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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. - PowerPoint PPT Presentation
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 3.3
Truth Tables for
the Conditional
and Bicondition
al
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
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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.
3.3-3
p q p → qCase 1 T T TCase 2 T F FCase 3 F T TCase 4 F F T
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example: Truth Table with a Conditional
Construct a truth table for the statement ~p → ~q.
3.3-4
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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
3.3-5
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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
3.3-6
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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.
3.3-7
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Example 4: A Truth Table Using a Biconditional
Construct a truth table for the statement ~p ↔ (~q → r).
3.3-8
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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
3.3-9
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.
3.3-10
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Example 7: Using Real Data in Compound Statements
3.3-11
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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.
3.3-12
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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
3.3-13
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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.
3.3-14
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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.
3.3-15
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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.
3.3-17
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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.
3.3-18
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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
3.3-20
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.
3.3-21
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Example 9: All Trues, a Tautology
Construct a truth table for the statement (p ⋀ q) → (p ⋁ r).
3.3-22
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Solution
3.3-23
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.
3.3-24
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Example 10: An Implication?
Determine whether the conditional statement [(p ⋀ q) ⋀ q] → q is an implication.
3.3-25
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 10: An Implication?Solution
3.3-26
The statement is a tautology, so it is an implication.
