Free online courseDiscrete Math
Duration of the online course: 9 hours and 38 minutes
New course
Master Discrete Math with this comprehensive online course. Dive into sets, logic, proofs, probability, and graph theory, building essential math skills in calculus.
In this free course, learn about
- Foundations: Sets, Relations, and Functions
- Propositional Logic and Logical Equivalence
- Implications, Conditionals, and Logical Arguments
- Predicates, Quantifiers, and Formal Statements
- Proof Techniques and Number Theory Basics
- Sequences and Mathematical Induction
- Advanced Set Theory and Relations
- Counting, Probability, and Combinatorics
- Conditional Probability, Bayes’ Theorem, and Markov Chains
- Optimization and Graph Theory
- Course Wrap-Up and Big-Picture Skills
Course Description
Explore the fascinating world of Discrete Mathematics, a foundational course that spans a diverse range of topics. Begin with an introduction to sets, learning about set notations like Set-Roster and Set-Builder, and grasp concepts such as the Cartesian Product of two sets, relations, and the fundamental idea of functions.
Delve into logical statements, understanding truth tables, tautologies, and the relationships between logical equivalences. Strengthen your reasoning with conditional and biconditional arguments, examining universal and existential quantifiers deeply, and not to forget, get guided through the first steps in mathematical proofs.
The course will lead you through sequences, modular arithmetic, proof strategies, and inspire confidence while tackling mathematical inductions and recursive sequences. A comprehensive exploration of set theory, including union, intersection, and power sets, equips learners with robust techniques for analyzing set relationships.
Transition into the intriguing world of probability, unveil the mystery of counting, and solidify knowledge with practical examples, like calculating permutations and Bayes' Theorem. Test probability principles with exciting questions like guessing a PIN code or understanding chess from a probabilistic viewpoint.
Advance into more technical realms such as graph theory, with topics covering complete and connected graphs, subgraphs, and the classic Euler Paths. As you reach course completion, you'll come away with holistic insight into logical reasoning, algorithmic thinking, and a mature mathematical perspective.
Course content
- Video class: Intro to Discrete Math - Welcome to the Course! 05m
- Exercise: What distinguishes discrete mathematics from continuous mathematics in this course context
- Video class: Intro to Sets | Examples, Notation 07m
- Video class: Set-Roster vs Set-Builder notation 05m
- Exercise: Which set builder notation matches the roster ..., -4, -2, 0, 2, 4, ...?
- Video class: The Empty Set 04m
- Video class: Cartesian Product of Two Sets A x B 07m
- Exercise: When are two ordered pairs equal
- Video class: Relations between two sets | Definition First Examples 06m
- Video class: The intuitive idea of a function 05m
- Exercise: Which statement correctly defines a function in terms of domain and range?
- Video class: Formal Definition of a Function using the Cartesian Product 05m
- Video class: Example: Is this relation a function? 05m
- Exercise: Is the relation C = {(x,y) in R×R | x^2 + y^2 = 1} a function from [-1,1] to [-1,1]?
- Video class: Intro to Logical Statements 06m
- Video class: Intro to Truth Tables | Negation, Conjunction, and Disjunction 05m
- Exercise: When is the conjunction P and Q true?
- Video class: Truth Table Example: ~p V ~q 03m
- Video class: Logical Equivalence of Two Statements 03m
- Exercise: Identify a logical equivalence using truth tables
- Video class: Tautologies and Contradictions 03m
- Video class: 3 Ways to Show a Logical Equivalence | Ex: DeMorgan's Laws 05m
- Exercise: Simplify the expression: ¬(p ∨ ¬q) ∧ T using logical equivalences
- Video class: Conditional Statements: if p then q 07m
- Video class: Vacuously True Statements 02m
- Exercise: Understanding vacuous truth in implications
- Video class: Negating a Conditional Statement 02m
- Video class: Contrapositive of a Conditional Statement 05m
- Exercise: Which statement is logically equivalent to the conditional P → Q?
- Video class: The converse and inverse of a conditional statement 05m
- Video class: Biconditional Statements | if and only if 02m
- Exercise: Which option is logically equivalent to p iff q
- Video class: Logical Arguments - Modus Ponens 08m
- Video class: Logical Argument Forms: Generalizations, Specialization, Contradiction 03m
- Exercise: Which inference rule is used in the step From P infer P ∨ Q
- Video class: Analyzing an argument for validity 02m
- Video class: Predicates and their Truth Sets 06m
- Exercise: Which set is the truth set of the predicate P(x): x is a factor of 12, with domain the positive integers?
- Video class: Universal and Existential Quantifiers, ? For All and ? There Exists 09m
- Video class: Negating Universal and Existential Quantifiers 08m
- Exercise: Negate the statement For all x in Z+, x > 3
- Video class: Negating Logical Statements with Multiple Quantifiers 08m
- Video class: Universal Conditionals P(x) implies Q(x) 03m
- Exercise: Which formalization correctly expresses the statement Every president of the United States is a US citizen given domain all people, P x means x is president of the United States, C x means x is a US citizen
- Video class: Necessary and Sufficient Conditions 07m
- Video class: Formal Definitions in Math | Ex: Even 03m
- Exercise: Formal definition of odd integers
- Video class: How to Prove Math Theorems | 1st Ex: Even Odd = Odd 08m
- Video class: Step-By-Step Guide to Proofs | Ex: product of two evens is even 18m
- Exercise: Choose the correct formalization of the statement The product of two even integers is even
- Video class: Rational Numbers | Definition First Proof 06m
- Video class: Proving that divisibility is transitive 11m
- Exercise: Transitivity of Divisibility
- Video class: Disproving implications with Counterexamples 08m
- Video class: Proof by Division Into Cases 05m
- Exercise: Which proof strategy shows that for any integer n, n squared has the same parity as n by analyzing the even and odd cases separately?
