Most of the videos support the following free discrete math textbook.

Discrete Mathematics, An Open Introduction (3rd Edition)

by Oscar Levin

**Introduction andPreliminaries**

**0.1 What is Discrete Math**

**What is Discrete Mathematics?**

**0.2 Mathematical Statements (Playlist) **

Mathematical Statements and Logic Connectives

Truth Conditions for Connectives

Implications and Truth Conditions for Implications

Write Math Statements As Symbols and Symbols as Math Statements

Determine Truth Values of Statements from a Given Implication

Introduction to The Converse and Contrapositive of an Implication

Determine the Converse, Inverse, and Contrapositive of an Implication

Converse and Contrapositive Example (Chinese Food and Milk)

Converse and Contrapositive: Drawing Conclusion from an Implication

If And Only If Statements

Given the Truth Value of an Implication and Converse, Find the Truth Value of If And Only If Statements

Necessary and Sufficient Statements

Equivalent Statements to an Implication

Introduction to Predicates and Quantifiers

Predicate and Quantifier Concept Check 1

Predicate and Quantifier Concept Check 2

Determine if Quantified Statements are True or False from a Table

**0.3: Introduction to Sets (Playlist)**

Introduction to Sets and Set Notation

Determine the Union, Intersection, and Difference of Two Set Given as Lists

Determine Sets Given Using Set Notation (Ex 1)

Determine Sets Given Using Set Notation (Ex 2)

Set Notation: Determine Which Statements are True or False

Determine the Least Element in a Set Given using Set Notation.

Determine the Power Set of a Set and the Cardinality of a Power Set

Determine the Cardinality of Sets Given as Lists

Determine the Cardinality of Sets: Set Notation, Intersection

Determine Sets Involving Unions, Intersections, and Compliments

Determine Sets Involving Unions, Intersections, and Compliments Using a Venn Diagram

The Cartesian Product of Two Sets

Shading Venn Diagrams with Two and Three Sets: Unions and Intersections

Shading Venn Diagrams with Two and Three Sets: Complements

Shading Venn Diagrams with Two and Three Sets: Set Differences

Shading Venn Diagrams with Two and Three Sets: Complements, Intersections, Unions

De Morgan's Laws with Venn Diagrams

De Morgan's Laws: Set Example

Find a Set with Greatest Cardinality that is a Subset of Two Given Sets (Lists)

Find a Set with Greatest Cardinality that is a Subset of Two Given Sets (Set Notation)

Find a Set with Least Cardinality that has Two Given Subsets (Lists)

Find How Many Sets Are a Subset of a Given Set and Have a Given Subset

**0.4: Functions (Playlist)**

Introduction to Functions (Discrete Math)

Two Line Notation for a Function: Inputs and Outputs

Describing Functions (Discrete Math)

Recursive Functions (Discrete Math)

Determine Function Values for a Recursive Functions

Surjective, Injective, and Bijective Functions

Determine if Functions Given in Two Line Notation are Surjective, Injective, and Bijective

Determine if Various Functions are Surjective, Injective, and Bijective

List All Possible Function Given Domain an Codomain Then Determine if Surjective, Injective, and Bijective (8)

List All Possible Function Given Domain an Codomain Then Determine if Surjective, Injective, and Bijective (9)

Image of a Subset of the Domain and Inverse Image of a Subset of the Codomain

**Chapter 1: Counting (Playlist)**

**1.1: Additive and Multiplicative Principles**

Introduction to Counting Using Additive and Multiplicative Principles

Counting Principle With Playing Cards: Picking 1 and 2 Cards (Not Disjoint)

Counting: Neck Wear and Outfits (Additive and Multiplicative Principles)

Counting: Number of Hexadecimal Numbers with Restrictions (And/Or)

Counting: Number of Hexadecimal Strings with Restrictions (And/Or)

