conditional p q p  q T T T F F T F F Conditional p q p  q T T T F F T F F p  q antecedent  consequent If p then q. A conditional is false only when the antecedent is true and the consequent is false. 10

Construct a truth table for ~q  ~p p q T T T F F T F F p q T T T F F T F F 11

Biconditional Statements p  q p if and only if q: p  q and q  p True only when the component statements have the same value. Truth table for the Biconditional p q T T T F F T F F p q T T T F F T F F 12

A Truth Table Using a Biconditional Construct a truth table for the statement ~p ↔ (~q → r). p q r T F T F T F 13

Using Real Data in Compound Statements The graph on the next slide represents the student population by age group in 2009 for the State College of Florida (SCF). Use this graph to determine the truth value of the following compound statements. 14

Using Real Data in Compound Statements If 37% of the SCF population is younger than 21 or 26% of the SCF population is age 21–30, then 13% of the SCF population is age 31–40. 15

Using Real Data in Compound Statements 3% of the SCF population is older than 50 and 8% of the SCF population is age 41–50, if and only if 19% of the SCF population is age 21–30. 16

Construct a truth table for p  ~ p Some compound statement are false in all possible cases. Such statements are called self-contradictions. 17

Tautology A tautology is a compound statement that is always true. When every truth value in the answer column of the truth table is true, the statement is a tautology. 18

Construct a truth table for the statement (p ⋀ q) → (p ⋁ r) Construct a truth table for the statement (p ⋀ q) → (p ⋁ r). Is this a tautology? p q r T F T F T F 19

Construct a truth table for [( p  q) ~ p]  q T T T F F T F F p q T T T F F T F F Conditional statements that are tautologies are called implications. 20