# Set Theory: Learn Subsets, Unions, Intersections and Differences

**Level 1**does not require any previous knowledge. Suitable for absolute beginners.

## Table of contents

Basically all complex mathematical structures are based on the concept of **sets**. These more complex structures are necessary to mathematically understand and formulate physical theories. Therefore, it is important to understand what sets are and what basic operations you can perform on sets.

## What is a set?

Each object of the set is called **element** of the set. The elements are specified between two *curly* brackets. To better indicate that it is a set, a *double bar* is made at the *label* of the set, as in the following examples.

In principle, you may take any label for a set, be it \(\mathbb{A}, \mathbb{B}, \mathbb{D} \) etc. Note, however, that the following labels are reserved for important sets:

\( \mathbb{N} \) stands for the

**set of natural numbers**.\( \mathbb{Z} \) stands for the

**set of integers**.\( \mathbb{Q} \) stands for the

**set of rational numbers**.\( \mathbb{R} \) stands for the

**set of real numbers**.\( \mathbb{C} \) stands for the

**set of complex numbers**.

For the numbers, for example, there is an operation "+" that adds two numbers together:\(1 + 3 = 4\). Or an operation "\(\cdot\)" that multiplies two numbers together: \( 2 \cdot 4 = 8\). As well as there are number operations, there are also **set operations**. Let's have a look at the most important set operations.

## Intersection of two sets

The **intersection** \( \cap \) of two sets \( \mathbb{A} \) and \( \mathbb{B} \) is the set of all elements contained in both \( \mathbb{A} \) and \( \mathbb{B} \).

## Union of two sets

The **union** \( \cup \) of two sets \( \mathbb{A} \) and \( \mathbb{B} \) are all elements of \( \mathbb{A} \) and all elements of \( \mathbb{B} \).

Consider again the two sets:

**Two example sets**

\mathbb{B} &~=~ \{ 3,~ 7,~ 9,~ 42 \} \end{align} $$

Now you simply combine all the elements of \( \mathbb{A} \) and \( \mathbb{B} \) into a new set, the union set:

**Union set is 2 3 5 6 7 9 42**

## Difference set

The difference set \( \mathbb{A} \backslash \mathbb{B} \) is the set of all elements belonging to \( \mathbb{A} \) only.

Consider again the two sets:

**Two example sets again**

\mathbb{B} &~=~ \{ 3,~ 7,~ 9,~ 42 \} \end{align} $$

Now you take only all elements which are in \( \mathbb{A} \) and these elements must not be in \( \mathbb{B} \):

**Difference set is 2 5 6**

Now you should know what sets are and how to do math with them, namely how to form union, intersection, and difference of two sets. Along with sets, **functions (mappings)** are one of the most fundamental concepts in mathematics and are hugely important to understand physics.