Discrete Mathematics

Course Introduction

Introducing the Course

0.1 - Introducing the Teacher

0.2 - Overview and Content of the Course

Chapter 1

Content and Learning Outcomes Chapter 1 - Logic and Proofs

Introduction to Chapter 1

1.1 - Content and Learning Outcomes Paragraph 1

1.1.1 - Introduction to Propositional Logic

1.1.2.1 - Defining Propositions

1.1.2.2 - Compound Propositions

1.1.3.1 - Definition of Conditional Statements

1.1.3.2 - Converse, Contrapositive, Inverse and Biconditional Statements

1.1.3.3 - Examples Related to Converse, Contrapositive, Inverse and Biconditionsl Statements

1.1.4 - Truth Tables of Compound Propositions

1.1.5 - Precedence of Logical Operations

1.1.6 - Logic and Bit Operations

Content of Paragraph 1.2: Applications of Propositional Logic

1.2.1 - Introduction to Propositional Logic and Translating English Sentences

1.2.2 - System Specification

1.2.3 - Boolean Searches

1.2.4 - Logic Puzzles

1.2.5 - Logic Circuits

Content of Paragraph 1.3: Propositional Equivalences

1.3.1 - Introduction to Propositional Equivalences - Logic Equivalences

1.3.2 - Using De Morgan's Laws

1.3.3 - Constructing New Logical Equivalences

1.3.4 - Application of Satisfiability

1.4.1 - Introduction and Definition of Predicates

1.4.2 - Definition and Application of Quantifiers

1.4.3 - Binding Variables and Logical Equivalences

1.4.4 - Negating Quantified Expressions

1.4.5 - Translating Sentences Into Logical Expressions

1.4.6 - Using Quantifiers in Systems Specifications

1.5.1 - Understanding Nested Quantifiers

1.5.2 - Order of Quantifiers

1.5.3 - Translating Statements

1.5.4 - Negating Nested Quantifiers

1.6.1 - Valid Arguments in Propositional Logic

1.6.2 - Rules of Inference for propositional Logic

1.6.3 - Using Rules of Inference to build Arguments

1.6.4 - Resolution and Fallacies

1.6.5 - Rules of Inference for Propositional Logic

1.7.1 - Terminology Related to Proofs and How to Write a Theorem

1.7.2 - Methods to Prove Theorems

1.7.3 - Mistakes in Proofs

1.8.1 - Exhaustive Proof by Cases

1.8.2 - Existence and Uniqueness Proofs

1.8.3 - Proof Strategies and Looking for Counter Examples

Chapter 3 - Algorithms

3.1.1 Introduction to Algorithms

3.1.2 Searching Algorithms

3.1.3 Sorting Algorithms

3.1.4 Greedy Algorithms

3.2.1 Introduction of Big O

3.2.2 Big O notation

3.2.3 Big O for some Important Functions

3.2.4 The Growth of a Combination of Functions

3.3.1 Time Complexity of Algorithms

3.3.2 Complexity of Matrix Multiplications

3.3.3 Algorithm Paradigms

3.3.4 Understanding the Complexity of Algorithms

Chapter 4 -

4.1.1 Division and Division Algorithm

4.1.2 Modular Arithmetics

4.1.3 Arithmetic Modulo m

4.2.1 Integer Representation and Algorithms

4.2.2 Algorithms for Integer Operations

4.2.3 Modular Exponentiation

4.3.1 Primes and Trial Division

4.3.2 The Sieve of Eratosthenes

4.3.3 Conjectures and Open Problems about Primes

4.3.4 Greatest Common Divisors and Least Common Multiples

4.3.5 The Euclidian Algorithm

4.3.6 GCDs as Linear Combinations

4.4.1 Linear Congruences

4.4.2 The Chinese Remainder Theorem

4.4.3 Fermat;s Little Theorem

4.4.4 Fermat'sLittle Theorem

4.4.5 Primitive roots and Discrete Logarithms

4.5.1 Hashing Function

4.5.2 Pseudorandom Numbers

4.5.3 Check Digits

1.5.2 - Order of Quantifiers

Discrete Mathematics Chapter 1

By: Luc De Ceuster, MSc, PMP

10 minutes

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Discrete Mathematics

1.5.2 - Order of Quantifiers

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