Chapter 3
Introduction
to Logic
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Chapter 3: Introduction to Logic
3.1
3.2
3.3
3.4
3.5
3.6
Statements and Quantifiers
Truth Tables and Equivalent Statements
The Conditional and Circuits
More on the Conditional
Analyzing Arguments with Euler Diagrams
Analyzing Arguments with Truth Tables
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Chapter 1
Section 3-4
More on the Conditional
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More on the Conditional
•
•
•
•
Converse, Inverse, and Contrapositive
Alternative Forms of “If p, then q”
Biconditionals
Summary of Truth Tables
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Converse, Inverse, and Contrapositive
Conditional
Statement
Converse
pq
If p, then q
q p
If q, then p
Inverse
p q
Contrapositive
q p
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If not p, then
not q
If not q, then
not p
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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.
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Equivalences
A conditional statement and its contrapositive
are equivalent, and the converse and inverse
are equivalent.
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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.
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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.
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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.
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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
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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)
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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.
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3-4-13