Proofs and concepts: the fundamentals of abstract mathematics

220 Pages·2009·0.98 MB·English

by  Morris D.V.

Proofs
and
Concepts
the fundamentals of abstract mathematics
by
Dave Witte Morris and Joy Morris
University of Lethbridge
incorporating material by
P.D. Magnus
University at Albany, State University of New York
Preliminary Version 0.78 of May 2009
This book is offered under the Creative Commons license.
(Attribution-NonCommercial-ShareAlike 2.0)
The presentation of logic in this textbook is adapted from
forallx
An Introduction to Formal Logic
P.D. Magnus
University at Albany, State University of New York
The most recent version of forallx is available on-line at
http://www.fecundity.com/logic
We thank Professor Magnus for making forallx freely available,
and for authorizing derivative works such as this one.
He was not involved in the preparation of this manuscript,
so he is not responsible for any errors or other shortcomings.
Please send comments and corrections to:
[email protected] or [email protected]
(cid:13)c 2006–2009 by Dave Witte Morris and Joy Morris. Some rights reserved.
Portions (cid:13)c 2005–2006 by P.D.Magnus. Some rights reserved.
Brief excerpts are quoted (with attribution) from copyrighted works of various authors.
You are free to copy this book, to distribute it, to display it, and to make derivative works, under the
following conditions: (1) Attribution. You must give the original author credit. (2) Noncommercial.
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to Harmony

