第 37 节，共 85 节
Proof by Division Into Cases
课时85 节课
- 1Intro to Discrete Math - Welcome to the Course!
- 2Intro to Sets | Examples, Notation & Properties
- 3Set-Roster vs Set-Builder notation
- 4The Empty Set & Vacuous Truth
- 5Cartesian Product of Two Sets A x B
- 6Relations between two sets | Definition + First Examples
- 7The intuitive idea of a function
- 8Formal Definition of a Function using the Cartesian Product
- 9Example: Is this relation a function?
- 10Intro to Logical Statements
- 11Intro to Truth Tables | Negation, Conjunction, and Disjunction
- 12Truth Table Example: ~p V ~q
- 13Logical Equivalence of Two Statements
- 14Tautologies and Contradictions
- 153 Ways to Show a Logical Equivalence | Ex: DeMorgan's Laws
- 16Conditional Statements: if p then q
- 17Vacuously True Statements
- 18Negating a Conditional Statement
- 19Contrapositive of a Conditional Statement
- 20The converse and inverse of a conditional statement
- 21Biconditional Statements | "if and only if"
- 22Logical Arguments - Modus Ponens & Modus Tollens
- 23Logical Argument Forms: Generalizations, Specialization, Contradiction
- 24Analyzing an argument for validity
- 25Predicates and their Truth Sets
- 26Universal and Existential Quantifiers, ∀ "For All" and ∃ "There Exists"
- 27Negating Universal and Existential Quantifiers
- 28Negating Logical Statements with Multiple Quantifiers
- 29Universal Conditionals P(x) implies Q(x)
- 30Necessary and Sufficient Conditions
- 31Formal Definitions in Math | Ex: Even & Odd Integers
- 32How to Prove Math Theorems | 1st Ex: Even + Odd = Odd
- 33Step-By-Step Guide to Proofs | Ex: product of two evens is even
- 34Rational Numbers | Definition + First Proof
- 35Proving that divisibility is transitive
- 36Disproving implications with Counterexamples
- 37Proof by Division Into Cases
- 38Proof by Contradiction | Method & First Example
- 39Proof by Contrapositive | Method & First Example
- 40Quotient-Remainder Theorem and Modular Arithmetic
- 41Proof: There are infinitely many primes numbers
- 42Introduction to sequences
- 43The formal definition of a sequence.
- 44The sum and product of finite sequences
- 45Intro to Mathematical Induction
- 46Induction Proofs Involving Inequalities.
- 47Strong Induction // Intro and Full Example
- 48Recursive Sequences
- 49The Miraculous Fibonacci Sequence
- 50Prove A is a subset of B with the ELEMENT METHOD
- 51Proving equalities of sets using the element method
- 52The union of two sets
- 53The Intersection of Two Sets
- 54Universes and Complements in Set Theory
- 55Using the Element Method to prove a Set Containment w/ Modus Tollens
- 56Power Sets and the Cardinality of the Continuum
- 57Relations and their Inverses
- 58Reflexive, Symmetric, and Transitive Relations on a Set
- 59Equivalence Relations - Reflexive, Symmetric, and Transitive
- 60You need to check EVERY spot for reflexivity, symmetry, and transitivity
- 61Introduction to probability // Events, Sample Space, Formula, Independence
- 62Example: Computing Probabilities using P(E)=N(E)/N(S)
- 63What is the probability of guessing a 4 digit pin code?
- 64Counting with Triple Intersections // Example & Formula
- 65Permutations: How many ways to rearrange the letters in a word?
- 66The summation rule for disjoint unions
- 67Counting formula for two intersecting sets: N(A union B)=N(A)+N(B)-N(A intersect B)
- 68Combinations Formula: Counting the number of ways to choose r items from n items.
- 69How many ways are there to reorder the word MISSISSIPPI? // Choose Formula Example
- 70Counting and Probability Walkthrough
- 71Intro to Conditional Probability
- 72Two Conditional Probability Examples (what's the difference???)
- 73Conditional Probability With Tables | Chance of an Orange M&M???
- 74Bayes' Theorem - The Simplest Case
- 75Bayes' Theorem Example: Surprising False Positives
- 76Bayes' Theorem - Example: A disjoint union
- 77IS CHESS A GAME OF CHANCE? Classical vs Frequentist vs Bayesian Probability
- 78Intro to Markov Chains & Transition Diagrams
- 79Markov Chains & Transition Matrices
- 80Intro to Linear Programming
- 81Intro to Graph Theory | Definitions & Ex: 7 Bridges of Konigsberg
- 82Properties in Graph Theory: Complete, Connected, Subgraph, Induced Subgraph
- 83Degree of Vertices | Definition, Theorem & Example | Graph Theory
- 84Euler Paths & the 7 Bridges of Konigsberg | Graph Theory
- 85The End of Discrete Math - Congrats! Some final thoughts...