An Introduction to Mathematical Proofs (Textbooks in Mathematics)

413 Pages·2019·2.881 MB·English

by  Nicholas A. Loehr

An Introduction to
Mathematical Proofs

An Introduction to
Mathematical Proofs
Nicholas A. Loehr
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Contents
Preface ix
1 Logic 1
1.1 Propositions, Logical Connectives, and Truth Tables . . . . . . . . . . . . . 1
1.2 Logical Equivalences and IF-Statements . . . . . . . . . . . . . . . . . . . . 9
1.3 IF, IFF, Tautologies, and Contradictions . . . . . . . . . . . . . . . . . . . 18
1.4 Tautologies, Quanti(cid:12)ers, and Universes . . . . . . . . . . . . . . . . . . . . 26
1.5 Quanti(cid:12)er Properties and Useful Denials . . . . . . . . . . . . . . . . . . . 32
1.6 Denial Practice and Uniqueness Statements . . . . . . . . . . . . . . . . . . 40
2 Proofs 47
2.1 De(cid:12)nitions, Axioms, Theorems, and Proofs . . . . . . . . . . . . . . . . . . 47
2.2 Proving Existence Statements and IF Statements . . . . . . . . . . . . . . 56
2.3 Contrapositive Proofs and IFF Proofs . . . . . . . . . . . . . . . . . . . . . 63
2.4 Proofs by Contradiction and Proofs of OR-Statements . . . . . . . . . . . . 70
2.5 Proofs by Cases and Disproofs . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.6 Proving Quanti(cid:12)ed Statements . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.7 More Quanti(cid:12)er Properties and Proofs (Optional) . . . . . . . . . . . . . . 90
Review of Logic and Proofs 97
3 Sets 103
3.1 Set Operations and Subset Proofs . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Subset Proofs and Set Equality Proofs . . . . . . . . . . . . . . . . . . . . 110
3.3 Set Equality Proofs, Circle Proofs, and Chain Proofs . . . . . . . . . . . . 117
3.4 Small Sets and Power Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.5 Ordered Pairs and Product Sets . . . . . . . . . . . . . . . . . . . . . . . . 131
3.6 General Unions and Intersections . . . . . . . . . . . . . . . . . . . . . . . 140
3.7 Axiomatic Set Theory (Optional) . . . . . . . . . . . . . . . . . . . . . . . 148
4 Integers 159
4.1 Recursive De(cid:12)nitions and Proofs by Induction . . . . . . . . . . . . . . . . 159
4.2 Induction Starting Anywhere and Backwards Induction . . . . . . . . . . . 166
4.3 Strong Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.4 Prime Numbers and Integer Division . . . . . . . . . . . . . . . . . . . . . 179
4.5 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
4.6 GCDs and Uniqueness of Prime Factorizations . . . . . . . . . . . . . . . . 192
4.7 Consequences of Prime Factorization (Optional) . . . . . . . . . . . . . . . 197
vii
viii Contents
Review of Set Theory and Integers 203
5 Relations and Functions 211
5.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
5.2 Inverses, Identity, and Composition of Relations . . . . . . . . . . . . . . . 219
5.3 Properties of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5.4 De(cid:12)nition of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
5.5 Examples of Functions and Function Equality . . . . . . . . . . . . . . . . 239
5.6 Composition, Restriction, and Gluing . . . . . . . . . . . . . . . . . . . . . 246
5.7 Direct Images and Preimages . . . . . . . . . . . . . . . . . . . . . . . . . . 254
5.8 Injective, Surjective, and Bijective Functions . . . . . . . . . . . . . . . . . 260
5.9 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
6 Equivalence Relations and Partial Orders 275
6.1 Re(cid:13)exive, Symmetric, and Transitive Relations . . . . . . . . . . . . . . . . 275
6.2 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
6.3 Equivalence Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
6.4 Set Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.5 Partially Ordered Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.6 Equivalence Relations and Algebraic Structures (Optional) . . . . . . . . . 308
7 Cardinality 315
7.1 Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
7.2 Countably In(cid:12)nite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.3 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
7.4 Uncountable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
Review of Functions, Relations, and Cardinality 339
8 Real Numbers (Optional) 349
8.1 Axioms for R and Properties of Addition . . . . . . . . . . . . . . . . . . . 349
8.2 Algebraic Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . 356
8.3 Natural Numbers, Integers, and Rational Numbers . . . . . . . . . . . . . . 362
8.4 Ordering, Absolute Value, and Distance . . . . . . . . . . . . . . . . . . . . 368
8.5 Greatest Elements, Least Upper Bounds, and Completeness . . . . . . . . . 375
Suggestions for Further Reading 383
Index 385
Preface
Thisbookcontainsanintroductiontomathematicalproofs,includingfundamentalmaterial
onlogic,proofmethods,settheory,numbertheory,relations,functions,cardinality,andthe
real number system. The book can serve as the main text for a proofs course taken by
undergraduatemathematicsmajors.Nospeci(cid:12)cprerequisitesareneededbeyondfamiliarity
with high school algebra. Most readers are likely to be college sophomores or juniors who
have taken calculus and perhaps some linear algebra, but we do not assume any knowledge
of these subjects. Anyone interested in learning advanced mathematics could use this text
for self-study.
Structure of the Book
This book evolved from classes given by the author over many years to students at the
College of William & Mary, Virginia Tech, and the United States Naval Academy. I have
divided the book into 8 chapters and 54 sections, including three review sections. Each
sectioncorrespondsverycloselytothematerialIcoverinasingle50-minutelecture.Sections
are further divided into many short subsections, so that my suggested pacing can readily
be adapted for classes that meet for 75 minutes, 80 minutes, or other time intervals. If the
instructor omits all sections and topics designated as \optional," it should be just possible
to (cid:12)nish all of the core material in a semester class that meets for 2250 minutes (typically
forty-(cid:12)ve 50-minute meetings or thirty 75-minute meetings). More suggestions for possible
course designs appear below.
Ihavetriedtocapturethebestfeaturesoflivemathematicslecturesinthepagesofthis
book. New material is presented to beginning students in small chunks that are easier to
digest in a single reading or class meeting. The book maintains the friendly conversational
style of a classroom presentation, without relinquishing the necessary level of precision and
rigor. Throughout this text, you will (cid:12)nd the personal pronouns \I" (the author), \you"
(the reader), and \we" (the author and the reader, working together), reminding us that
teachingandlearningarefundamentallyhumanactivities.Teachingthismateriale(cid:11)ectively
can be as di(cid:14)cult as learning it, and new instructors are often unsure how much time to
spend on the fundamentals of logic and proof techniques. The organization of this book
shows at a glance how one experienced teacher of proofs allocates time among the various
core topics. The text develops mathematical ideas through a continual cycle of examples,
theorems, proofs, summaries, and reviews. A new concept may be introduced brie(cid:13)y via
an example near the end of one section, then examined in detail in the next section, then
recalled as needed in later sections. Every section ends with an immediate review of the
key points just covered, and three review sections give detailed summaries of each major
section of the book. The essential core material is supplemented by more advanced topics
that appear in clearly labeled optional sections.
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