1 Intro to Discrete Math - Welcome to the Course!
2 Intro to Sets | Examples, Notation & Properties
3 Set-Roster vs Set-Builder notation
4 The Empty Set & Vacuous Truth
5 Cartesian Product of Two Sets A x B
6 Relations between two sets | Definition + First Examples
7 The intuitive idea of a function
8 Formal Definition of a Function using the Cartesian Product
9 Example: Is this relation a function?
10 Intro to Logical Statements
11 Intro to Truth Tables | Negation, Conjunction, and Disjunction
12 Truth Table Example: ~p V ~q
13 Logical Equivalence of Two Statements
14 Tautologies and Contradictions
15 3 Ways to Show a Logical Equivalence | Ex: DeMorgan's Laws
16 Conditional Statements: if p then q
17 Vacuously True Statements
18 Negating a Conditional Statement
19 Contrapositive of a Conditional Statement
20 The converse and inverse of a conditional statement
21 Biconditional Statements | "if and only if"
22 Logical Arguments - Modus Ponens & Modus Tollens
23 Logical Argument Forms: Generalizations, Specialization, Contradiction
24 Analyzing an argument for validity
25 Predicates and their Truth Sets
26 Universal and Existential Quantifiers, ∀ "For All" and ∃ "There Exists"
27 Negating Universal and Existential Quantifiers
28 Negating Logical Statements with Multiple Quantifiers
29 Universal Conditionals P(x) implies Q(x)
30 Necessary and Sufficient Conditions
31 Formal Definitions in Math | Ex: Even & Odd Integers
32 How to Prove Math Theorems | 1st Ex: Even + Odd = Odd
33 Step-By-Step Guide to Proofs | Ex: product of two evens is even
34 Rational Numbers | Definition + First Proof
35 Proving that divisibility is transitive
36 Disproving implications with Counterexamples
37 Proof by Division Into Cases
38 Proof by Contradiction | Method & First Example
39 Proof by Contrapositive | Method & First Example
40 Quotient-Remainder Theorem and Modular Arithmetic
41 Proof: There are infinitely many primes numbers
42 Introduction to sequences
43 The formal definition of a sequence.
44 The sum and product of finite sequences
45 Intro to Mathematical Induction
46 Induction Proofs Involving Inequalities.
47 Strong Induction
48 Recursive Sequences
49 The Miraculous Fibonacci Sequence
50 Prove A is a subset of B with the ELEMENT METHOD
51 Proving equalities of sets using the element method
52 The union of two sets
53 The Intersection of Two Sets
54 Universes and Complements in Set Theory
55 Using the Element Method to prove a Set Containment w/ Modus Tollens
56 Relations and their Inverses
57 Reflexive, Symmetric, and Transitive Relations on a Set
58 Equivalence Relations - Reflexive, Symmetric, and Transitive
59 You need to check EVERY spot for reflexivity, symmetry, and transitivity
60 Introduction to probability // Events, Sample Space, Formula, Independence
61 Example: Computing Probabilities using P(E)=N(E)/N(S)
62 What is the probability of guessing a 4 digit pin code?
63 Permutations: How many ways to rearrange the letters in a word?
64 The summation rule for disjoint unions
65 Counting formula for two intersecting sets: N(A union B)=N(A)+N(B)-N(A intersect B)
66 Counting with Triple Intersections // Example & Formula
67 Combinations Formula: Counting the number of ways to choose r items from n items.
68 How many ways are there to reorder the word MISSISSIPPI? // Choose Formula Example
69 Counting and Probability Walkthrough
70 Intro to Conditional Probability
71 Two Conditional Probability Examples (what's the difference???)
72 Conditional Probability With Tables | Chance of an Orange M&M???
73 Bayes' Theorem - The Simplest Case
74 Bayes' Theorem Example: Surprising False Positives
75 Bayes' Theorem - Example: A disjoint union
76 Intro to Markov Chains & Transition Diagrams
77 Markov Chains & Transition Matrices
78 Intro to Linear Programming and the Simplex Method
79 Intro to Graph Theory | Definitions & Ex: 7 Bridges of Konigsberg
80 Properties in Graph Theory: Complete, Connected, Subgraph, Induced Subgraph
81 Degree of Vertices | Definition, Theorem & Example | Graph Theory
82 Euler Paths & the 7 Bridges of Konigsberg | Graph Theory
83 The End of Discrete Math - Congrats! Some final thoughts...