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Methods of Proof

Math Academy official course import: Methods of Proof

135 节课

课程大纲

  1. 1.Logic
    1. 1.1.1. Statements
      1. 1
        Statements and Predicates
      2. 2
        The "And" and "Or" Connectives
      3. 3
        Truth Tables
      4. 4
        The "Not" Connective
      5. 5
        Logical Equivalence
      6. 6
        Logical Associative and Commutative Laws
      7. 7
        The Distributive Laws
      8. 8
        The Absorption Laws
      9. 9
        De Morgan's Laws for Logic
    2. 2.1.2. Implications and Biconditionals
      1. 1
        Conditional Statements
      2. 2
        Logical Equivalence with Conditional Statements
      3. 3
        Biconditional Statements
      4. 4
        Tautologies and Contradictions
      5. 5
        Converse, Inverse, and Contrapositive
    3. 3.1.3. Predicates
      1. 1
        Truth Sets of Predicates
      2. 2
        The "And" and "Or" Connectives With Predicates
      3. 3
        The "Not" Connective With Predicates
      4. 4
        Simplifying Predicate Expressions Using De Morgan's Laws
      5. 5
        Conditional Statements With Predicates
      6. 6
        Necessary and Sufficient Conditions
      7. 7
        Grammatical Constructions for Conditional Statements
      8. 8
        Translating Between Logical and Set Operations
    4. 4.1.4. Quantifiers
      1. 1
        Universal and Existential Quantifiers
      2. 2
        Negating Quantified Statements
      3. 3
        Nested Quantifiers
      4. 4
        Formal and Informal Language
      5. 5
        Negating Statements With Nested Quantifiers
      6. 6
        Prenex Normal Form
    5. 5.1.5. Logical Inference
      1. 1
        Implication Elimination and Denying the Consequent
      2. 2
        Disjunctive Syllogism and Transitivity of Implication
      3. 3
        Additional Rules of Logical Inference
  2. 2.Set Theory
    1. 1.2.1. Introduction to Set Theory
      1. 1
        Special Sets
      2. 2
        Equivalent Sets
      3. 3
        The Constructive Definition of a Set
      4. 4
        The Conditional Definition of a Set
      5. 5
        Describing Sets Using Set-Builder Notation
      6. 6
        Describing Planar Regions Using Set-Builder Notation
      7. 7
        Cardinality of Finite Sets
      8. 8
        Infinite Sets
      9. 9
        The Maximum and Minimum of a Set
    2. 2.2.2. Subsets
      1. 1
        Subsets
      2. 2
        Power Sets
      3. 3
        Partitions of Sets
      4. 4
        Indicator Functions
      5. 5
        Proving Subset Relations
    3. 3.2.3. Set Operations
      1. 1
        The Difference of Sets
      2. 2
        Set Complements
      3. 3
        The Cartesian Product
      4. 4
        Visualizing Cartesian Products
      5. 5
        Indexed Sets
      6. 6
        Disjoint Sets
    4. 4.2.4. Properties of Set Operations
      1. 1
        Elementary Properties of Set Operations
      2. 2
        Proving Elementary Properties of Set Operations
      3. 3
        De Morgan's Laws for Sets
      4. 4
        Proving De Morgan's Laws for Sets
      5. 5
        Distributive Properties of Set Operations
      6. 6
        Proving Distributive Properties of Union and Intersection
      7. 7
        Proving Distributive Properties of the Cartesian Product
  3. 3.Mathematical Proof
    1. 1.3.1. Introduction to Mathematical Proof
      1. 1
        Direct Proof
      2. 2
        Proving Parity
      3. 3
        Proving Divisibility
      4. 4
        Proof by Cases
      5. 5
        Disproving Universal Statements
      6. 6
        Disproving Implications
      7. 7
        Trivial and Vacuous Proofs
      8. 8
        Proving Biconditional Statements
    2. 2.3.2. Proof by Contrapositive and Contradiction
