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Chapter 1
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Prologue
Quantitative Reasoning
Daily Life
Quantitative
Skills
College Course
Work
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Career
Experiences
Slide 1-3
Prologue
Six Math Misconceptions
1.
2.
3.
4.
5.
6.
Math requires a special brain.
Math in modern issues is too complex.
Math makes you less sensitive.
Math makes no allowance for creativity.
Math provides exact answers.
Math is irrelevant to my life.
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Prologue
What is mathematics?
Language
Math
Sum of its branches
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Way to model the world
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1-A
Importance of Logic
Parade Magazine
Ask Marilyn Column
Question:
“What is the most important thing a person can do to
improve his or her critical thinking skills?”
Phyllis Evitch, Rice Lake, Wisconsin
Answer:
“Study logic. Without a sound foundation in the principles
of reasoning, you’ll be less able to understand your world,
and the ramifications of this will ripple through everything
from work to play. Even worse, you won’t realize what
you’re missing.”
Marilyn Vos Savant
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Slide 1-6
1-A
Definitions



Logic is the study of the methods and principles
of reasoning.
An argument uses a set of facts or assumptions,
called premises, to support a conclusion.
A fallacy is a deceptive argument—an argument
in which the conclusion is not well supported
by the premises.
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Slide 1-7
1-A
Fallacy Structures
Appeal to Popularity
Many people believe p is true; therefore ... p is true.
False Cause
A came before B; therefore ... A caused B.
Appeal to Ignorance
There is no proof that p is true;
therefore ... p is false.
Hasty Generalization
A and B are linked one or a few times;
therefore ... A causes B or vice versa.
Limited Choice
p is false; therefore ... only q can be true.
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Slide 1-8
1-A
Fallacy Structures
Appeal to Emotion
p is associated with a positive emotional response;
therefore . . . p is true.
Personal Attack
I have a problem with the person or group claiming p.
p is not true.
Circular Reasoning
p is true.
p is restated in different words.
Diversion (Red Herring)
p is related to q and I have an argument concerning q;
therefore . . . p is true.
Straw Man
I have an argument concerning a distorted version of p;
therefore . . . I hope you are fooled into concluding I
have an argument concerning the real version of p.
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1-B
Negations (Opposites)
p
T
F
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not p
F
T
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1-B
And Statements (Conjunctions)
p
q
p and q
T
T
F
F
T
F
T
F
T
F
F
F
Note: Conjunction is false unless both p and q are true.
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1-B
Or Statements (Disjunctions)
p
q
p or q
T
T
F
F
T
F
T
F
T
T
T
F
Note: Disjunction is true unless both p and q are false.
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Slide 1-12
If . . . Then Statements
(Conditionals)
1-B
p
q
if p, then q
T
T
F
F
T
F
T
F
T
F
T
T
Note: Conditional is true unless p is true and q is false.
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Slide 1-13
1-B
Truth Table Practice
p
q
T
T
T
F
F
T
F
F
~ p
~ q
Note: ~ signifies
NEGATION
 signifies AND
 signifies OR
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p q
~ p  ~ q ~ ( p  q)
Practice by writing the
truth values of each
row in the table
above.
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1-B
Truth Table Practice
p
q
~ p
~ q
p q
T
T
F
F
T
T
F
F
T
F
F
Note: ~ signifies
NEGATION
 signifies AND
 signifies OR
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~ p  ~ q ~ ( p  q)
F
F
Practice by writing the
truth values of each
row in the table
above.
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1-B
Truth Table Practice
p
q
~ p
~ q
p q
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
F
F
Note: ~ signifies
NEGATION
 signifies AND
 signifies OR
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~ p  ~ q ~ ( p  q)
Practice by writing the
truth values of each
row in the table
above.
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1-B
Truth Table Practice
p
q
~ p
~ q
p q
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
Note: ~ signifies
NEGATION
 signifies AND
 signifies OR
Copyright © 2005 Pearson Education, Inc.
~ p  ~ q ~ ( p  q)
Practice by writing the
truth values of each
row in the table
above.
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1-B
Truth Table Practice
p
q
~ p
~ q
p q
T
T
F
F
T
F
F
T
F
F
T
F
T
T
F
T
T
F
F
T
T
F
F
T
T
F
T
T
Note: ~ signifies
NEGATION
 signifies AND
 signifies OR
Copyright © 2005 Pearson Education, Inc.
~ p  ~ q ~ ( p  q)
Practice by writing the
truth values of each
row in the table
above.
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Converse, Inverse, and
Contrapositive
1-B
Conditional:
If it is raining,
then I will bring an umbrella to work.
Converse:
If I bring an umbrella to work,
then it must be raining.
Inverse:
If it is not raining,
then I will not bring an umbrella to work.
Contrapositive:
If I do not bring an umbrella to work,
then it must not be raining.
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Slide 1-19
1-C
Definitions

