Gain insights into fundamental set theory concepts and operations. Explore relations, functions, set ordering, and cardinality using Python. Delve into Cantor’s diagonalization and its applications in various fields.

This course aims to introduce fundamental concepts of set theory. It provides hands-on experience through practical tasks using Python programming language (Python 3.8) to enhance engagement.

You’ll start with the basics of sets, their properties, and set operations. Additionally, you’ll learn about relations, functions, ordering of sets, and cardinality of sets. You’ll also explore Cantor’s diagonalization method as a tool to demonstrate the uncountability of sets. 

After completing this course, you’ll have a solid understanding of the foundations of set theory that are vital for understanding many mathematics and computer science concepts, such as discrete mathematics, design of algorithms, theory of automata, and problem-solving. These concepts can also be applied in other significant areas including Physics, Statistics, and Probability.

An Introduction to Basic Set Theory



If a universal set is $\mathbb{U}$, then the **complement** of the set $A$, denoted by $\overline{A}$,  is the set of elements from $\mathbb{U}$ that are not elements of $A$. We can write it as follows:

$$\overline{A} =  \mathbb{U} \setminus A$$

By this definition, the union of a set and its complement always yields the universal set, which is:

$$A \cup \overline{A} =\mathbb{U}$$

Similarly, we can extract the following facts from the definition: 
$$\begin{align*} \overline{\emptyset} &= \mathbb{U} \\ \overline{\mathbb{U}} &= \emptyset
\\ \overline{\overline{A}} &= A\end{align*}$$

Furthermore, it's important to note the following: $A\ne B \Rightarrow \overline{A} \ne \overline{B}.$






Introduction

Fundamentals

Basic Set Operations

Assessment 1

Relations

Assessment 2

Functions

Assessment 3

Indexing and Ordering of Sets

Cardinality of Sets

Assessment 4

Concluding Remarks

Complement