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Discrete Mathematics

Math Academy official course import: Discrete Mathematics

143 节课

课程大纲

  1. 1.Preliminaries
    1. 1.1.1. Matrices
      1. 1
        Systems of Equations as Augmented Matrices
      2. 2
        Row Echelon Form
      3. 3
        Solving Systems of Equations Using Back Substitution
      4. 4
        Symmetric Matrices
  2. 2.Boolean Algebra
    1. 1.2.1. Boolean Functions
      1. 1
        Boolean Functions
      2. 2
        Boolean Functions And Logical Operations
      3. 3
        Disjunctive Normal Forms
      4. 4
        Principal Disjunctive Normal Forms
      5. 5
        Conjunctive Normal Forms
      6. 6
        Principal Conjunctive Normal Forms
      7. 7
        Boolean Polynomials
      8. 8
        Simplifying Logical Expressions Using Karnaugh Maps
    2. 2.2.2. Classes of Boolean Functions
      1. 1
        Truth-Preserving Boolean Functions
      2. 2
        Self-Dual Boolean Functions
      3. 3
        Monotonic Boolean Functions
      4. 4
        Affine Boolean Functions
      5. 5
        Functionally Complete Sets
      6. 6
        Post's Functional Completeness Theorem
    3. 3.2.3. Logic Circuits
      1. 1
        Introduction to Logic Circuits
      2. 2
        Logic Gates and Combinational Circuits
      3. 3
        Simplifying Logic Circuits
  3. 3.Number Theory
    1. 1.3.1. Numbers in Different Bases
      1. 1
        Binary Integers
      2. 2
        Adding and Subtracting Binary Integers
      3. 3
        Multiplying and Dividing Binary Integers
      4. 4
        Binary Fractions
      5. 5
        Hexadecimal Integers
      6. 6
        Converting Between Binary and Hexadecimal Integers
      7. 7
        Integers in Arbitrary Bases
      8. 8
        Fractions in Arbitrary Bases
    2. 2.3.2. Fermat and Euler's Theorems
      1. 1
        Fermat's Little Theorem
      2. 2
        Euler's Totient Function
      3. 3
        Euler's Theorem
    3. 3.3.3. Cryptography
      1. 1
        Introduction to Cryptography
      2. 2
        The Linear Cipher
      3. 3
        The Diffie-Hellman Protocol
      4. 4
        The RSA Cryptosystem
  4. 4.Probability & Combinatorics
    1. 1.4.1. Discrete Random Variables
      1. 1
        Probability Mass Functions of Discrete Random Variables
      2. 2
        Cumulative Distribution Functions for Discrete Random Variables
      3. 3
        One-to-One Transformations of Discrete Random Variables
      4. 4
        Many-to-One Transformations of Discrete Random Variables
      5. 5
        Expected Values of Discrete Random Variables
      6. 6
        Properties of Expectation for Discrete Random Variables
      7. 7
        Variance of Discrete Random Variables
      8. 8
        Moments of Discrete Random Variables
      9. 9
        Properties of Variance for Discrete Random Variables
    2. 2.4.2. The Discrete Uniform Distribution
      1. 1
        The Discrete Uniform Distribution
      2. 2
        Mean and Variance of the Discrete Uniform Distribution
      3. 3
        Modeling With Discrete Uniform Distributions
    3. 3.4.3. The Bernoulli Distribution
      1. 1
        The Bernoulli Distribution
      2. 2
        Mean and Variance of the Bernoulli Distribution
    4. 4.4.4. The Binomial Distribution
      1. 1
        The Binomial Distribution
      2. 2
        Modeling With the Binomial Distribution
      3. 3
        Mean and Variance of the Binomial Distribution
    5. 5.4.5. The Poisson Distribution
      1. 1
        The Poisson Distribution
      2. 2
        Modeling With the Poisson Distribution
      3. 3
        Mean and Variance of the Poisson Distribution
      4. 4
        The Poisson Approximation of the Binomial Distribution
    6. 6.4.6. The Geometric Distribution
      1. 1
        The Geometric Distribution
      2. 2
        Modeling With the Geometric Distribution
      3. 3
        Mean and Variance of the Geometric Distribution
    7. 7.4.7. The Negative Binomial Distribution
      1. 1
        The Negative Binomial Distribution
      2. 2
        Modeling With the Negative Binomial Distribution
    8. 8.4.8. The Hypergeometric Distribution
      1. 1
        The Hypergeometric Distribution
      2. 2
        The PMF of the Hypergeometric Distribution
    9. 9.4.9. Bayes' Theorem
      1. 1
        Bayes' Theorem
      2. 2
        Extending Bayes' Theorem
      3. 3
        The Law of Total Probability
      4. 4
        Extending the Law of Total Probability
    10. 10.4.10. Combinatorics
      1. 1
        Permutations With Repetition
      2. 2
        K Permutations of N With Repetition
