离散数学、验证技巧及数学结构DISCRETE MATHEMATICS, PROOF TECHNIQUES AND MATHEMATICAL STRUCTURES

Based on a first course in mathematics for students from diverse disciplines developed and taught by Robert Penner over a period of fifteen years at the University of Southern California, where the author is Professor of Mathematics and Physics. DLC: Mathematics.

General Preface
Preface for Instructors
PART 1
Chapter 1 Proof Techniques
Part A Elements of Logic and Induction
A.1 Definitions
A.2 Propositions and Predicates
A.3 Conjunction and Negation
A.4 Implication
A.5 Proofs by Induction for Natural Numbers
Part B Methods of Proof
B.1 Chains of Implications
B.2 Proof by Contradiction
B.3 Instantiation
B.4 Constructive and Non-Constructive Proofs
B.5 Disproofs
Chapter 2 Predicate Calculus
2.1 Logical Operators
2.2 Propositional Forms
2.3 Parentheses
2.4 Standard Logical Identities
2.5 Standard Rules of Inference
(*)2.6 Aristotelean Logic
2.7 Logical Equivalence
(*)2.8 Commutativity and Associativity
2.9 Proving Propositional Forms
2.10 Predicate Forms and Quantification
2.11 Standard Valid Predicate Forms
2.12 Proving Predicate Forms
2.13 Disproofs
Chapter 3 Set Theory
3.1 Axioms and the Primitives of Set Theory
3.2 The Basics of Set Theory
(*)3.3 ZFC Set Theory
3.4 Binary Operations on Sets
3.5 Cartesian Products
3.6 Absolute Complements and DeMorgan's Laws
(*)3.7 The Set of Non-Negative Integers
3.8 Inductive Definitions
3.9 Sets of Numbers
Chapter 4 Elementary Number Theory
4.1 Common Multiples
4.2 The Division Algorithm
4.3 Common Divisors
4.4 Relatively Prime Pairs
4.5 Linear Diophantine Equations
4.6 The Fundamental Theorem of Arithmetic
4.7 The Euclidean Algorithm
(*)4.8 Continued Fractions
Chapter 5 Relations
5.1 N-ary Relations
5.2 Binary Relations and Digraphs
5.3 Properties of Relations
5.4 Set-Theoretic Operations
5.5 Inversion
5.6 Composition
(*)5.7 Iterations
5.8 Posets
5.9 Linear and Well Orders
(*)5.10 Bounds in Posets
(*)5.11 Axiom of Choice Revisited
5.12 Equivalence Relations
5.13 Partitions
(*)5.14 Closure Operations
(*)5.15 Meets and Joins of Partitions
(*)5.16 Lattices
Chapter 6 Functions
6.1 Definitions and Examples
6.2 Composition
6.3 Restriction and Extension
6.4 Injectivity, Surjeetivity, and Bijectivity
6.5 Inverses
6.6 Images and Pre-Images
……
PART 2