2012 Pearson Education, Inc. Slide 3-4-1 Chapter 3 Introduction to Logic.
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2012 Pearson Education, Inc. Slide 3-4-1 Chapter 3 Chapter 3 Introduction Introduction to Logic to Logic
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2012 Pearson Education, Inc. Slide Section 3-4 More on the Conditional
Transcript of 2012 Pearson Education, Inc. Slide 3-4-1 Chapter 3 Introduction to Logic.
2012 Pearson Education, Inc. Slide 3-4-1
Chapter 3Chapter 3Introduction Introduction to Logicto Logic
2012 Pearson Education, Inc. Slide 3-4-2
Chapter 3: Chapter 3: Introduction to LogicIntroduction to Logic
3.1 Statements and Quantifiers3.2 Truth Tables and Equivalent Statements3.3 The Conditional and Circuits 3.4 More on the Conditional3.5 Analyzing Arguments with Euler
Diagrams3.6 Analyzing Arguments with Truth Tables
2012 Pearson Education, Inc. Slide 3-4-3
Section 3-4Section 3-4More on the Conditional
2012 Pearson Education, Inc. Slide 3-4-4
• Converse, Inverse, and Contrapositive• Alternative Forms of “If p, then q”• Biconditionals• Summary of Truth Tables
More on the ConditionalMore on the Conditional
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Conditional Statement
If p, then q
Converse If q, then p
Inverse If not p, then not q
Contrapositive If not q, then not p
p q
q p
q p
p q
Converse, Inverse, and ContrapositiveConverse, Inverse, and Contrapositive
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Given the conditional statementIf I live in Wisconsin, then I shovel snow, determine each of the following:
a) the converse b) the inverse c) the contrapositive
Solutiona) If I shovel snow, then I live in Wisconsin.b) If I don’t live in Wisconsin, then I don’t shovel snow.c) If I don’t shovel snow, then I don’t live in Wisconsin.
Example: Determining Related Example: Determining Related Conditional StatementsConditional Statements
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A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent.
EquivalencesEquivalences
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The conditional can be translated in any of the following ways.
If p, then q. p is sufficient for q.
If p, q. q is necessary for p.
p implies q. All p are q.
p only if q. q if p.
p q
Alternative Forms of “If Alternative Forms of “If pp, then , then qq””
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Write each statement in the form “if p, then q.”a) You’ll be sorry if I go.b) Today is Sunday only if yesterday was Saturday.c) All Chemists wear lab coats.
Solutiona) If I go, then you’ll be sorry.b) If today is Sunday, then yesterday was Saturday.c) If you are a Chemist, then you wear a lab coat.
Example: Rewording Conditional Example: Rewording Conditional StatementsStatements
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The compound statement p if and only if q (often abbreviated p iff q) is called a biconditional. It is symbolized , and is interpreted as the conjunction of the two conditionals
p q and .p q q p
BiconditionalsBiconditionals
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p if and only if q p q T T T
T F F
F T F
F F T
p q
Truth Table for the BiconditionalTruth Table for the Biconditional
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Determine whether each biconditional statement is true or false.
a) 5 + 2 = 7 if and only if 3 + 2 = 5.b) 3 = 7 if and only if 4 = 3 + 1.c) 7 + 6 = 12 if and only if 9 + 7 = 11.
Solutiona) True (both component statements are true)b) False (one component is true, one false) c) True (both component statements are false)
Example: Determining Whether Example: Determining Whether Biconditionals are True or FalseBiconditionals are True or False
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1. The negation of a statement has truth value opposite of the statement.
2. The conjunction is true only when both statements are true.
3. The disjunction is false only when both statements are false.
4. The biconditional is true only when both statements have the same truth value.
Summary of Truth TablesSummary of Truth Tables