The Cardinality of the Union of Two Sets

Determine the Number of Elements in a Large Set that are Multiples of 3 or 5

The Cardinality of the Union of Three Sets

Determine the Number of Elements in a Large Set that Multiples of 3 or 5 or 8

Determine the Greatest and Least Values of the Cardinality of an Intersection and Union

Determine Sum of the Cardinality of the Union and Intersection of Two Sets

The Cardinality of the Union of Three Sets Application: TV Shows

**1.2: Binomial Coefficients**

Counting Subsets and Subsets of a Specific Cardinality

Determine How Many Subsets Meet Various Conditions (1)

Determine How Many Subsets Meet Various Conditions (2)

Determine How Many Subsets Have More Than a Given Cardinality

Introduction to Bit Strings

Determine the Number of 10-Bit Strings Under Various Conditions

Problem Solving with Bit Strings - Coin Problems

Determine the Number of 9-Bit Strings That Have More Than a Given Number of 1's

Introduction to Lattice Paths

Determine the Number of Shortest Lattice Paths Under Various Conditions

Lattice Paths Application: Driving

Introduction to Binomial Coefficients

Determine Binomial Coefficients

Determine a Coefficient of a Expression of Binomials Raised to Powers

Evaluating Combinations Using Pascal's Triangle

Making Connections: The Many Applications of Combinations

Combinations: The Number of 4 Topping Pizzas

**1.3: Combinations and Permutations**

Counting: Find the Number of Functions and Bijective Functions (Discrete Math)

Counting: Find the Number of Functions, Injective Functions, and Increasing Functions (Discrete Math)

Counting: Number of Pizza with 3 Toppings and All Possible Pizzas

Counting: Find the Number of Lock Combinations

Counting: Find the Number of 5-Digit Numbers Under Various Conditions

Counting: Books on a Shelf - Arrangements and Alphabetized

Counting: The Number of Types of Quadrilaterals from Two Rows of Points

Counting: The Number of Triangles from an L-Shape of Points

Counting: The Number of Anagrams of Words. (No repeats and repeats)

Counting: The Number of Ordered Groups of Four

**1.4: Combinatorial Proofs**

Algebraic Proof: C(n,k)=C(n-1,k-1)+C(n-1,k)

Combinatorial Proofs: C(n,k)=C(n-1,k-1)+C(n-1,k)

Algebraic and Combinatorial Proofs: C(n,k)=C(n,n-k)

Combinatorial Proof: 1n+2(n-1)+3(n-2)+…+(n-1)2+n1

**1.5: Stars and Bars**

Introduction to Stars and Bars Counting Method

Stars and Bars: The Number of Multisets

Stars and Bars: The Number of Ways of Putting Basketballs in Bins

Stars and Bars: The Number of 4-Digit Number with Repetition in Nondecreasing Order

Stars and Bars: The Number of Integers Solutions to x+y+z=8 With Conditions

Stars and Bars: Determine the Number of Outcomes of Rolling 5 Dice (Yahtzee)

Stars and Bars: Determine the Number of Outcomes of 7 Coins of 4 Types

Stars and Bars: The Number of Integer Solutions to w+x+y+z=25 with Different Lower Bounds

Stars and Bars: Number of Discrete Functions, Increasing Functions, and Nondecreasing Functions

**1.6: Advanced Counting Using PIE**

Stars and Bars with PIE: Find the Number of Meals from a Dollar Menu with Conditions

Introduction to Derangements

Find the Number of Permutations with One Fixed Element Using Derangements

Stars and Bars with PIE: Determine the Number of Ways of Putting Balls in Bins with Upper Bound

Stars and Bars with PIE: Assigning Stars to Students

**Chapter 2: Sequences (Playlist)**

**2.1: Describing Sequences**

Introduction to Sequences (Discrete Math)

Determine a Closed Formula for a Given Sequence (1)

Determine a Closed Formula for a Given Sequence (2)

