# Discrete Mathematics

Discrete Mathematics is an important subject for any software developers. I’ve used my time duirng the COVID pandemic to learn about this subject.

The resources that I have been using to learn Discrete Mathematics:

- Discrete Mathematics and its Applications - Rosen 7th Edition
- thetrevtutor
- Professor Kimberly Brehm Videos

- I highly recommend watching Professor Kimberly Brehm videos, they go along with the book, Discrete Mathematics and its Applications

All my notes can be viewed here or download via PDF Format on my Github account Link to Notes via Github

# Chapter 1: The Foundations: Logic and Proofs

Includes:

- Propositions
- Negations
- Conjunctions
- Disjunctions
- Truth Tables
- Direct Proofs
- Proof by Contradiction
- Proof by Cases

# Chapter 2: Sets, Functions, Sequences, Sums, and Matrices

Includes:

- Introduction to Sets
- Set Relationships
- Operations on Sets
- Set Identities
- Proving Set Identities
- Introduction to Functions
- One to One and Onto Functions
- Inverse Functions and Composition Functions
- Introduction to Sequences
- Recurrence Relations
- Summations and Sigma Notation
- Cardinality of Sets
- Matrices and Matrox Operations

# Chapter 3: Algorithms

Includes:

- Growth rate of Functions

# Chapter 4: Number Theory

Includes:

- Divisibility
- Modular Arithmetic
- Decimal Expansions from Binary, Octal and Hexadecimal
- Binary, Octal and Hexadecimal Expansions from Decimal
- Conversions between Binary, Octal and Hexadecimal Expansions
- Algorithms for Integer Operations
- Prime Numbers and their Properties
- Greatest Common Divisors and Least Common Multiples
- The Euclidean Algorithm
- The Greatest Common Divisor as Linear Combinations
- Solving Linear Congruences using Inverse

# Chapter 5: Induction and Recursion

Includes:

- Proof using Mathematical Induction -Summation Formula -Inequalities -Divisibilty
- The Well-Ordering Principle and Strong Induction
- Structural Induction
- Recursive Algorithms

# Chapter 6: Counting Rules

Includes:

- The Basics of Counting
- Sum Rule
- Subtraction Rules
- Division Rule

- The Pigeonhole Principle
- Permutations and Combinations
- The Binomial Theorem
- Generalized Permutations and Combinations

# Chapter 7: Introduction to Probability

Includes:

- LaPlace Definition
- Probability Rules
- Probability Theory
- Random Variables and the Binomial Distribution

# Chapter 9: Introduction to Relations

Includes:

- Introduction to Relations
- Properties of Relations
- Combining Relations
- Matrix Representations of Relations and Properties
- Representing Relations using Digraphs
- Closures of Relations
- Equivalence Relations

# Chapter 13: Language

- notes coming soon

# Homework Problems

## Homework 1:

- Sections 1.1

## Homework 2:

- Sections 1.1, 1.2, 1.3, 1.4

## Homework 3:

- Sections 1.5, 1.6, 1.7

## Homework 4:

- Sections 2.1, 2.2, 2.3

## Homework 5:

- Sections 2.3, 2.4

## Homework 6:

- Sections 2.6, 3.2

## Homework 7:

- Sections 4.1, 4.2, 4.3

## Homework 8:

- Sections 4.4, 5.1, 5.2

## Homework 9:

- Sections 6.1, 6.2, 6.3, 6.4, 6.5, 7.1

## Homework 10:

- Sections 7.2, 9.1, 9.2, 9.3, 9.4, 9.5