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Applied Discrete Structures

Publisher: lulu.com
Keywords: structures, discrete, applied
Number of Pages: 496
Published: 2013-03-08
ISBN-10: 1105559297
ISBN-13: 9781105559297

Book Description:
Applied Discrete Structures is a two semester undergraduate text in discrete mathematics, focusing on the structural properties of mathematical objects. These include matrices, functions, graphs, trees, lattices and algebraic structures. The algebraic structures that are discussed are monoids, groups, rings, fields and vector spaces. Applied Discrete Structures has been approved by the American Institute of Mathematics as part of their Open Textbook Initiative. For more on open textbooks, visit http://www.aimath.org/textbooks/.
Table of Contents
Preface
Introduction
Chapter 1 Set Theory
1.1 Set Notation and Relations
1.2 Basic Set Operations
1.3 Cartesian Products and Power Sets
1.4 Binary Representation of Positive Integers
1.5 Summation Notation and Generalizations
Supplementary Exercises for Chapter 1
Chapter 2 Combinatorics
2.1 Basic Counting Techniques— the Rule of Products
2.2 Permutations
2.3 Partitions of Sets and the Laws of Addition
2.4 Combinations and the Binomial Theorem
Chapter 3 Logic
3.1 Propositions and Logical Operations
3.2 Truth Tables and Propositions Generated by a Set
3.3 Equivalence and Implication
3.4 The Laws of Logic
3.5 Mathematical Systems
3.6 Propositions Over a Universe
3.7 Mathematical Induction
3.8 Quantifiers
3.9 A Review of Methods of Proof
Supplementary Exercises for Chapter 3
Chapter 4 More on Sets
4.1 Methods of Proof for Sets
4.2 Laws of Set Theory
4.3 Minsets
4.4 The Duality Principle
Supplementary Exercises for Chapter 4
Applied Discrete Structures
Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License.
3Chapter 5 Introduction to Matrix Algebra
5.1 Basic Definition
5.2 Addition and Scalar Multiplication
5.3 Multiplication of Matrices
5.4 Special Types of Matrices
5.5 Laws of Matrix Algebra
5.6 Matrix Oddities
Supplementary Exercises for Chapter 5
Chapter 6 Relations and Graphs
6.1 Basic Definitions
6.2 Graphs of Relations
6.3 Properties of Relations
6.4 Matrices of Relations
6.5 Closure Operations on Relations
Supplementary Exercises for Chapter 6
Chapter 7 Functions
7.1 Definition of a Function and Notation
7.2 Injective, Surjective, and Bijective Functions
7.3 Composition, Identity, and Inverses
Supplementary Exercises for Chapter 7
Chapter 8 Recursion and Recurrence Relations
8.1 The Many Faces of Recursion
8.2 Sequences
8.3 Recurrence Relations
8.4 Some Common Recurrence Relations
8.5 Generating Functions
8.6 Recursion and Computer Algebra Systems
Supplementary Exercises for Chapter 8
Chapter 9 Graph Theory
9.1 Graphs—a General Introduction
9.2 Data Structures and Computer Generation of Graphs
9.3 Connectivity
9.4 Traversals : Eulerian and Hamiltonian
9.5 Graph Optimization
9.6 Planarlty and Colorings
Supplementary Exercises for Chapter 9
Applied Discrete Structures
Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License.
4Chapter 10 Trees
10.1 What is a Tree?
10.2 Spanning Trees
10.3 Rooted Trees
10.4 Binary Trees
Supplementary Exercises for Chapter 10
Chapter 11 AlgebraIc Systems
11.1 Operations
11.2 Algebraic Systems
11.3 Some General Properties of Groups
11.4 Zn, the Integers Modulo n
11.5 Subsystems
11.6 Direct Products
11.7 Isomorphisms
11.8 Using Computers to Study Groups
Supplementary Exercises for Chapter 11
Chapter 12 More Matrix Algebra
12.1 Systems of Linear Equations
12.2 Matrix Inversion
12.3 An Introduction to Vector Spaces
12.4 The Diagonialization Process
12.5 Some Applications
Supplementary Exercises for Chapter 12
Chapter 13 Boolean Algebra
13.1 Posets Revisited
13.2 Lattices
13.3 Boolean Algebras
13.4 Atoms of a Boolean Algebra
13.5 Finite Boolean Algebras as n-tuples of Zeros and Ones
13.6 Boolean Expressions
13.7 A Brief Introduction to the Application of Boolean Algebra to Switching Theory
Supplementary Exercises for Chapter 13
Applied Discrete Structures
Applied Discrete Structures by Alan Doerr & Kenneth Levasseur is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 3.0 United States License.
5Chapter 14 Monoids and Automata
14.1 Monoids
14.2 Free Monoids and Languages
14.3 Automata, Finite-state Machines
14.4 The Monoid of A Finite-state Machine
14.5 The Machine of A Monoid
Supplementary Exercises for Chapter 14
Chapter 15 Groups Theory and Applications
15.1 Cyclic Groups
15.2 Cosets and Factor Groups
15.3 Permutation Groups
15.4 Normal Subgroups and Group Homomorphisms
15.5 Coding Theory—Group Codes
Supplementary Exercises for Chapter 15
Chapter 16 An Introduction to Rings and Fields
16.1 Rings—Basic Definitions and Concepts
16.2 Fields
16.3 Polynomial Rings
16.4 Field Extensions
16.5 Power Series
Supplementary Exercises for Chapter 16
Solutions and Hints to Selected Exercises