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Set Theory: Learn Subsets, Unions, Intersections and Differences

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.

Examples for sets

Set with arbitrary numbers: $ \mathbb{A} = \{ 1, 5, \pi, 0 \} $
Set with names: $ \mathbb{B} = \{ \text{Anna}, \text{Alexander}, \text{Dima} \} $
Empty set: $ \mathbb{D} = \{ \} $
Set of natural numbers: $ \mathbb{N} = \{ 1,2,3,4 ...\} $
Set of geometric objects: $ \mathbb{E} = \{ \Box,\Delta,\nabla \} $

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} $.

Example of an intersection

For example, consider the following two sets:

$$ \begin{align} \mathbb{A} &~=~ \{ 2,~ 5,~ 6,~ 7,~ 9 \}\\\\
\mathbb{B} &~=~ \{ 3,~ 7,~ 9,~ 42 \} \end{align} $$

Now you have to see which objects, in this case natural numbers, are contained in both sets. These form the intersection of $ \mathbb{A} $ and $ \mathbb{B} $:

$$ \begin{align} \mathbb{A} ~\cap~ \mathbb{B} ~=~ \{ 7,~ 9 \} \end{align} $$

Schnittmenge zweier Mengen.

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} $.

Example of a union set

Consider again the two sets:

$$ \begin{align} \mathbb{A} &~=~ \{ 2,~ 5,~ 6,~ 7,~ 9 \}\\\\
\mathbb{B} &~=~ \{ 3,~ 7,~ 9,~ 42 \} \end{align} $$

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

$$ \begin{align} \mathbb{A} ~\cup~ \mathbb{B} ~=~ \{2,~ 3,~ 5,~ 6,~ 7,~ 9,~ 42\} \end{align} $$

Vereinigung zweier Mengen.

Difference set

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

Example of a difference set

Consider again the two sets:

$$ \begin{align} \mathbb{A} &~=~ \{ 2,~ 5,~ 6,~ 7,~ 9 \}\\\\
\mathbb{B} &~=~ \{ 3,~ 7,~ 9,~ 42 \} \end{align} $$

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

$$ \begin{align} \mathbb{A} ~\backslash~ \mathbb{B} ~=~ \{2,~ 5,~ 6\} \end{align} $$

Differenzmenge. Alle Elemente von $ \mathbb{A} $, jedoch ohne, dass sie in $ \mathbb{B} $ sind.

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.