Advanced Diploma in Discrete Structures
This free online course introduces you to the elements of mathematics that form the basis of computer science.
Description
In this free online course, you will be introduced to the fundamentals of logic and proposition. You will learn how to construct truth tables and how to verify the validity of arguments. You will also learn about propositional logic and predicate logic and understand how laws of inference are used on quantified statements to solve practical problems.
Next, you will learn about mathematical systems, proof techniques and the concept of mathematical induction. You will be exposed to set theory, including the different unary and binary set operations and representations of set relationships by Venn and Arrow diagrams. Following set theory, you will learn about relations and their properties, including equivalence relation and POSET. You will also learn how to define set relations using recursion and how to frame or design recursive algorithms. Next, you will delve into recurrence relations, the pigeonhole principle and the inclusionexclusion principle.
The next part of the course will introduce you to combinatorics and permutations and combinations of sets and multisets. You will also get to know about different algebraic systems and their properties, including ring, group, and field. Finally, you will be exposed to modular and polynomial arithmetic, that governs the construction of finite fields. This course will be of interest to students of mathematics, engineering and science, or those interested in learning more about the advanced principles of mathematical and logical systems.
Start Course NowModules
Introduction to Propositional Logic

Introduction to Propositional Logic  Learning Outcomes

Fundamentals of Logic

Conditional Propositions

Relation Between Two Propositions

Laws of Propositions

Validity of Propositions

Rules of Inference

Introduction to Propositional Logic  Lesson Summary
Predicate Logic

Predicate Logic  Learning Outcomes

Introduction to Predicate Logic

Generalized Predicate Logic

Nested Quantifiers

Laws of Inference for Quantified Statements

Predicate Logic  Lesson Summary
Methods of Proof and Induction

Methods of Proof and Induction  Learning Outcomes

Different Proof Techniques

Variations of Direct and Indirect Proof Techniques

Proof by Induction

Strong Form of Mathematical Induction

Problem Solving Using Mathematical Induction Techniques

Methods of Proof and Induction  Lesson Summary
Sets and Functions

Sets and Functions  Learning Outcomes

What are Sets?

Basic Set Operations

More Set Operations

Laws of Sets

Functions of Sets

Sets and Functions  Lesson Summary
Relations and their Properties

Relations and their Properties  Learning Outcomes

What is a Relation?

Properties of a Relation

Ordered Sets

Hasse Diagrams

Properties of Poset

Relations and their Properties  Lesson Summary
Recursion

Recursion  Learning Outcomes

What is Recursion?

Recursively Defined Sequences

Recursively Defined Functions and Sets

Recursive Algorithms

Application of Recursive Algorithms

Recursion  Lesson Summary
Recurrence Relations

Recurrence Relations  Learning Outcomes

Identifying Recurrence Relations

Framing Recurrence Relations

Solving Recurrence Relations

Solving Linear Homogeneous Recurrence Relations

Linear Nonhomogeneous Recurrence Relations

Recurrence Relations  Lesson Summary
Counting Techniques and Pigeonhole Principle

Counting Techniques and Pigeonhole Principle  Learning Outcomes

The Pigeonhole Principle

Solving Numerical Problems Using Pigeonhole Principle

Using the Simple Form of the Pigeonhole Principle

Strong Form of the Pigeonhole Principle

InclusionExclusion Principle

Counting Techniques and Pigeonhole Principle  Lesson Summary
Combinatorics

Combinatorics  Learning Outcomes

What is Combinatorics?

Combinations of Sets

Permutations of Multisets

Combinations of Multisets

Derangement

Combinatorics  Lesson Summary
Algebraic Structures

Algebraic Structures  Learning Outcomes

Introduction to Algebraic Systems

Semigroup and Monoid

Subsemigroup and Cyclic Monoid

Group

Group Theory

Algebraic Structures  Lesson Summary
Rings and Modular Arithmetic

Rings and Modular Arithmetic  Learning Outcomes

Overview of Ring

Ring Properties

Modular Arithmetic

Modular Operations  Part 1

Modular Operations  Part 2

Rings and Modular Arithmetic  Lesson Summary
Finite Fields and Applications

Finite Fields and Applications  Learning Outcomes

Introduction to Fields

Polynomial Arithmetic

Modular Polynomial Arithmetic  Part 1

Modular Polynomial Arithmetic  Part 2

Modular Polynomial Arithmetic  Part 3

Finite Fields and Applications  Lesson Summary
Course assessment
Learning Outcomes
Having completed this course you will be able to:
 Define logic and proposition
 Explain the concept of Predicate Logic
 Explain how to solve problems using mathematical induction techniques
 List the different unary and binary set operations
 Define relations and how they are expressed in mathematical notation
 Distinguish between functions and relations
 List the properties of a partially ordered set (POSET)
 Explain how a POSET is represented by a Hasse diagram
 Define functions, sets and sequences recursively
 Explain how to frame and solve recurrence relations
 Apply the pigeonhole principle to solve problems
 Compute permutations and combinations of sets and multisets
 Distinguish between the different algebraic systems based on their properties
 Explain how modular polynomial arithmetic governs the construction of a field
Certification
All Alison courses are free to enrol, study and complete. To successfully complete this Diploma course and become an Alison Graduate, you need to achieve 80% or higher in each course assessment. Once you have completed this Diploma course, you have the option to acquire an official Diploma, which is a great way to share your achievement with the world. Your Alison Diploma is:
Ideal for sharing with potential employers  include it in your CV, professional social media profiles and job applications
An indication of your commitment to continuously learn, upskill and achieve high results
An incentive for you to continue empowering yourself through lifelong learning
Alison offers 3 types of Diplomas for completed Diploma courses:
Digital Diploma  a downloadable Diploma in PDF format, immediately available to you when you complete your purchase
Diploma  a physical version of your officially branded and securitymarked Diploma, posted to you with FREE shipping
Framed Diploma  a physical version of your officially branded and securitymarked Diploma in a stylish frame, posted to you with FREE shipping
All Diplomas are available to purchase through the Alison Shop. For more information on purchasing Alison Diplomas, please visit our FAQs. If you decide not to purchase your Alison Diploma, you can still demonstrate your achievement by sharing your Learner Record or Learner Achievement Verification, both of which are accessible from your Dashboard. For more details on our Diploma pricing, please visit our Pricing Page.