1. Chapter 3 Introduction to Logic © 2008 Pearson Addison-Wesley.
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2. Chapter 3: Introduction to Logic
3.1 Statements and Quantifiers 3.2 Truth Tables and Equivalent Statements 3.3 The Conditional and Circuits 3.4 More on the Conditional 3.5 Analyzing Arguments with Euler Diagrams 3.6 Analyzing Arguments with Truth Tables
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3. Section 3-4 Chapter 1 More on the Conditional
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4. More on the Conditional
Converse, Inverse, and Contrapositive
Alternative Forms of “If p, then q”
Biconditionals
Summary of Truth Tables
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5. Converse, Inverse, and Contrapositive
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
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6. Example: Determining Related Conditional Statements
Given the conditional statement
If I live in Wisconsin, then I shovel snow,
determine each of the following:
a) the converse b) the inverse c) the contrapositive
Solution
a) 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.
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7. © 2008 Pearson Addison-Wesley. All rights reserved
Equivalences
A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent.
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8. Alternative Forms of “If p, then q”
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.
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9. Example: Rewording Conditional Statements
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.
Solution
a) 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.
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10. © 2008 Pearson Addison-Wesley. All rights reserved
Biconditionals
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
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11. Truth Table for the Biconditional
p if and only if q
p q
T T T
T F F
F T
F F
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12. Example: Determining Whether Biconditionals are True or False
Determine whether each biconditional statement is true or false.
a) = 7 if and only if = 5.
b) 3 = 7 if and only if 4 =
c) = 12 if and only if = 11.
Solution
a) True (both component statements are true)
b) False (one component is true, one false)
c) True (both component statements are false)
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13. Summary of Truth Tables
1. The negation of a statement has truth value opposite of the statement.
The conjunction is true only when both statements are true.
The disjunction is false only when both statements are false.
The biconditional is true only when both statements have the same truth value.
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