An Introduction to Higher Mathematics - Whitman College

An Introduction to Higher

Mathematics

Patrick Keef

David Guichard

with modi?cations by

Russ Gordon

Whitman College

c 2024

?

Contents

1

Logic

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1

Logical Operations . . .

George Boole . . . .

Quanti?ers . . . . . . .

De Morgan¡¯s Laws . . .

Augustus De Morgan

Mixed Quanti?ers . . . .

Logic and Sets . . . . .

Rene? Descartes . . .

Families of Sets . . . . .

Equivalence Relations . .

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. . . 1

. 6

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. . 11

. 13

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. 20

. . 22

. . 24

2

Proofs

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

29

Direct Proofs . . . . . .

Divisibility . . . . . . .

Existence proofs . . . .

Mathematical Induction

Two Important Results .

Strong Induction . . . .

Well-Ordering Property .

Indirect Proof . . . . .

Euclid of Alexandria

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. 61

32

36

38

41

45

50

54

59

iii

iv

Contents

3

Number Theory

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

Congruence . . . . . . . . . . . . . . .

Carl Friedrich Gauss . . . . . . . .

The spaces Zn . . . . . . . . . . . . .

The Euclidean Algorithm . . . . . . . .

The spaces Un . . . . . . . . . . . . .

The GCD and the LCM . . . . . . . .

The Fundamental Theorem of Arithmetic

Wilson¡¯s Theorem and Euler¡¯s Theorem .

Leonhard Euler . . . . . . . . . . .

Quadratic Residues . . . . . . . . . . .

Gotthold Eisenstein . . . . . . . . .

Sums of Two Squares . . . . . . . . . .

63

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101

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63

68

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77

81

83

87

93

102

4

Functions

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

De?nition and Examples . . . . . . . . . .

Induced Set Functions . . . . . . . . . . .

Injections and Surjections . . . . . . . . . .

More Properties of Injections and Surjections

Pseudo-Inverses . . . . . . . . . . . . . . .

Bijections and Inverse Functions . . . . . .

Cardinality and Countability . . . . . . . .

Uncountability of the Reals . . . . . . . . .

Georg Cantor . . . . . . . . . . . . . .

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134

107

111

114

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119

121

123

130

Bibliography

137

Index

139

1

Logic

Although mathematical ability and opinions about mathematics vary widely, even among educated

people, there is certainly widespread agreement that mathematics is logical. Indeed, properly

conceived, this may be one of the most important de?ning properties of mathematics.

Logical thought and logical arguments are not easy to come by (ponder some of the current

discussions on topics such as abortion, climate change, evolution, immigration, gun restrictions,

or LGBTQI rights to appreciate this statement), nor is it always clear whether a given argument

is logical (that is, logically correct). Logic itself deserves study; the right tools and concepts can

make logical arguments easier to discover and to discern. In fact, logic is a major and active area

of mathematics; for our purposes, a brief introduction will give us the means to investigate more

traditional mathematics with con?dence.

1.1

Logical Operations

Mathematics typically involves combining true (or hypothetically true) statements in various ways

to produce (or prove) new true statements. We begin by clarifying some of these fundamental

ideas.

By a sentence, we mean a statement that has a de?nite truth value of either true (T) or

false (F). For example,

¡°In terms of area, Pennsylvania is larger than Iowa.¡± (F)

¡°The integer 289 is a perfect square.¡± (T)

Because we insist that our sentences have a truth value, we are not allowing sentences such as

¡°Chocolate ice cream is the best.¡±

¡°This statement is false.¡±

1

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