A Discrete Transition to Advanced Mathematics (2023)

Copyright
Contents
Preface
Preface to the Second Edition
Chapter 1. Sets and Logic
	1.1. Sets
	1.2. Set Operations
	1.3. Partitions
	1.4. Logic and Truth Tables
	1.5. Quantifiers
	1.6. Implications
Chapter 2. Proofs
	2.1. Proof Techniques
	2.2. Mathematical Induction
	2.3. The Pigeonhole Principle
Chapter 3. Number Theory
	3.1. Divisibility
	3.2. The Euclidean Algorithm
	3.3. The Fundamental Theorem of Arithmetic
	3.4. Divisibility Tests
	3.5. Number Patterns
Chapter 4. Combinatorics
	4.1. Getting from Point A to Point B
	4.2. The Fundamental Principle of Counting
	4.3. A Formula for the Binomial Coefficients
	4.4. Permutations with Indistinguishable Objects
	4.5. Combinations with Indistinguishable Objects
	4.6. The Inclusion-Exclusion Principle
	4.7. Circular Permutations
	4.8. Probability
Chapter 5. Relations
	5.1. Relations
	5.2. Equivalence Relations
	5.3. Partial Orders
	5.4. Quotient Spaces
Chapter 6. Functions and Cardinality
	6.1. Functions
	6.2. Inverse Relations and Inverse Functions
	6.3. Cardinality of Infinite Sets
	6.4. An Order Relation for Cardinal Numbers
Chapter 7. Graph Theory
	7.1. Graphs
	7.2. Matrices, Digraphs, and Relations
	7.3. Shortest Paths in Weighted Graphs
	7.4. Trees
Chapter 8. Sequences
	8.1. Sequences
	8.2. Finite Differences
	8.3. Limits of Sequences of Real Numbers
	8.4. Some Convergence Properties
	8.5. Infinite Arithmetic
	8.6. Recurrence Relations
Chapter 9. Fibonacci Numbers and Pascal’s Triangle
	9.1. Pascal’s Triangle
	9.2. The Fibonacci Numbers
	9.3. The Golden Ratio
	9.4. Fibonacci Numbers and the Golden Ratio
	9.5. Pascal’s Triangle and the Fibonacci Numbers
Chapter 10. Combinatorial Geometry in the Plane
	10.1. Polygons and Convex Sets
	10.2. Pick’s Theorem
	10.3. Irrational Approximations of