Embark on a comprehensive 9-hour journey through Discrete Mathematics, covering essential topics such as sets, logic, proofs, probability, and graph theory. Learn from Dr. Trefor Bazett as he guides you through logical statements, operations, and truth tables. Explore set theory, functions, and relations before delving into various proof methods including contrapositive, contradiction, and induction. Discover the fundamentals of probability, permutations, and combinations. Investigate Markov chains, transition diagrams, and an introduction to linear programming. Conclude with an exploration of graph theory, examining properties, degree of vertices, and Euler paths. Master the core concepts of discrete mathematics to build a strong foundation for further study in computer science and mathematics.
Overview
Syllabus
Intro to Discrete Math - Welcome to the Course!.
Intro to Sets | Examples, Notation & Properties.
Set-Roster vs Set-Builder notation.
The Empty Set & Vacuous Truth.
Cartesian Product of Two Sets A x B.
Relations between two sets | Definition + First Examples.
The intuitive idea of a function.
Formal Definition of a Function using the Cartesian Product.
Example: Is this relation a function?.
Intro to Logical Statements.
Intro to Truth Tables | Negation, Conjunction, and Disjunction.
Truth Table Example: ~p V ~q.
Logical Equivalence of Two Statements.
Tautologies and Contradictions.
3 Ways to Show a Logical Equivalence | Ex: DeMorgan's Laws.
Conditional Statements: if p then q.
Vacuously True Statements.
Negating a Conditional Statement.
Contrapositive of a Conditional Statement.
The converse and inverse of a conditional statement.
Biconditional Statements | "if and only if".
Logical Arguments - Modus Ponens & Modus Tollens.
Logical Argument Forms: Generalizations, Specialization, Contradiction.
Analyzing an argument for validity.
Predicates and their Truth Sets.
Universal and Existential Quantifiers, ∀ "For All" and ∃ "There Exists".
Negating Universal and Existential Quantifiers.
Negating Logical Statements with Multiple Quantifiers.
Universal Conditionals P(x) implies Q(x).
Necessary and Sufficient Conditions.
Formal Definitions in Math | Ex: Even & Odd Integers.
How to Prove Math Theorems | 1st Ex: Even + Odd = Odd.
Step-By-Step Guide to Proofs | Ex: product of two evens is even.
Rational Numbers | Definition + First Proof.
Proving that divisibility is transitive.
Disproving implications with Counterexamples.
Proof by Division Into Cases.
Proof by Contradiction | Method & First Example.
Proof by Contrapositive | Method & First Example.
Quotient-Remainder Theorem and Modular Arithmetic.
Proof: There are infinitely many primes numbers.
Introduction to sequences.
The formal definition of a sequence..
The sum and product of finite sequences.
Intro to Mathematical Induction.
Induction Proofs Involving Inequalities..
Strong Induction.
Recursive Sequences.
The Miraculous Fibonacci Sequence.
Prove A is a subset of B with the ELEMENT METHOD.
Proving equalities of sets using the element method.
The union of two sets.
The Intersection of Two Sets.
Universes and Complements in Set Theory.
Using the Element Method to prove a Set Containment w/ Modus Tollens.
Relations and their Inverses.
Reflexive, Symmetric, and Transitive Relations on a Set.
Equivalence Relations - Reflexive, Symmetric, and Transitive.
You need to check EVERY spot for reflexivity, symmetry, and transitivity.
Introduction to probability // Events, Sample Space, Formula, Independence.
Example: Computing Probabilities using P(E)=N(E)/N(S).
What is the probability of guessing a 4 digit pin code?.
Permutations: How many ways to rearrange the letters in a word?.
The summation rule for disjoint unions.
Counting formula for two intersecting sets: N(A union B)=N(A)+N(B)-N(A intersect B).
Counting with Triple Intersections // Example & Formula.
Combinations Formula: Counting the number of ways to choose r items from n items..
How many ways are there to reorder the word MISSISSIPPI? // Choose Formula Example.
Counting and Probability Walkthrough.
Intro to Conditional Probability.
Two Conditional Probability Examples (what's the difference???).
Conditional Probability With Tables | Chance of an Orange M&M???.
Bayes' Theorem - The Simplest Case.
Bayes' Theorem Example: Surprising False Positives.
Bayes' Theorem - Example: A disjoint union.
Intro to Markov Chains & Transition Diagrams.
Markov Chains & Transition Matrices.
Intro to Linear Programming and the Simplex Method.
Intro to Graph Theory | Definitions & Ex: 7 Bridges of Konigsberg.
Properties in Graph Theory: Complete, Connected, Subgraph, Induced Subgraph.
Degree of Vertices | Definition, Theorem & Example | Graph Theory.
Euler Paths & the 7 Bridges of Konigsberg | Graph Theory.
The End of Discrete Math - Congrats! Some final thoughts....
Taught by
Dr. Trefor Bazett