A set is a collection of objects; the individual
objects are the members of the set. We often
describe sets by listing their members within a
pair of braces, {}. If there are too many members
to list, we can use three dots, …, to indicate a
continuing pattern.
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Slide 1-20
1-C
Real Number Venn Diagram
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Slide 1-21
Venn Diagram for
Categorical Propositions
1-C
All S are P
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Slide 1-22
Venn Diagram for
Categorical Propositions
1-C
No S are P
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Slide 1-23
Venn Diagram for
Categorical Propositions
1-C
Some S are P
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Slide 1-24
Venn Diagram for
Categorical Propositions
1-C
Some S are not P
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Negations for
Categorical Propositions
Proposition
Negation
All S are P
Some S are not P
No S are P
Some S are P
Some S are P
No S are P
Some S are not P
All S are P
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1-C
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1-C
Venn Diagram of Blood Types
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Slide 1-27
1-D
Two Types of Arguments
Inductive Reasoning
Deductive Reasoning
specific premises
general premises
general conclusion
specific conclusion
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Slide 1-28
Basic Forms of Conditional
Deductive Arguments
1-D
Affirming the Antecedent:
If one gets a college degree, then one can get a
good job.
Marilyn has a college degree.
Marilyn can get a good job.
Valid
(modus ponens)
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Slide 1-29
Basic Forms of Conditional
Deductive Arguments
1-D
Affirming the Consequent:
If one gets a college degree, then one can get a
good job.
Marilyn gets a good job.
Marilyn has a college degree.
Invalid
(inverse fallacy)
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Slide 1-30
Basic Forms of Conditional
Deductive Arguments
1-D
Denying the Antecedent:
If one gets a college degree, then one can get a
good job.
Marilyn does not have a college degree.
Marilyn cannot get a good job.
Invalid
(converse fallacy)
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Slide 1-31
Basic Forms of Conditional
Deductive Arguments
1-D
Denying the Consequent:
If one gets a college degree, then one can get a
good job.
Marilyn does not have a good job.
Marilyn does not a college degree.
Valid
(modus tollens)
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Slide 1-32
1-D
Inductive Counterexample
Consider the following algebraic expression: n2  n  11
It appears that
n2  n  11
will always equal a
prime number
when n ≥ 0.
n
n2  n  11
0
02  0  11  11
(prime)
1
12  1  11  11
(prime)
2
22  2  11  13
(prime)
Or does it?
3
32  3  11  17
(prime)
How about n = 11?
4
42  4  11  23
(prime)
5
52  5  11  31
(prime)
112  11 + 11 = 121
(a non-prime counterexample)
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Slide 1-33
1-E
Critical Thinking In Everyday Life

General Guidelines.
1. Read (or listen) carefully.
2. Look for hidden assumptions.
3. Identify the real issue.
4. Use visual aids.
5. Understand all the options.
6. Watch for fine print and missing information.
7. Are other conclusions possible?
Copyright © 2005 Pearson Education, Inc.
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