      1. 1
        Proof by Contrapositive
      2. 2
        Proving Parity by Contrapositive
      3. 3
        Proving Divisibility by Contrapositive
      4. 4
        Proof by Contradiction
      5. 5
        Proving Irrationality by Contradiction
      6. 6
        Proving Properties of Irrationals by Contradiction
      7. 7
        Proving Divisibility by Contradiction
  4. 4.Discrete Mathematics & Number Theory
    1. 1.4.1. Divisibility
      1. 1
        Parity
      2. 2
        Integer Divisibility
      3. 3
        The Division Algorithm
      4. 4
        The Euclidean Algorithm
      5. 5
        The Extended Euclidean Algorithm
      6. 6
        Linear Diophantine Equations
      7. 7
        Properties of Integer Divisibility
      8. 8
        Properties of Prime Divisibility
    2. 2.4.2. Proof by Induction
      1. 1
        Mathematical Induction
      2. 2
        Proving Inequalities Using Induction
      3. 3
        Proving Divisibility Using Induction
      4. 4
        Proving Matrix Identities Using Induction
      5. 5
        Strong Induction and Recurrence Relations
  5. 5.Sequences & Functions
    1. 1.5.1. Sequences
      1. 1
        The Limit of a Null Sequence
      2. 2
        Proving the Limit of a Null Sequence
      3. 3
        Proving the Finite Limit of a Sequence
      4. 4
        Infinite Limits of Sequences
      5. 5
        Proving a Sequence Has an Infinite Limit
    2. 2.5.2. Functions
      1. 1
        Sets and Functions
      2. 2
        Injections
      3. 3
        Proving Injectivity
      4. 4
        Surjections
      5. 5
        Into Functions
      6. 6
        Proving Surjectivity
      7. 7
        Bijections
      8. 8
        The Floor and Ceiling Functions
    3. 3.5.3. Cardinality
      1. 1
        Discrete Infinite Sets With Equal Cardinality
      2. 2
        Continuous Infinite Sets With Equal Cardinality
      3. 3
        Cardinality of the Natural Numbers, Integers, and Rationals
      4. 4
        Cantor's Diagonal Argument
      5. 5
        The Cantor-Bernstein-Schröder Theorem
      6. 6
        The Cardinality of the Power Set of Natural Numbers
      7. 7
        Cantor's Theorem
  6. 6.Congruence of Integers
    1. 1.6.1. Modular Congruence
      1. 1
        Modular Congruence
      2. 2
        The Addition Property of Modular Arithmetic
      3. 3
        Modular Residues
      4. 4
        The Multiplication Properties of Modular Arithmetic
      5. 5
        The Division Properties of Modular Arithmetic
      6. 6
        Proving Properties of Modular Congruence
      7. 7
        Proving Divisibility Using Congruence
      8. 8
        Proving Congruence by Contrapositive
    2. 2.6.2. Linear Congruences
      1. 1
        Additive Inverses Modulo N
      2. 2
        Multiplicative Inverses Modulo N
      3. 3
        Solving Linear Congruences
      4. 4
        Solving Advanced Linear Congruences
      5. 5
        The Chinese Remainder Theorem
      6. 6
        Proving Properties of Linear Congruences
  7. 7.Equivalence Relations
    1. 1.7.1. Relations
      1. 1
        Relations on Finite Sets
      2. 2
        Relations on Infinite Sets
      3. 3
        The Domain and Range of a Relation
      4. 4
        Graphical Representations of Relations
      5. 5
        Operations on Relations
      6. 6
        N-ary Relations
      7. 7
        Functional Relations
    2. 2.7.2. Equivalence Relations
      1. 1
        Reflexive and Irreflexive Relations
      2. 2
        Symmetric and Antisymmetric Relations
      3. 3
        Transitive Relations
      4. 4
        Equivalence Relations
      5. 5
        Proving a Relation Is an Equivalence Relation
      6. 6
        Residue Classes
      7. 7
        Equivalence Classes
      8. 8
        The Integers Modulo N