Mathematical Foundations for Computing

  1. Counting and Combinatorics
    1. The Basics of Counting
      1. The Product Rule
        1. Statement and Applications
          1. Multiple-Step Processes
          2. The Sum Rule
            1. Statement and Applications
              1. Mutually Exclusive Events
              2. The Subtraction Rule
                1. Inclusion-Exclusion for Complements
                2. The Inclusion-Exclusion Principle
                  1. Two Sets
                    1. Three Sets
                      1. General Form
                      2. The Division Rule
                        1. Counting with Symmetry
                          1. Overcounting Corrections
                        2. The Pigeonhole Principle
                          1. Basic Pigeonhole Principle
                            1. Statement and Proof
                              1. Simple Applications
                              2. Generalized Pigeonhole Principle
                                1. Statement and Applications
                                  1. Average Case Analysis
                                  2. Applications of the Pigeonhole Principle
                                    1. Number Theory Applications
                                      1. Graph Theory Applications
                                        1. Computer Science Applications
                                      2. Permutations and Combinations
                                        1. Permutations
                                          1. Definition and Notation
                                            1. Counting Permutations
                                              1. Permutations of Distinct Objects
                                                1. Permutations with Repetition
                                                  1. Circular Permutations
                                                  2. Combinations
                                                    1. Definition and Notation
                                                      1. Counting Combinations
                                                        1. Combinations of Distinct Objects
                                                          1. Relationship to Permutations
                                                        2. Binomial Coefficients and Identities
                                                          1. Definition of Binomial Coefficient
                                                            1. Combinatorial Interpretation
                                                              1. Factorial Formula
                                                              2. The Binomial Theorem
                                                                1. Statement and Proof
                                                                2. Pascal's Identity
                                                                  1. Statement and Proof
                                                                    1. Recursive Formula
                                                                    2. Pascal's Triangle
                                                                      1. Construction
                                                                      2. Other Binomial Identities
                                                                        1. Vandermonde's Identity
                                                                          1. Hockey Stick Identity
                                                                            1. Symmetry Identity
                                                                          2. Generalized Permutations and Combinations
                                                                            1. Permutations with Repetition
                                                                              1. Unlimited Repetition
                                                                                1. Limited Repetition
                                                                                2. Combinations with Repetition
                                                                                  1. Stars and Bars Method
                                                                                  2. Multisets
                                                                                    1. Definition and Properties
                                                                                      1. Counting Multisets
                                                                                      2. Derangements
                                                                                        1. Definition and Formula

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                                                                                    5. Induction and Recursion

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                                                                                    7. Discrete Probability