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Applied Discrete Structures

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Applied Discrete Structures

Applied Discrete Structures is designed for use in a university course in discrete mathematics spanning up to two semesters. Its original design was for computer science majors to be introduced to the mathematical topics that are useful in computer science. It can also serve the same purpose for mathematics majors, providing a first exposure to many essential topics.

Combinatorics and Discrete MathematicsMathematics
Author
Al Doerr & Ken Levasseur
Publisher
University of Massachusetts Lowell
Source
LibreTexts
License
CC BY-NC-SA
Language
EN
Category
Basic Sciences
Read online Download PDF (open licence)
Contents· 112 sections
  1. 01: Set Theory
  2. 1.01: Set Notation and Relations
  3. 1.02: Basic Set Operations
  4. 1.03: Cartesian Products and Power Sets
  5. 1.04: Binary Representation of Positive Integers
  6. 1.05: Summation Notation and Generalizations
  7. 02: Combinatorics
  8. 2.01: Basic Counting Techniques - The Rule of Products
  9. 2.02: Permutations
  10. 2.03: Partitions of Sets and the Law of Addition
  11. 2.04: Combinations and the Binomial Theorem
  12. 03: Logic
  13. 3.01: Propositions and Logical Operators
  14. 3.02: Truth Tables and Propositions Generated by a Set
  15. 3.03: Equivalence and Implication
  16. 3.04: The Laws of Logic
  17. 3.05: Mathematical Systems and Proofs
  18. 3.06: Propositions over a Universe
  19. 3.07: Mathematical Induction
  20. 3.08: Quantifiers
  21. 3.09: A Review of Methods of Proof
  22. 04: More on Sets
  23. 4.01: Methods of Proof for Sets
  24. 4.02: Laws of Set Theory
  25. 4.03: Minsets
  26. 4.04: The Duality Principle
  27. 05: Introduction to Matrix Algebra
  28. 5.01: Basic Definitions and Operations
  29. 5.02: Special Types of Matrices
  30. 5.03: Laws of Matrix Algebra
  31. 5.04: Matrix Oddities
  32. 06: Relations
  33. 6.01: Basic Definitions
  34. 6.02: Graphs of Relations on a Set
  35. 6.03: Properties of Relations
  36. 6.04: Matrices of Relations
  37. 6.05: Closure Operations on Relations
  38. 07: Functions
  39. 7.01: Definition and Notation
  40. 7.02: Properties of Functions
  41. 7.03: Function Composition
  42. 08: Recursion and Recurrence Relations
  43. 8.01: The Many Faces of Recursion
  44. 8.02: Sequences
  45. 8.03: Recurrence Relations
  46. 8.04: Some Common Recurrence Relations
  47. 8.05: Generating Functions
  48. 09: Graph Theory
  49. 9.01: Graphs - General Introduction
  50. 9.02: Data Structures for Graphs
  51. 9.03: Connectivity
  52. 9.04: Traversals- Eulerian and Hamiltonian Graphs
  53. 9.05: Graph Optimization
  54. 9.06: Planarity and Colorings
  55. 10: Trees
  56. 10.01: What is a Tree
  57. 10.02: Spanning Trees
  58. 10.03: Rooted Trees
  59. 10.04: Binary Trees
  60. 11: Algebraic Structures
  61. 11.01: Operations
  62. 11.02: Algebraic Systems
  63. 11.03: Some General Properties of Groups
  64. 11.04: Greatest Common Divisors and the Integers Modulo n
  65. 11.05: Subsystems
  66. 11.06: Direct Products
  67. 11.07: Isomorphisms
  68. 12: More Matrix Algebra
  69. 12.01: Systems of Linear Equations
  70. 12.02: Matrix Inversion
  71. 12.03: An Introduction to Vector Spaces
  72. 12.04: The Diagonalization Process
  73. 12.05: Some Applications
  74. 12.06: Linear Equations over the Integers Mod 2
  75. 13: Boolean Algebra
  76. 13.01: Posets Revisited
  77. 13.02: Lattices
  78. 13.03: Boolean Algebras
  79. 13.04: Atoms of a Boolean Algebra
  80. 13.05: Finite Boolean Algebras as n-tuples of 0's and 1's
  81. 13.06: Boolean Expressions
  82. 13.07: A Brief Introduction to Switching Theory and Logic Design
  83. 14: Monoids and Automata
  84. 14.01: Monoids
  85. 14.02: Free Monoids and Languages
  86. 14.03: Automata Finite-State Machines
  87. 14.04: The Monoid of a Finite-State Machine
  88. 14.05: The Machine of a Monoid
  89. 15: Group Theory and Applications
  90. 15.01: Cyclic Groups
  91. 15.02: Cosets and Factor Groups
  92. 15.03: Permutation Groups
  93. 15.04: Normal Subgroups and Group Homomorphisms
  94. 15.05: Coding Theory Group Codes
  95. 16: An Introduction to Rings and Fields
  96. 16.01: Rings Basic Definitions and Concepts
  97. 16.02: Fields
  98. 16.03: Polynomial Rings
  99. 16.04: Field Extensions
  100. 16.05: Power Series
  101. 17: Appendix
  102. 17.01: A - Algorithms
  103. 17.1.01: An Introduction to Algorithms
  104. 17.1.02: The Invariant Relation Theorem
  105. 17.02: B - Python and SageMath
  106. 17.2.01: Python Iterators
  107. 17.2.02: Dictionaries
  108. 17.03: C - Determinants
  109. 17.3.01: Definition
  110. 17.3.02: Computation
  111. 17.04: D - Notation
  112. 17.05: E - References