Applications of Cartesian Products and Relations

Cartesian products and relations are not only theoretical concepts but also have a wide range of practical applications in mathematics, computer science, and everyday problems.

 

 

Coordinate systems

The classical two-dimensional coordinate system is built on the Cartesian product. If we take two sets of numbers, for example \( \large \mathbb{R} \times \mathbb{R}\), we obtain the plane, where each point is represented by an ordered pair \((x,y)\).

 

Example: The point \( \large (3,5)\) represents the coordinates \( \large x=3\) and \( \large y=5\) in the plane.

 

 

Graphs and networks

A graph can be seen as a relation on a set of vertices.

If \( \large V\) is the set of vertices, an edge can be described as an element of \( \large V \times V\). The entire edge set \( \large E\) is therefore a relation that shows which vertices are connected.

 

Example: If \( \large V = \{\text{A},\text{B},\text{C}\}\) and \( \large E = \{(A,B),(B,C)\}\), it means there is a connection from A to B and from B to C.

 

 

Databases

In databases, a table represents a relation. If we have a set of customers and a set of products, we can define a relation that shows which customers have ordered which products.

 

Example: If \( \large K = \{\text{Ida}, \text{Bo}\}\) and \( \large V = \{101, 102\}\), a relation can be \( \large R = \{(\text{Ida},101),(\text{Bo},102),(\text{Bo},101)\}\).

This means that Ida has ordered product 101, while Bo has ordered both 101 and 102.

 

 

Mathematical logic

Relations are used in logic to express connections between statements. For example, “implication” can be seen as a relation between two truth values.

 

Example: In the set of truth values \( \large \{\text{true}, \text{false}\}\), the implication \( \large p \Rightarrow q\) can be seen as a relation that is false only in the case \( \large (p=\text{true}, q=\text{false})\).

 

 

Combinatorics and probability

When calculating possible combinations, for example in probability theory, Cartesian products are used to describe all possible outcomes.

 

Example: If we roll a die \( \large T = \{1,2,3,4,5,6\}\) and spin a wheel with three sections \( \large H = \{A,B,C\}\), the sample space is:

 

$$ \large T \times H = \{(1,A),(1,B),(1,C),\ldots,(6,A),(6,B),(6,C)\} $$

 

Here there are \( \large 6 \times 3 = 18\) possible outcomes.

 

 

Significance

Cartesian products and relations serve as a bridge between the abstract and the concrete.

They provide mathematics with a formal framework to describe connections between objects, and they are also indispensable tools in modern technology, computer science, and statistics.