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

# Union
The **union** of the two sets $A$ and $B$ is the set $C$ that contains every element that is a member of either $A$ or $B$ (or both). We can write this definition as follows:

$$C=A \cup B = \{x \mid x\in A \lor x\in B\}$$

> **Note:** If an object is a member of both $A$ and $B$, it will appear only once in the union set $C$  because repetition of an element is not allowed in a set. 



 The following facts are obvious from the definition of a union operation where $\mathbb{U}$ is the universal set, $A$ is a set, and $\emptyset$ is the empty set:

$$\begin{align*}
A\cup \emptyset = \emptyset \cup A &= A \\
A\cup \mathbb{U} = \mathbb{U} \cup A &= \mathbb{U} \\

A\cup A &= A 
\end{align*}$$

## Cardinality 
Let’s consider that $C = A\cup B$, where $A$ and $B$ are finite sets.

The cardinality of $C$ can't be greater than the sum of cardinalities of $A$ and $B$. 

At most, $|C|=|A|+|B|$ if there is no common element in $A$ and $B$. But the general relationship of $|C|$ with $|A|$ and $|B|$ can be written as follows:

$$\begin{align*}
\newline \left| A\cup B\right| &\leq |A| +|B|\\
 \left| A\cup B\right| &\ge |A| \\
\left| A\cup B\right| &\ge |B|
\end{align*}$$
 

Learn about the union operation defined for sets. 

Union

Learn about the union operation defined for sets.

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

Union

Union