Discrete Mathematics. An Open Introduction

Acknowledgements Preface How to use this book Introduction and Preliminaries What is Discrete Mathematics? Mathematical Statements Atomic and Molecular Statements Implications Predicates and Quantifiers Exercises Sets Notation Relationships Between Sets Operations On Sets Venn Diagrams Exercises Functions Describing Functions Surjections, Injections, and Bijections Image and Inverse Image Exercises Counting Additive and Multiplicative Principles Counting With Sets Principle of Inclusion/Exclusion Exercises Binomial Coefficients Subsets Bit Strings Lattice Paths Binomial Coefficients Pascal's Triangle Exercises Combinations and Permutations Exercises Combinatorial Proofs Patterns in Pascal's Triangle More Proofs Exercises Stars and Bars Exercises Advanced Counting Using PIE Counting Derangements Counting Functions Exercises Chapter Summary Chapter Review Sequences Describing Sequences Exercises Arithmetic and Geometric Sequences Sums of Arithmetic and Geometric Sequences Exercises Polynomial Fitting Exercises Solving Recurrence Relations The Characteristic Root Technique Exercises Induction Stamps Formalizing Proofs Examples Strong Induction Exercises Chapter Summary Chapter Review Symbolic Logic and Proofs Propositional Logic Truth Tables Logical Equivalence Deductions Beyond Propositions Exercises Proofs Direct Proof Proof by Contrapositive Proof by Contradiction Proof by (counter) Example Proof by Cases Exercises Chapter Summary Chapter Review Graph Theory Definitions Exercises Trees Properties of Trees Rooted Trees Spanning Trees Exercises Planar Graphs Non-planar Graphs Polyhedra Exercises Coloring Coloring in General Coloring Edges Exercises Euler Paths and Circuits Hamilton Paths Exercises Matching in Bipartite Graphs Exercises Chapter Summary Chapter Review Additional Topics Generating Functions Building Generating Functions Differencing Multiplication and Partial Sums Solving Recurrence Relations with Generating Functions Exercises Introduction to Number Theory Divisibility Remainder Classes Properties of Congruence Solving Congruences Solving Linear Diophantine Equations Exercises Selected Hints Selected Solutions List of Symbols Index