      3. 3
        Combinations With Repetition
      4. 4
        The Inclusion-Exclusion Principle
      5. 5
        Counting Integer Solutions of a Constrained Equation
      6. 6
        Partitions
      7. 7
        The Pigeonhole Principle
  5. 5.Sequences
    1. 1.5.1. Finite Series
      1. 1
        Finite Linear Series
      2. 2
        Finite Quadratic Series
      3. 3
        Finite Cubic Series
    2. 2.5.2. Geometric Series
      1. 1
        Infinite Series and Partial Sums
      2. 2
        Convergent and Divergent Infinite Series
      3. 3
        Finding the Sum of an Infinite Geometric Series
      4. 4
        Writing an Infinite Geometric Series in Sigma Notation
      5. 5
        Sums of Infinite Geometric Series Given in Sigma Notation
    3. 3.5.3. The Binomial Theorem
      1. 1
        The Generalized Binomial Theorem
      2. 2
        Working With the Generalized Binomial Theorem
      3. 3
        Determining Ranges of Validity for Generalized Binomial Expansions
      4. 4
        Approximating Values Using the Generalized Binomial Theorem
      5. 5
        The Multinomial Theorem
      6. 6
        Counting Terms in Multinomial Expansions
  6. 6.Recursion
    1. 1.6.1. Recurrence Relations
      1. 1
        Introduction to Recurrence Relations
      2. 2
        First-Order Recurrence Relations
      3. 3
        First-Order Recurrence Relations With Polynomial Forcing
      4. 4
        First-Order Recurrence Relations With Exponential Forcing
    2. 2.6.2. Second-Order Recurrence Relations
      1. 1
        Second-Order Homogeneous Recurrence Relations: Characteristic Equations with Distinct Real Roots
      2. 2
        Second-Order Homogeneous Recurrence Relations: Characteristic Equations with Repeated Roots
      3. 3
        Second-Order Homogeneous Recurrence Relations: Characteristic Equations with Complex Roots
      4. 4
        Second-Order Recurrence Relations with Polynomial Forcing
      5. 5
        Second-Order Recurrence Relations with Exponential Forcing
    3. 3.6.3. Generating Functions
      1. 1
        Generating Functions
      2. 2
        Generating Functions of Homogeneous Recurrence Relations
      3. 3
        Determining the General Term of a Sequence Given Its Generating Function
      4. 4
        Solving Homogeneous Recurrence Relations Using Generating Functions
      5. 5
        Generating Functions for Inhomogeneous Recurrence Relations
  7. 7.Graph Theory
    1. 1.7.1. Introduction to Graph Theory
      1. 1
        Introduction to Graphs
      2. 2
        Walks, Trails, Paths, and Distances
      3. 3
        Walks, Paths, and Distances in Directed Graphs
      4. 4
        The Degree of a Vertex
      5. 5
        The Handshaking Lemma
      6. 6
        Subgraphs and Graph Complements
      7. 7
        Complete and Bipartite Graphs
      8. 8
        Regular Graphs
      9. 9
        Planar Graphs
    2. 2.7.2. Connectivity in Graphs
      1. 1
        Cycles in Graphs
      2. 2
        Connected Graphs
      3. 3
        Cut Vertices and Cut Edges
    3. 3.7.3. Trees
      1. 1
        Trees and Forests
      2. 2
        Counting Trees
    4. 4.7.4. Matrix Representations of Graphs
      1. 1
        Incidence Matrices
      2. 2
        Adjacency Matrices
      3. 3
        Adjacency Matrices of Directed Graphs
      4. 4
        Distance Matrices
    5. 5.7.5. DAGs and Topological Ordering
      1. 1
        Directed Acyclic Graphs
      2. 2
        Topological Ordering
      3. 3
        Kahn's Algorithm
    6. 6.7.6. Algorithms on Graphs
      1. 1
        Breadth-First Search
      2. 2
        Kruskal's Algorithm
      3. 3
        Prim's Algorithm
      4. 4
        Applying Prim's Algorithm to a Distance Matrix
      5. 5
        Shortest Paths in Unweighted Graphs
      6. 6
        Dijkstra's Algorithm
  8. 8.The Theory of Algorithms
    1. 1.8.1. Finite-State Transducers
      1. 1
        Mealy Machines
      2. 2
        Moore Machines
    2. 2.8.2. Finite-State Interceptors
      1. 1
        Deterministic Finite Automata
      2. 2
        Processing Strings Using Deterministic Finite Automata
      3. 3
        Nondeterministic Finite Automata
      4. 4
        The Language of Automata
      5. 5
        Regular Expressions
    3. 3.8.3. Turing Machines
      1. 1
        Introduction to Turing Machines
      2. 2
        Processing Strings Using Turing Machines
      3. 3
        Nondeterministic Turing Machines
    4. 4.8.4. Algorithmic Complexity
      1. 1
        Big-O Notation
      2. 2
        Big-Omega Notation
      3. 3
        Big-Theta Notation