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Chapter 3
Introduction
to Logic
 2012 Pearson Education, Inc.
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Chapter 3: Introduction to Logic
3.1
3.2
3.3
3.4
3.5
Statements and Quantifiers
Truth Tables and Equivalent Statements
The Conditional and Circuits
More on the Conditional
Analyzing Arguments with Euler
Diagrams
3.6 Analyzing Arguments with Truth Tables
 2012 Pearson Education, Inc.
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Section 3-4
More on the Conditional
 2012 Pearson Education, Inc.
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More on the Conditional
•
•
•
•
Converse, Inverse, and Contrapositive
Alternative Forms of “If p, then q”
Biconditionals
Summary of Truth Tables
 2012 Pearson Education, Inc.
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Converse, Inverse, and Contrapositive
Conditional
Statement
Converse
Inverse
Contrapositive
p→q
If p, then q
q→ p
If q, then p
∼ p →∼ q
∼ q →∼ p
If not p, then
not q
If not q, then
not p
 2012 Pearson Education, Inc.
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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.
 2012 Pearson Education, Inc.
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Equivalences
A conditional statement and its contrapositive are
equivalent, and the converse and inverse are
equivalent.
 2012 Pearson Education, Inc.
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Alternative Forms of “If p, then q”
The conditional p → q 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.
 2012 Pearson Education, Inc.
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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.
 2012 Pearson Education, Inc.
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Biconditionals
The compound statement p if and only if q (often
abbreviated p iff q) is called a biconditional. It is
symbolized p ↔ q, and is interpreted as the
conjunction of the two conditionals p → q and q → p.
 2012 Pearson Education, Inc.
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Truth Table for the Biconditional
p if and only if q
q
p↔q
T
T
T
T
F
F
F
T
F
F
F
T
p
 2012 Pearson Education, Inc.
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Example: Determining Whether
Biconditionals are True or False
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.
Solution
a) True (both component statements are true)
b) False (one component is true, one false)
c) True (both component statements are false)
 2012 Pearson Education, Inc.
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Summary of Truth Tables
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.
 2012 Pearson Education, Inc.
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