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

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Section 3.3

Truth Tables for

the Conditional

and Bicondition

al

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What You Will Learn

Truth tables for conditional and

biconditional

Self-contradictions, Tautologies,

and Implications

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Conditional

The conditional statement p → q is true in every case except when p is a true statement and q is a false statement.

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p q p → qCase 1 T T TCase 2 T F FCase 3 F T TCase 4 F F T

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Example: Truth Table with a Conditional

Construct a truth table for the statement ~p → ~q.

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Example: Truth Table with a Conditional SolutionConstruct a standard four case truth

table.p q ~p → ~q

TTFF

TFTF

FFTT

TTFT

FTFT

It’s a conditional, the answer lies under →.

231

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BiconditionalThe biconditional statement, p ↔ q means that p → q and q → p or, symbolically (p → q) (⋀ q → p).

5647231order of steps

FTFTFTFFFcase 4

FFTFTTFTFcase 3TTFFFFTFT

case 2

TTTTTTTTTcase 1p)(qq)(pqp

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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.

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Example 4: A Truth Table Using a Biconditional

Construct a truth table for the statement ~p ↔ (~q → r).

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Example 4: A Truth Table Using a Biconditionalp q r ~p ↔ (~q → r)TTTTFFFF

TTFFTTFF

TTTFTTTF

FFFFTTTT

TFTFTFTF

FFTTFFTT2 31 4

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TFTFTFTF

FFFTTTTF5

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Example 7: Using Real Data in Compound StatementsThe 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.

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Example 7: Using Real Data in Compound Statements

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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.

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Example 7: Using Real Data in Compound StatementsSolutionLetp: 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

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Example 7: Using Real Data in Compound StatementsSolutionOriginal statement:

(p ⋁ q) → rp and r are true, q is false

(T ⋁ F) → TT → T T

The original statement is true.

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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.

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Example 7: Using Real Data in Compound StatementsSolutionLetp: 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) ↔ r3.3-16

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Example 7: Using Real Data in Compound StatementsSolutionOriginal statement:

(p ⋀ q) ↔ rp and q are true, r is false

(T ⋀ T) ↔ FT ↔ F F

The original statement is false.

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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.

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Example 8: All Falses, a Self-Contradiction

Construct a truth table for the statement (p ↔ q) (⋀ p ↔ ~q).

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Example 8: All Falses, a Self-ContradictionSolution

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The statement is a self-contradiction or a logically false statement.

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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.

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Example 9: All Trues, a Tautology

Construct a truth table for the statement (p ⋀ q) → (p ⋁ r).

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Solution

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The statement is a tautology or a logically true statement.

Example 9: All Trues, a Tautology

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Implication

An implication is a conditional statement that is a tautology.

The consequent will be true whenever the antecedent is true.

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Example 10: An Implication?

Determine whether the conditional statement [(p ⋀ q) ⋀ q] → q is an implication.

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Example 10: An Implication?Solution

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The statement is a tautology, so it is an implication.