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

A\cap (B\cup C) = (A\cap B)\cup (A\cap C)

Take $x\in A\cap (B\cup C)$  as an arbitrary element. By the definition of the union and intersection operations, we can conclude that $x\in A$ $\land$ $x\in (B\cup C)$. We can further conclude that $x\in B$ $\lor$ $x\in C$.  Therefore, either  ($x\in A$ $\land$ $x\in B$) or ($x\in A$ $\land$ $x\in C$). Again, using definitions of the union and the intersection, we can write this information as
 $x\in (A\cap B)$ $\lor$ $x\in (A\cap C)$. As a result, $x\in (A\cap B)\cup (A\cap C)$. Until this point, we have shown that $A\cap (B\cup C) \sube (A\cap B)\cup (A\cap C)$. 

Learn about the second distributive law for sets and its proof.

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

The Second Distributive Law

The distribution of an intersection over a union