Determine Partial Sums and Recursive and Closed Formulas for Sequences

Given a Recursive Definition of a Sequence, Find Terms and a Closed Formula (1)

Given a Recursive Definition of a Sequence, Find Terms and a Closed Formula (2)

Given a Closed Formula for a Sequence, Find Terms and a Recursive Definition

Introduction to Partial Sums and Partial Products

Given a Sequence, Find Partial Sums and Find a Formula For S_n Involving a_n

Determine Recursive Formulas for Sequences

Given a Recursive Definition, Find Terms of the Sequence

Given a Closed Formula for a Sequence, Find Terms and Find a Closed Formula for a Sequence

Show a Closed Formula for a Sequence Satisfies a Recurrence Relation

**2.2: Arithmetic and Geometric Sequences**

Introduction to Arithmetic and Geometric Sequences (Discrete Math)

Arithmetic Sequence: Find Recursive Definition, Close Formula, and Number of Terms

Find the Sum of an Arithmetic Sequence and a Closed Formula for a Sequence of Partial Sums (Formula Used)

Find the Sum of an Arithmetic Sequence and a Closed Formula for a Sequence of Partial Sums (Reverse and Add)

Arithmetic Sequence: Find Next Term, Closed Formula, and Partial Sum

Sum of Arithmetic Sequence: Find Number of Terms and Sum with No Formula

Arithmetic Sequence: Find Number of Terms, Second to Last Term, and Partial Sum (Reverse/Add)

Find a Sum of a Geometric Sequence Using Multiply, Shift, and Subtract Method (1)

Find a Sum of a Geometric Sequence Using Multiply, Shift, and Subtract Method (2)

Given 5,x,y,625, Find x and y if the Sequence is Arithmetic or Geometric

Find the Closed Formula for a Sequence that is the Partial Sums of an Arithmetic Sequence

Show that 0.33333… Equals 1/3

**2.3: Polynomial Fitting**

Introduction to Polynomial Fitting to Find a Closed Formula for a Sequence

Find a Closed Formula for a Sequence Using Polynomial Fitting (Degree 2, by hand)

Find a Closed Formula for a Sequence Using Polynomial Fitting (Degree 2, Aug Matrix)

Find a Closed Formula for a Sequence Using Polynomial Fitting (Degree 3, Aug Matrix)

Find a Closed Formula for a Sequence of Difference Given a_n

**2.4: Solving Recurrence Relations**

Sequences: Introduction to Solving Recurrence Relations

Solve a Recurrence Relation Using Inspection

Solve a Recurrence Relation Using the Telescoping Technique

Solve a Recurrence Relation Using the Characteristic Root Technique (1 Repeated Root)

Solve a Recurrence Relation Using the Characteristic Root Technique (2 Distinct Roots)

Find a Recurrence Definition and Solve the Recurrence Relation of a Given Sequence

Determine a Recursive Definition and Closed Formula for a Geometric Sequence

Determine and Solve a Recurrence Relation: Path of Tiles Problem

**2.5: Induction**

Introduction to Proof by Induction: Prove 1+3+5+…+(2n-1)=n^2

Mathematical Induction (older)

Proof by Induction: Prove The Sum of n Squares Formula

Proof by Induction: 4^n - 1 is a Multiple of 3

Proof by Induction: Prove The Sum of n Counting Numbers Formula

Proof by Induction: Prove n^2 < 2^n with n >= 5.

Proof by Strong Induction: If x + 1/x is an Integer Then x^n+1/x^n is an Integer

**Chapter 3 Symbolic Logic and Proofs (Playlist)**

**3.1: Propositional Logic**Introduction to Propositional Logic and Truth Tables

Make Truth Tables for If (P and Q) Then (P or Q) and If (P or Q) Then (P and Q)

Make Truth Tables for P and (If Q Then P) and If (Not P) and (If Q Then P)

Make a Truth Table for If (Not P) Then (Q and R)

Introduction to Logically Equivalent Statements

Determining if Two Statements Are Equivalent Using a Truth Table

Deduction Rules: Modus Ponens and Modul Tollens

Verify the Deduction Rule: If P then Q. If (Not P) Then Q. Therefore Q.

