內容大鋼
The book is mainly aimed at the bilingual curriculum design of discrete mathematics. It can meet the needs of the types of an introduction to the fundamental ideas of discrete mathematics, and as a foundation for the development of more advanced mathematical concepts.
The book includes four parts. The first part is mathematical logic which covers propositional logic in Chapter 1 and predicate logic in Chapter 2. It gives a method how to express natural language statements using symbols as the foundation to discrete mathematics, especially for sets. The logic parts include equivalent calculation and the discussion of proof. The second part is set theory with sets in Chapter 3, relations in Chapter 4, and functions in Chapter 5, which is the basics of the whole mathematics. Chapter 3 presents basic types of sets and their operations. Chapter 4 presents definition, properties and operations of relations, along with their representation as directed graphs and connections with matrices. Chapter 5 restates the function from the perspective of relations, including countable sets and uncountable sets. The third part is graph theory in Chapter 6 which is the sources for data structures and algorithms. It covers terminology, representation and special structure of graphs. The last part, that is a highly abstract part, is algebraic systems in Chapter 7, which covers semigroups, groups rings and fields. By algebraic systems we can find commonalities of many familiar rules.
This book can be used as a discrete mathematics course's reference for computer, mathematics, big data technology and some related disciplines in Institutions of Higher Learning, and can also provide useful help for other readers of discrete mathematics, especially for bilingual learning.
目錄
Chapter 1 Propositional Logic
1.1 The Basics of Propositional Logic
1.1.1 Propositions and Operators
1.1.2 Propositional Formulas
1.2 Propositional Equivalence
1.2.1 Equivalence and Some Basic Properties
1.2.2 Normal Form
1.3 Rules of Inference and Proof for Propositional Logic
1.3.1 Valid Arguments
1.3.2 Building Arguments and Proofs
Exercise 1
Chapter 2 Predicate Logic
2.1 The Basics of Predicate Logic
2.1.1 Language of Predicate Logic
2.1.2 Predicate Formulas
2.2 Predicate Equivalences
2.2.1 Equivalences
2.2.2 Prenex Normal Form
2.3 Rules of Inference for Predicate Logic
Exercise 2
Chapter 3 Sets
3.1 The Basics of Sets
3.1.1 The Set Concept and Set Description
3.1.2 Set Equality and Relationship between Sets
3.1.3 Some Special Sets
3.2 Operations on Sets
3.2.1 The Basic Operation
3.2.2 Set Identities
3.3 Principle of Inclusion-Exclusion
Exercise 3
Chapter 4 Relations
4.1 The Basics of Relations
4.1.1 Order Pairs and Cartesian Product
4.1.2 Concept and Representations of Binary Relations
4.1.3 Operations on Relations
4.1.4 Properties of Relations
4.1.5 Closures of Relations
4.2 Equivalence Relations
4.2.1 Definition of Equivalence Relations
4.2.2 Partitions
4.3 Partial Order Relations
4.3.1 Definition of Partial Order
4.3.2 Hasse Diagrams
4.3.3 Special Elements in Posets
Exercise 4
Chapter 5 Functions
5.1 The Basics of Function
5.1.1 Concept and Properties of Functions
5.1.2 Inverse Function and Composition of Functions
5.2 Countability of Sets
5.2.1 Cardinality of Sets
5.2.2 Countable Sets and Uncountable Sets
Exercise 5
Chapter 6 Graphs
6.1 The Basics of Graphs
6.2 Graph Terminology
6.2.1 The Path
6.2.2 Connectivity
6.3 Representing Graph Using Matrices
6.3.1 Incidence Matrices
6.3.2 Adjacency Matrices
6.3.3 Reachability Matrices
6.4 Tree
6.5 Euler Graphs and Hamilton Graphs
6.5.1 Euler Graphs
6.5.2 Hamilton Graphs
6.6 Planar
6.6.1 The Concepts of Planar
6.6.2 Euler's Formulas
6.6.3 Kuratowski's Theorem
Exercise 6
Chapter 7 Algebra
7.1 Algebraic Structures
7.1.1 Binary Operations
7.1.2 Algebra
7.2 Groups
7.2.1 Concept of a Group
7.2.2 Subgroups
7.3 Rings and Fields
Exercise 7
References
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