Contents
Part I. Introduction to Logic and Proofs
Chapter 1. What is Logic? 3
§1A. Assertions and deductions ....................................................3
§1B. Two ways that deductions can go wrong --------------------------------- 4
§1C. Deductive validity............................................................5
§1D. Other logical notions -------------------------------------------------- 6
§1D.1. Truth-values .............................................................6
§1D.2. Logical truth ----------------------------------------------------- 6
§1D.3. Logical equivalence.......................................................7
§1E. Logic puzzles --------------------------------------------------------- 8
Summary ...........................................................................9
Chapter 2. Propositional Logic 11
§2A. Using letters to symbolize assertions.........................................11
§2B. Connectives --------------------------------------------------------- 12
§2B.1. Not (¬) .................................................................13
§2B.2. And (&)--------------------------------------------------------- 14
§2B.3. Or (∨) ..................................................................16
§2B.4. Implies (⇒)------------------------------------------------------ 18
§2B.5. Iff (⇔) ..................................................................21
Summary ----------------------------------------------------------------- 22
Chapter 3. Basic Theorems of Propositional Logic 23
§3A. Calculating the truth-value of an assertion...................................23
§3B. Identifying tautologies, contradictions, and contingent sentences------------ 25
§3C. Logical equivalence..........................................................26
§3D. Converse and contrapositive ------------------------------------------- 30
§3E. Some valid deductions .......................................................31
§3F. Counterexamples ----------------------------------------------------- 34
Summary ..........................................................................35
i
ii
Chapter 4. Two-Column Proofs 37
§4A. First example of a two-column proof.........................................37
§4B. Hypotheses and theorems in two-column proofs -------------------------- 40
§4C. Subproofs for ⇒-introduction................................................43
§4D. Proof by contradiction ------------------------------------------------ 49
§4E. Proof strategies .............................................................53
§4F. What is a proof? ----------------------------------------------------- 54
Summary ..........................................................................56
Part II. Sets and First-Order Logic
Chapter 5. Sets, Subsets, and Predicates 59
§5A. Propositional Logic is not enough ...........................................59
§5B. Sets and their elements ----------------------------------------------- 60
§5C. Subsets .....................................................................64
§5D. Predicates ---------------------------------------------------------- 65
§5E. Using predicates to specify subsets...........................................68
Summary ----------------------------------------------------------------- 70
Chapter 6. Operations on Sets 71
§6A. Union and intersection ......................................................71
§6B. Set difference and complement ----------------------------------------- 73
§6C. Cartesian product ...........................................................74
§6D. Disjoint sets --------------------------------------------------------- 75
§6E. The power set ...............................................................76
Summary ----------------------------------------------------------------- 78
Chapter 7. First-Order Logic 79
§7A. Quantifiers..................................................................79
§7B. Translating to First-Order Logic --------------------------------------- 81
§7C. Multiple quantifiers .........................................................84
§7D. Negations ----------------------------------------------------------- 85
§7E. Equality ....................................................................88
§7F. Vacuous truth ------------------------------------------------------- 89
§7G. Uniqueness .................................................................89
§7H. Bound variables------------------------------------------------------ 90
§7I. Counterexamples in First-Order Logic ........................................91
Summary ----------------------------------------------------------------- 93
iii
Chapter 8. Quantifier Proofs 95
§8A. The introduction and elimination rules for quantifiers ........................95
§8A.1. ∃-introduction --------------------------------------------------- 95
§8A.2. ∃-elimination............................................................96
§8A.3. ∀-elimination ---------------------------------------------------- 97
§8A.4. ∀-introduction ..........................................................98
§8A.5. Proof strategies revisited ----------------------------------------- 101
§8B. Some proofs about sets .....................................................101
§8C. Theorems, Propositions, Corollaries, and Lemmas ----------------------- 104
Summary .........................................................................105
Part III. Functions
Chapter 9. Functions 109
§9A. Informal introduction to functions ..........................................109
§9B. Official definition---------------------------------------------------- 112
Summary .........................................................................115
Chapter 10. One-to-One Functions 117
Summary .........................................................................121
Chapter 11. Onto Functions 123
§11A. Concept and definition ....................................................123
§11B. How to prove that a function is onto---------------------------------- 124
§11C. Image and pre-image ......................................................126
Summary ---------------------------------------------------------------- 127
Chapter 12. Bijections 129
Summary .........................................................................132
Chapter 13. Inverse Functions 133
Summary .........................................................................135
Chapter 14. Composition of Functions 137
Summary .........................................................................140
iv
Part IV. Other Fundamental Concepts
Chapter 15. Cardinality 143
§15A. Definition and basic properties ............................................143
§15B. The Pigeonhole Principle-------------------------------------------- 146
§15C. Cardinality of a union.....................................................148
§15D. Hotel Infinity and the cardinality of infinite sets ----------------------- 149
§15E. Countable sets ............................................................152
§15F. Uncountable sets --------------------------------------------------- 156
§15F.1. The reals are uncountable .............................................156
§15F.2. The cardinality of power sets ------------------------------------- 157
§15F.3. Examples of irrational numbers........................................157
Summary ---------------------------------------------------------------- 159
Chapter 16. Proof by Induction 161
§16A. The Principle of Mathematical Induction ..................................161
§16B. Proofs about sets -------------------------------------------------- 165
§16C. Other versions of Induction ...............................................168
Summary ---------------------------------------------------------------- 170
Chapter 17. Divisibility and Congruence 171
§17A. Divisibility................................................................171
§17B. Congruence modulo n ---------------------------------------------- 173
Summary .........................................................................176
Chapter 18. Equivalence Relations 177
§18A. Binary relations...........................................................177
§18B. Definition and basic properties of equivalence relations ------------------ 180
§18C. Equivalence classes........................................................182
§18D. Modular arithmetic ------------------------------------------------ 183
§18D.1. The integers modulo 3 ................................................183
§18D.2. The integers modulo n ------------------------------------------ 184
§18E. Functions need to be well defined..........................................185
§18F. Partitions --------------------------------------------------------- 185
Summary .........................................................................187
Part V. Topics
Chapter 19. Elementary Graph Theory 191
§19A. Basic definitions ..........................................................191
§19B. Isomorphic graphs-------------------------------------------------- 195
§19C. Digraphs..................................................................196
§19D. Sum of the valences ------------------------------------------------ 198
Summary .........................................................................201
v
Chapter 20. Isomorphisms 203
§20A. Definition and examples...................................................203
§20B. Proofs that isomorphisms preserve graph-theoretic properties ------------ 204
Summary .........................................................................207
Index of Definitions 209