Determine if an Argument is a Deduction Rule or Not: P -> R. Q -> R. R. Therefore P or Q

Determine if an Argument is a Deduction Rule or Not: If (P and Q), Then R. Not P or Not Q. Therefore Not R

Draw Conclusions From the Negation of Statements: Eat and Drink Application

Simplify Statements Using Logically Equivalent Statements

Logically Equivalent Statements: Draw a Conclusion from a False Implication

Introduction to Predicate Logic

Simplify the Negation of Statements with Quantifiers and Predicates

**3.2: Proofs**Introduction to Common Mathematical Proof Methods

Introduction to Direct Proofs: If n is even, then n squared is even

Introduction to Proof by Contrapositive: If n squared is even, then n is even

Proof by Contrapositive: If a + b is odd, then a is odd or b is odd

Introduction to Proof by Contradiction: sqrt(2) is irrational

Proof by Contradiction: There are infinitely many primes

Proof by Contradiction: There are no integers x and y such that x^2 = 4y + 2

The Pigeonhole Principle (Proof by Contrapositive)

Introduction to Proof by Counter Example

Proof by Counter Example: Prove a Converse is False

Determine the Negation, Converse, and Contrapositive of a Quantifier Statement (Symbols)

Proof by Cases: For Any Integer, n^3-n is Odd

Proof Exercise: State the Contrapositive, Converse and Negation, Then Prove the Truth Value

Proof Exercise: Determine the Type of Proof to be Used

**Chapter 4: Graph Theory (Playlist)**

**4.1: Definitions**

Introduction to Graph Theory

The Definition of a Graph (Graph Theory)

Isomorphic Graphs

Subgraphs and Induced Subgraphs

More Graph Theory Definitions

The Handshake Lemma

Bipartite Graphs and Named Graphs

Handshake Lemma Exercises: Possible Number of Friends

Largest Possible Number of Edges for Various Types of Graphs

**4.2: Trees**

Introduction to Trees and Properties of Trees

Introduction to Rooted Trees

Given a Rooted Tree, Determine Relationships

Every Tree is a Bipartite Graph

Introduction to Spanning Trees

Given the Definition of a Graph, Determine if the Graph is a Tree

Given a Degree Sequence, Determine if the Graph is a Tree

Given the Graph of a Rooted Tree List the Children, Parents and Siblings of All Vertices

Given a Tree, Does Changing the Root Change the Number of Children and Grandchildren?

**4.3: Planar Graphs**

Introduction to Planar Graphs and Euler's Formula

Non-Planar Graphs: Proving K5 and K33 are Non-Planar

Introduction Polyhedra Using Euler's Formula

Prove There Are Exactly 5 Regular Polyhedra

Is a Planar Graph Possible Given the Number of Vertices, Edges, and Faces?

Is a Planar Graph Possible Given a Degree Sequence?

Is it Possible for a Connected Graph to be Planar Given Vertices, Edges, and Faces?

Is it Possible for a Graph to be Planar with the Same Number of Vertices and Edges?

Euler's Formula: Find a Missing Face of a Polyhedron

**4.4: Coloring**

Introduction to Vertex Coloring and the Chromatic Number of a Graph

Upper and Lower Bounds for the Chromatic Number of a Graph

Edge Coloring and the Chromatic Index of a Graph

Determine Which Graphs have a Given Chromatic Number

Find the Chromatic Number of the Given Graphs

Find the Chromatic Number of a Graph that Represents a Cube

**4.5: Euler Paths and Circuits**

Introduction to Euler Paths and Euler Circuits

Introduction to Hamilton Paths and Hamilton Circuits

Euler Path Application: Road Trip

Determine if Named Graphs Have Euler Paths or Euler Circuits

Euler Path Application: Home Tour Passing Though each Doorway Once

Hamilton Path Application: Home Tour Visiting Each Room Once

Vertex Degree Application: Home Remodel with Odd Number of Doors

For Which Values of n Does K_n have an Euler Path or an Euler Circuit

For Which Values of m and n Does K_m,n have an Euler Path or an Euler Circuit

For Which Values of n Does K_n have a Hamilton Path or a Hamilton Circuit

**4.6: Matching in Bipartite Graphs**

Introduction to Matching in Bipartite Graphs (Hall's Marriage Theorem)