- Video class: Proof by Contradiction | Method 09m
- Video class: Proof by Contrapositive | Method 03m
- Exercise: Which statement is the contrapositive of the claim If n^2 is even then n is even?
- Video class: Quotient-Remainder Theorem and Modular Arithmetic 09m
- Video class: Proof: There are infinitely many primes numbers 07m
- Exercise: Infinitely Many Primes via Product Plus One
- Video class: Introduction to sequences 06m
- Video class: The formal definition of a sequence. 03m
- Exercise: Formal definition of a sequence
- Video class: The sum and product of finite sequences 09m
- Video class: Intro to Mathematical Induction 12m
- Exercise: Identify the two required components of a valid mathematical induction proof for statements Pn for all n greater or equal to a
- Video class: Induction Proofs Involving Inequalities. 06m
- Video class: Strong Induction // Intro and Full Example 10m
- Exercise: What is the inductive hypothesis used in strong induction to prove P(n) for n ≥ a?
- Video class: Recursive Sequences 07m
- Video class: The Miraculous Fibonacci Sequence 06m
- Exercise: Given F1=1, F2=1 and for n>=3, F_n = F_{n-1} + F_{n-2}, what is F6?
- Video class: Prove A is a subset of B with the ELEMENT METHOD 06m
- Video class: Proving equalities of sets using the element method 03m
- Exercise: Which set identity is correct based on logical equivalences?
- Video class: The union of two sets 05m
- Video class: The Intersection of Two Sets 05m
- Exercise: Find A ∩ B where A = {n in Z : n = 2p} and B = {n in Z : n = 3q}
- Video class: Universes and Complements in Set Theory 02m
- Video class: Using the Element Method to prove a Set Containment w/ Modus Tollens 03m
- Exercise: Given A ⊆ B, which containment must be true?
- Video class: Power Sets and the Cardinality of the Continuum 14m
- Video class: Relations and their Inverses 02m
- Exercise: Definition of an Inverse Relation
- Video class: Reflexive, Symmetric, and Transitive Relations on a Set 06m
- Video class: Equivalence Relations - Reflexive, Symmetric, and Transitive 04m
- Exercise: Transitivity in Modular Arithmetic Equivalence
- Video class: You need to check EVERY spot for reflexivity, symmetry, and transitivity 03m
- Video class: Introduction to probability // Events, Sample Space, Formula, Independence 08m
- Exercise: Probability of exactly one head in two fair coin flips
- Video class: Example: Computing Probabilities using P(E)=N(E)/N(S) 02m
- Video class: What is the probability of guessing a 4 digit pin code? 06m
- Exercise: What is the probability of correctly guessing a 4-digit PIN on the first try?
- Video class: Counting with Triple Intersections // Example 11m
- Video class: Permutations: How many ways to rearrange the letters in a word? 06m
- Exercise: How many length-4 permutations can be formed from 7 distinct letters without repetition?
- Video class: The summation rule for disjoint unions 05m
- Video class: Counting formula for two intersecting sets: N(A union B)=N(A) N(B)-N(A intersect B) 07m
- Exercise: How many integers from 1 to 100 are multiples of 4 or 6?
- Video class: Combinations Formula: Counting the number of ways to choose r items from n items. 06m
- Video class: How many ways are there to reorder the word MISSISSIPPI? // Choose Formula Example 07m
- Exercise: How many distinct rearrangements of the letters in MISSISSIPPI are there?
- Video class: Counting and Probability Walkthrough 17m
- Video class: Intro to Conditional Probability 06m
- Exercise: Compute conditional probability from intersection and event probability
- Video class: Two Conditional Probability Examples (what's the difference???) 06m
- Video class: Conditional Probability With Tables | Chance of an Orange M 09m
- Exercise: Compute a conditional probability from a count table
- Video class: Bayes' Theorem - The Simplest Case 05m
- Video class: Bayes' Theorem Example: Surprising False Positives 12m
- Exercise: Bayes theorem: Interpreting a positive medical test
- Video class: Bayes' Theorem - Example: A disjoint union 08m
- Video class: IS CHESS A GAME OF CHANCE? Classical vs Frequentist vs Bayesian Probability 13m
- Exercise: Which interpretation fits estimating P(heart) by repeated draws from an unknown deck?
- Video class: Intro to Markov Chains 11m
- Video class: Markov Chains 06m
- Exercise: Given a transition matrix P and an initial state vector s0, which expression gives the state vector after n iterations in a Markov chain
- Video class: Intro to Linear Programming 14m
- Video class: Intro to Graph Theory | Definitions 05m
- Exercise: Why is the edge collection modeled as a multiset in the bridge graph?
- Video class: Properties in Graph Theory: Complete, Connected, Subgraph, Induced Subgraph 04m
- Video class: Degree of Vertices | Definition, Theorem 04m
- Exercise: Effect of adding an edge on sum of degrees
- Video class: Euler Paths 06m
- Video class: The End of Discrete Math - Congrats! Some final thoughts... 04m
- Exercise: Which statement best captures the core skill emphasized in a discrete mathematics course focused on rigorous thinking?
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