Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 3.3 Truth Tables for the Conditional and Biconditional

Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Truth tables for conditional and biconditional Self-contradictions, Tautologies, and Implications 3.3-2

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Conditional The conditional statement p → q is true in every case except when p is a true statement and q is a false statement pqp → q Case 1TTT Case 2TFF Case 3FTT Case 4FFT

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example: Truth Table with a Conditional Construct a truth table for the statement ~p → ~q

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example: Truth Table with a Conditional Solution Construct a standard four case truth table. pq~p~p → ~q~q TTFFTTFF TFTFTFTF F F T T T T F T F T F T It’s a conditional, the answer lies under →

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Biconditional The biconditional statement, p ↔ q means that p → q and q → p or, symbolically (p → q) ⋀ (q → p) order of steps FTFTFTFFFcase 4 FFTFTTFTFcase 3 TTFFFFTFTcase 2 TTTTTTTTT case 1 p)p)(q(qq)q)(p(pqp 3.3-6

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: A Truth Table Using a Biconditional Construct a truth table for the statement ~p ↔ (~q → r)

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: A Truth Table Using a Biconditional pqr~p~p ↔ (~q → r)r) TTTTFFFFTTTTFFFF TTFFTTFFTTFFTTFF T T T F T T T F F F F F T T T T T F T F T F T F F F T T F F T T TFTFTFTFTFTFTFTF F F F T T T T F 5

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: Using Real Data in Compound Statements

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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–

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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–

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: All Falses, a Self- Contradiction Construct a truth table for the statement (p ↔ q) ⋀ (p ↔ ~q)

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: All Falses, a Self- Contradiction Solution The statement is a self-contradiction or a logically false statement.

Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: All Trues, a Tautology Construct a truth table for the statement (p ⋀ q) → (p ⋁ r)

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution The statement is a tautology or a logically true statement. Example 9: All Trues, a Tautology

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Implication An implication is a conditional statement that is a tautology. The consequent will be true whenever the antecedent is true

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 10: An Implication? Determine whether the conditional statement [(p ⋀ q) ⋀ q] → q is an implication

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 10: An Implication? Solution The statement is a tautology, so it is an implication.