Proof by Contradiction Using Hall's Marriage Theorem (Playing Cards)

Bipartite Graphs: Determine a Matching of A if Possible

Matching in Bipartite Graphs: Marriage Arrangements

**Chapter 5: Additional Topics (Playlist)**

**5.1: Generating Functions**

Introduction to Generating Functions for Sequences

Introduction to Building Generating Functions for Sequences

Building Generating Functions for Sequences Using Differencing

Determine a Generating Function for the Sequence: 1,3,5,7,9,… Using Differencing

Determine a Generating Function for the Sequence: 1,4,9,16,25,… Using Differencing

Determine a Generating Function for the Sequence: 4,5,7,10,14,… Using Differencing

Determine a Generating Function for a Recursively Defined Sequence (a_n=3a_(n-1)-2a_(n-2)

Determine a Generating Function for a Recursively Defined Sequence (a_n=4a_(n-1)-3a_(n-2)

Multiplying Generating Series to Determine a New Sequence and New Generating Function

Find a Closed Formula for a Recursively Defined Sequence Using the Generating Function

Determine Generating Functions of Sequences from Known Generating Functions (Part 1)

Determine Generating Functions of Sequences from Known Generating Functions (Part 2)

Determine Sequences from Given Generating Functions (Part 1)

Determine a Sequences from Given Generating Functions (Part 2)

Find a Generating Function of a Sequence Given a Closed Formula

Find the Closed Formula for a Sequence Given a Generating Function

**5.2: Introduction to Number Theory**

Intro to Number Theory and The Divisibility Relation

Introduction to Addition Using Clock Arithmetic: n (mod 12) and n + m (mod 12)

Clock Arithmetic (mod 12): Find a Time in the Future

Clock Arithmetic (mod 12): Find a Previous Time

Clock Arithmetic: Adding Numbers Modulo 12

Clock Arithmetic: Subtracting Numbers Modulo 12

Clock Arithmetic: Multiplying Numbers Modulo 12

Clock Arithmetic: Evaluate Modulo 12 Using the Desmos Graphing Calculator

The Division Algorithm and Remainder Classes

Introduction to the Modulo Operator: a mod b with a positive

The Modulo Operator: a mod b with a negative

Introduction to Congruence Modulo n

Congruence Modulo n Properties: Equivalent Relation

Find a Remainder Using Congruences: 3491/9

Use Congruence to Determine Remainders of 2^2019 when Divided by 2, 5, 7, and 9

Find a Remainder Using Congruences: 2^(124)/7

Find a Remainder Using Congruences: 3^(123)/7 (Two Versions)

Simplify a Congruence Using Division

Solving Congruences

Introduction to Solving Linear Diophantine Equations Using Congruence

Application of a Linear Diophantine Equation: Number of Stamps

**5.3: Algorithm Analysis**Introduction to Big-O Notation

Introduction to Big-Omega Notation

Introduction to Big-Theta Notation

Compare Algorithm Complexity Given The Execution Time as a Function

Rank the Complexity of Functions

Determine Time Complexity of Code Using Big-O Notation: O(1), O(n), O(n^2)

Determine Time Complexity of Code Using Big-O Notation: O(n+m), O(n*m), O(log(n))

Determine Time Complexity Function and Time Complexity Using Big-O Notation: f(n)=(cn(n-1))/2