Section 3.3 Truth Tables for the Conditional and Biconditional

What You Will Learn Truth tables for conditional and biconditional Self-contradictions, Tautologies, and Implications

Conditional p q p → q Case 1 T Case 2 F Case 3 Case 4 The conditional statement p → q is true in every case except when p is a true statement and q is a false statement.

Example: Truth Table with a Conditional Construct a truth table for the statement ~p → ~q.

Example: Truth Table with a Conditional Solution Construct a standard four case truth table. p q ~p → ~q T F T F F T T F F T 1 3 2 It’s a conditional, the answer lies under →.

Biconditional The biconditional statement, p ↔ q means that p → q and q → p or, symbolically (p → q) ⋀ (q → p). 5 6 4 7 2 3 1 order of steps F T case 4 case 3 case 2 case 1 p) (q q) (p q p

Biconditional The biconditional statement, p ↔ q is true only when p and q have the same truth value, that is, when both are true or both are false.

Example 4: A Truth Table Using a Biconditional Construct a truth table for the statement ~p ↔ (~q → r).

Example 4: A Truth Table Using a Biconditional q r ~p ↔ (~q → r) T F T F T F F T F T F T T F T F 1 5 2 4 3

Example 7: 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.

Example 7: Using Real Data in Compound Statements

Example 7: 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.

Example 7: Using Real Data in Compound Statements Solution Let p: 37% of the SCF population is younger than 21. q: 26% of the SCF population is age 21–30. r: 13% of the SCF population is age 31–40. Original statement can be written: (p ⋁ q) → r

Example 7: Using Real Data in Compound Statements Solution Original statement: (p ⋁ q) → r p and r are true, q is false (T ⋁ F) → T T → T T The original statement is true.

Example 7: 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.

Example 7: Using Real Data in Compound Statements Solution Let p: 3% of the SCF population is older than 50. q: 8% of the SCF population is age 41–50. r: 19% of the SCF population is age 21–30. Original statement can be written: (p ⋀ q) ↔ r

Example 7: Using Real Data in Compound Statements Solution Original statement: (p ⋀ q) ↔ r p and q are true, r is false (T ⋀ T) ↔ F T ↔ F F The original statement is false.

Self-Contradiction A self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is false, then the statement is a self-contradiction.

Example 8: All Falses, a Self-Contradiction Construct a truth table for the statement (p ↔ q) ⋀ (p ↔ ~q).

Example 8: All Falses, a Self-Contradiction Solution The statement is a self-contradiction or a logically false statement.

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.

Example 9: All Trues, a Tautology Construct a truth table for the statement (p ⋀ q) → (p ⋁ r).

Example 9: All Trues, a Tautology Solution The statement is a tautology or a logically true statement.

Implication An implication is a conditional statement that is a tautology. The consequent will be true whenever the antecedent is true.

Example 10: An Implication? Determine whether the conditional statement [(p ⋀ q) ⋀ q] → q is an implication.

Example 10: An Implication? Solution The statement is a tautology, so it is an implication.