Book of Proof
Book of Proof
Richard Hammack Virginia Commonwealth University
Mathematics Textbook Series. Editor: Lon Mitchell 1. Book of Proof by Richard Hammack 2. Linear Algebra by Jim Hefferon 3. Abstract Algebra: Theory and Applications by Thomas Judson
Department of Mathematics & Applied Mathematics Virginia Commonwealth University Richmond, Virginia, 23284
Book of Proof Edition 1.0 © 2009 by Richard Hammack This work is licensed under the Creative Commons Attribution-No Derivative Works 3.0 License
Cover art by Eleni Kanakis, © 2009, all rights reserved, used by permission
A Typeset in 11pt TEX Gyre Schola using PDFL TEX
iii
To Micol, Adriana and Sage
Contents
Preface Introduction
ix x
I
Fundamentals
1. Sets
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. Introduction to Sets The Cartesian Product Subsets Power Sets Union, Intersection, Difference Complement Venn Diagrams Indexed Sets
3
3 8 11 14 17 19 21 24
2. Logic
2.1. Statements 2.2. And, Or, Not 2.3. Conditional Statements 2.4. Biconditional Statements 2.5. Truth Tables for Statements 2.6. Logical Equivalence 2.7. Quantifiers 2.8. More on Conditional Statements 2.9. Translating English to Symbolic Logic 2.10. Negating Statements 2.11. Logical Inference 2.12. An Important Note
28
29 33 36 39 41 44 46 48 50 52 55 56
3. Counting
3.1. 3.2. 3.3. 3.4. 3.5. Counting Lists Factorials Counting Subsets Pascal’s Triangle and the Binomial Theorem Inclusion-Exclusion
57
57 64 67 72 75
vi
II
How to Prove Conditional Statements
4. Direct Proof
4.1. 4.2. 4.3. 4.4. 4.5. Theorems Definitions Direct Proof Using Cases Treating Similar Cases
81
81 83 85 90 92
5. Contrapositive Proof
5.1. Contrapositive Proof 5.2. Congruence of Integers 5.3. Mathematical Writing
94
94 97 99
6. Proof by Contradiction
6.1. 6.2. 6.3. 6.4. Proving Statements with Contradiction Proving Conditional Statements by Contradiction Combining Techniques Some Words of Advice
103
104 107
Richard Hammack Virginia Commonwealth University
Mathematics Textbook Series. Editor: Lon Mitchell 1. Book of Proof by Richard Hammack 2. Linear Algebra by Jim Hefferon 3. Abstract Algebra: Theory and Applications by Thomas Judson
Department of Mathematics & Applied Mathematics Virginia Commonwealth University Richmond, Virginia, 23284
Book of Proof Edition 1.0 © 2009 by Richard Hammack This work is licensed under the Creative Commons Attribution-No Derivative Works 3.0 License
Cover art by Eleni Kanakis, © 2009, all rights reserved, used by permission
A Typeset in 11pt TEX Gyre Schola using PDFL TEX
iii
To Micol, Adriana and Sage
Contents
Preface Introduction
ix x
I
Fundamentals
1. Sets
1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. Introduction to Sets The Cartesian Product Subsets Power Sets Union, Intersection, Difference Complement Venn Diagrams Indexed Sets
3
3 8 11 14 17 19 21 24
2. Logic
2.1. Statements 2.2. And, Or, Not 2.3. Conditional Statements 2.4. Biconditional Statements 2.5. Truth Tables for Statements 2.6. Logical Equivalence 2.7. Quantifiers 2.8. More on Conditional Statements 2.9. Translating English to Symbolic Logic 2.10. Negating Statements 2.11. Logical Inference 2.12. An Important Note
28
29 33 36 39 41 44 46 48 50 52 55 56
3. Counting
3.1. 3.2. 3.3. 3.4. 3.5. Counting Lists Factorials Counting Subsets Pascal’s Triangle and the Binomial Theorem Inclusion-Exclusion
57
57 64 67 72 75
vi
II
How to Prove Conditional Statements
4. Direct Proof
4.1. 4.2. 4.3. 4.4. 4.5. Theorems Definitions Direct Proof Using Cases Treating Similar Cases
81
81 83 85 90 92
5. Contrapositive Proof
5.1. Contrapositive Proof 5.2. Congruence of Integers 5.3. Mathematical Writing
94
94 97 99
6. Proof by Contradiction
6.1. 6.2. 6.3. 6.4. Proving Statements with Contradiction Proving Conditional Statements by Contradiction Combining Techniques Some Words of Advice
103
104 107