CE 215 • Final • Discrete Mathematics for Computer Science

Definition of Divisibility

Example 1

Example 2

Example 3

Division Algorithm

Definition of Modular Arithmetic

Example 1

Properties of Modular Arithmetic

Example 1

Example 2

GCD Greatest Common Divisor

LCM Least Commun Multiple

Euclidian Algorithm

Example 1

Bézout Identity

Example 1

Example 2

Example 3

Inverse of a Number in Modular Arithmetic

Fermat's Little Theorem

Solving Linear Congruences

Example 1

Example 2

Chinese Remainder Theorem

Example 1

Example 2

Example 3

Example 4

Introduction

Proof of Formulas by Induction

Example 1

Example 2

Exam-like Question 1

Exam-like Question 2

Proof of Divisibility by Induction

Example 1

Example 2

Exam-like Question 1

Exam-like Question 2

Proof of Inequality by Induction

Example 1

Example 2

Example 3

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Induction Related to Fibonacci

Exam-like Question 1

Strong Induction

Example 1

Basic Principles of Counting

Counting Examples

Permutations

Permutations Example

Groups and Circular Permutation Example

Identical Objects Example 1

Identical Objects Example 2

Identical Objects Example 3

Combination

n choose r

Committee Example 1

Committee Example 2

Ball Example 1

Ball Example 2

Binomial Theorem

Example 1

Example 3

Example 2

Pascal Identity

Vandermonde's Identity

Example 1

Combinatorial Proof

Example 1

Principle Inclusion-Exclusion

Example 1

Example 2

Pigeonhole Principle

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Number Theory 1

Number Theory 2

Number Theory 3

Number Theory 4

Number Theory 5

Number Theory 6

Number Theory 7

Induction for Formulas 1

Induction for Formulas 2

Induction for Formulas 3

Induction for Formulas 4

Induction for Formulas 5

Induction Proof for Divisibility 1

Induction Proof for Divisibility 2

Induction for Inequalities 1

Induction for Inequalities 2

Induction with Fibonacci 1

Counting 1

Counting 2

Counting 3

Counting 4

Counting 5

Counting 6

Counting 7

Counting & Binomial Coefficients 1

Principle Inclusion - Exclusion 1

Pigeonhole Principle 1

Pigeonhole Principle 2

Pigeonhole Principle 3

Pigeonhole Principle 4

Pigeonhole Principle 5

Introduction

Homogenous Recurrence Relations

Example 1

Example 2

Example 3

Example 4

Non Homogenous Recurrence Relations

Example5

Example 6

Creating Recurrence Relations

Example 7

Example 8

Example 9

Definition

Example 1

Example 2

Example 3

Example 4

Extended Binomial Theorem

Counting Problems with Generating Functions

Example 5

Example 6

Example 7

Definition of Relation

Example 1

Example 2

Example 3

Example 4

Reflexive Relations

Example 5

Example 6

Symmetric Relations

Antisymmetric Relations

Example 7

Transitive Relations

Example 8

Properties of Relations

Example 9

Example 10

Example 11

Example 12

Composition of Relations

Example 13

Example 14

Combining Relations

Inverse and Complementary of a Relation

Example 15

Matrix Representation of a Relation 1

Matrix Representation of a Relation 2

Example 16

Operations on Zero - One Matrices

Example 17

Equivalence Relation

Example 18

Example 19

Example 20

Partially Ordered Set

Example 21

Example 22

Example 23

Totally Ordered Set

Example 24

Hasse Diagram

Example 25

Maximal and Minimal Elements

Greatest and Least Elements

Example 26

Upper and Lower Bound

Least Upper and Greatest Lower Bound

Example 27

Recurrence Relations 1

Recurrence Relations 2

Recurrence Relations 3

Recurrence Relations 4

Recurrence Relations 5

Recurrence Relations 6 (Hard)

Generating Functions 1

Generating Functions 2

Generating Functions 3

Generating Functions 4

Generating Functions 5

Generating Functions 6

Generating Functions 7

Generating Functions 8

Relations 1

Relations 2

Relations 3

Relations 4

Relations 5

Relations 6

Relations 7

Relations 8

Relations 9

Relations 10

Relations 11

Introduction

Graph Terminology

Handshaking Theorem

Example 1

Special Graphs

Example 2

Bipartite Graphs

Example 3

Example 4

Complete Bipartite Graph

Matching

Subgraph

Example 5

Subgraph Induced

Edge Contraction

Example 6

Complementary Graph

Adjacency Matrices - Undirected Graphs

Adjacency Matrices - Directed Graphs

Example 1

Example 2

Incidence Matrices

Example 3

Isomorphism of Graphs

Example 4

Example 5

Example 6

Example 7

Definition of Paths and Circuits

Connected Graphs

Euler Paths and Circuits

Example 8

Hamilton Paths and Circuits

Example 9

Example 10

Planar Graph

Planar Graph - Euler Formula

Example 11

Corollaries about Planar Graph

Example 12

Adjacency Matrices

Incidence Matrices

Bipartite Graphs & Euler Circuits (Spring 2023)

Bipartite Graphs, Euler Circuit & Planar Graphs (Fall 2022)

Euler Path, Induced Subgraph & Incidence Matrix (Fall 2022)

Chromatic Number 1

Chromatic Number 2 (Spring 2023)

Isomorphism 1 (Spring 2023)

Isomorphism 2

Isomorphism 3 (Fall 2022)

Isomorphism 4 (Fall 2022)

Isomorphism 5 (Fall 2022)

True/False Problems 1 (Spring 2023)

True/False Problems 2 (Spring 2023)