Introduction to Mathematical Proofs A Transition to Advanced Mathematics

1. Logic
1.1 Statements, Negation, and Compound Statements
1.2 Truth Tables and Logical Equivalences
1.3 Conditional and Biconditional Statements
1.4 Logical Arguments
1.5 Open Statements and Quantifiers
1.6 Chapter Review

2. Deductive Mathematical Systems and Proofs
2.1 Deductive Mathematical Systems
2.2 Euclidean and Non-Euclidean Geometry Systems
2.3 Mathematical Proofs
2.4 Chapter Review

3. Set Theory
3.1 Sets and Subsets
3.2 Set Operations
3.3 Additional Set Operations
3.4 Generalized Set Union and Intersection
3.5 Chapter Review

4. Relations
4.1 Relations
4.2 The Order Relations
4.3 Reflexive, Symmetric, Transitive, and Equivalence Relations
4.4 Equivalence Relations, Equivalence Classes, and Partitions
4.5 Chapter Review

5. Functions
5.1 Functions
5.2 Onto Functions, One-to-One Functions, and One-to-One Correspondences
5.3 Inverse of a Function
5.4 Images and Inverse Images of Sets
5.5 Chapter Review

6. Mathematical Induction
6.1 Mathematical Induction
6.2 The Well-Ordering Principle and the Fundamental Theorem of Arithmetic

7. Cardinalities of Sets
7.1 Finite Sets
7.2 Denumerable and Countable Sets
7.3 Uncountable Sets

8. Proofs from Real Analysis
8.1 Sequences
8.2 Limit Theorems for Sequences
8.3 Monotone Sequences and Subsequences
8.4 Cauchy Sequences

9. Proofs from Group Theory
9.1 Binary Operations and Algebraic Structures
9.2 Groups
9.3 Subgroups and Cyclic Groups

10. Proofs from Combinatorics
10.1 Proofs from Enumerative Combinatorics
10.2 Proofs of Combinatorial Identities
10.3 Proofs from Graph Theory
10.4 Chapter Review

Appendix: Reading and Writing Mathematical Proofs
Answers to Selected Exercises

Biography

Charles E. Roberts Jr. (deceased) was Professor Emeritus in the Department of Mathematics and Computer Science at Indiana State University. He is remembered as a master teacher and prolific researcher. His book, A Mathematical Introduction to Proofs, is also published by CRC Press.

Miklós Bóna received his Ph.D in mathematics from the Massachusetts Institute of Technology in 1997. Since 1999, he has taught at the University of Florida, where, in 2010, he was inducted into the Academy of Distinguished Teaching Scholars. Professor Bóna has mentored numerous graduate and undergraduate students. He is the author of six books and more than 100 research articles, mostly focusing on enumerative and analytic combinatorics. His book, Combinatorics of Permutations, won a 2006 Outstanding Title Award from Choice, the journal of the American Library Association. He is also an Editor-in-Chief for the Electronic Journal of Combinatorics, and for two book series at CRC Press.