Set Theory and Number Systems - Mathematics (Undergraduate Foundation)

This course establishes the foundational language of mathematics. It covers the core principles of set theory and applies them directly to the structure of the real number system. Mastery of this material is non-negotiable for further study in any quantitative field. The principles of set theory are the structural basis for formal logic, database theory, and computer programming. Understanding number systems as structured sets is the prerequisite for all higher mathematics, including algebra and calculus. This course provides the essential logical framework for these disciplines. By the end of this course, you will be able to use formal notation to define sets and their relationships, perform standard set operations such as union and intersection, use Venn diagrams to analyse set identities, and classify numbers within the real number system. This course is designed for first-year university students of mathematics, computer science, and engineering. It is also required foundational material for any professional in data analysis, software development, or other technical fields requiring rigorous logical thought.

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

Course Chapters

1. Introduction

This chapter establishes the essential groundwork for the course. It provides precise definitions of a set and its fundamental properties, establishing the correct notation and terminology required for all subsequent material on the subject. Key topics covered are the formal definition of a set, determining a set's cardinality, the distinction between finite and infinite sets, and the roles of the empty and universal sets.

Chapter lessons

1-1. Welcome

A direct statement on the course's purpose. It establishes the critical role of set theory as the foundational language for all quantitative and logical disciplines.

1-2. What is a set?

9:02

Provides a precise definition of a mathematical set. The lesson covers the two essential methods for describing a set's elements: the roster form and set-builder notation.

1-3. Finite and infinite sets

A finite set has countable elements; an infinite set does not. This lesson defines both, provides examples, and explains why the distinction is critical.

1-4. Order and cardinality

Defines the order, or cardinality, of a set as the number of elements it contains. This is a fundamental property used in all subsequent set analysis.

1-5. The empty and universal sets

Introduces two critical concepts: the empty set, which contains no elements, and the universal set, which contains all elements relevant to a particular problem.

2. Subsets, Equality, and Power Sets

This chapter examines the relationships between sets. It covers the concepts of subsets and set equality, which are essential for comparing sets, and introduces the power set, a fundamental structure in advanced mathematics. Topics include defining subsets and proper subsets, verifying set equality, and constructing the power set of a given set. Worked examples demonstrate these core skills.

Chapter lessons

2-1. Subsets and proper subsets

Explains the concept of a subset, where all elements of one set are contained within another. The distinction between a subset and a proper subset is clarified.

2-2. Equality of sets

Defines the conditions required for two sets to be considered equal. This lesson establishes the principle of mutual inclusion as the basis for proving set equality.

2-3. The power set

Introduces the power set of a given set A as the set of all possible subsets of A. The notation and its relationship to cardinality are covered.

3. Set Operations

This chapter details the primary operations used to combine and modify sets. It provides the formal framework for manipulating sets, a skill essential for logic and problem-solving. Key operations covered are union, intersection, and complement. Their formal definitions, properties, and direct applications in worked calculations are demonstrated.

Chapter lessons

3-1. The union of sets

Defines the union of two or more sets as the set containing all elements from the original sets. It covers the notation and fundamental properties of the union operation.

3-2. The intersection of sets

Defines the intersection of sets as the set containing only the elements common to all original sets. Notation and properties are detailed.

3-3. The complement of a set

Covers the definition of the complement of a set relative to a universal set. It explains how to determine the elements not present in a given set.

4. Venn Diagrams

This chapter introduces Venn diagrams as the standard method for the visual representation of sets. This tool is critical for translating abstract set notation into a concrete graphical form for analysis. Topics include representing sets visually, analysing relationships, solving cardinality problems for two and three sets, and using diagrams to prove identities.

Chapter lessons

4-1. Representing sets visually

Introduces the components of a Venn diagram. This lesson explains how to represent the universal set, individual sets, and their elements in a standard graphical format.

4-2. Analysing set relationships

Demonstrates how to use Venn diagrams to illustrate set operations. The lesson covers graphical representation of union, intersection, complement, and subsets.

5. Real Number Systems

This chapter applies set theory to classify the real numbers. It formally defines the sets of integers, rational numbers, and irrational numbers, establishing the structured hierarchy of the number system used in all of mathematics. Key topics include the formal definitions of major number sets, the distinction between rational and irrational numbers, and the properties that define each classification.

Chapter lessons

5-1. Classifying number sets

Introduces the standard set notation for natural numbers, integers, rational numbers, and real numbers. This lesson establishes the hierarchy and inclusion relationships among them.

5-2. Integers and rational numbers

Provides the formal definitions for the set of integers and the set of rational numbers. It covers the properties of these numbers and their representation.

5-3. Irrational numbers

Defines irrational numbers as real numbers that cannot be expressed as a ratio of two integers. This lesson presents standard examples and explains their place in the real number system.

6. Conclusion

This chapter consolidates the core concepts of the course. It provides a structured summary of set theory and the real number system, reinforcing the essential principles required for progression. The conclusion summarises key definitions and operations. It also outlines how these concepts will be applied directly in subsequent mathematics and computer science courses.

Chapter lessons

6-1. Summary of concepts

A concise review of the key definitions, notations, and operations covered in the course. This lesson ensures all foundational material has been consolidated.

6-2. Applying these foundations

Explains how the principles of set theory and number systems are prerequisites for the next courses in the learning track. It directly links this course to future topics.