Equivalence Relations

In mathematics, relations are a way of describing the relationship between two or more objects. When dealing with sets, there are special types of relations that have different properties. One important type is known as an "equivalence relation." Equivalence relations are important to understand because they help us organize data and classify objects into categories that share common characteristics.

What is an equivalence relation?

An equivalence relation is a relationship between the elements of a set that groups them into equivalence classes. If you have a set of elements, the equivalence relation will help determine which elements are related and can thus be grouped together.

For a relation ~ on a set X to be an equivalence relation, it must satisfy three specific properties:

1. Reflexivity: For every element a in a set X, a is related to itself. In mathematical terms, this means a ~ a.
2. Symmetry: For every pair of elements a and b in a set X, if a belongs to b, then b belongs to a. This is written as: if a ~ b, then b ~ a.
3. Transitivity: For any elements a, b, and c in a set X, if a is related to b and b is related to c, then a is related to c. This can be expressed as: if a ~ b and b ~ c, then a ~ c.

Visualizing equivalence relations

To understand equivalence relations, let's take a simple example of a set with three elements: { A, B, C }.

Example 1: Visual representation

Imagine that we have the following equivalence relation:

a ~ a
B ~ B
C ~ C
A ~ B
B ~ A

This relationship can be represented by using circles to represent elements and lines to represent relationships:

In this diagram:

• The circles represent elements A, B and C.
• The solid lines indicate the symmetric relationship between A and B, and the dashed line indicates that each element is related to itself (reflexivity).

Note that C is related only to itself. Therefore, it forms its own equivalence class, while A and B form another equivalence class due to their mutual relationship.

Text-based examples

Example 2: Congruence of integers

A common equivalence relation in mathematics is congruence of integers. For this, consider the relation ≡, which is defined as follows: Two integers a and b are congruent modulo n if n divides their difference. This is written as a ≡ b (mod n).

Let us check whether this relation is reflexive, symmetric and transitive:

• Reflexive: For any integer a, the difference a - a = 0 is divisible by n. Thus, a ≡ a (mod n).
• Symmetric: If a ≡ b (mod n), then n divides a - b. This means that n also divides b - a, so b ≡ a (mod n).
• Transitive: If a ≡ b (mod n) and b ≡ c (mod n), then n divides both a - b and b - c. Adding these, n divides (a - b) + (b - c) = a - c, which means a ≡ c (mod n).

So congruence on the modulus of n is an equivalence relation.

Example 3: Equality as an equivalence relation

The most intuitive example of an equivalence relation is the equality relation. The equals sign = itself acts as an equivalence relation. Consider any set of objects. The relation that states that two elements are equal is an equivalence relation:

For any elements a, b, and c in a set X:
a = a (reflexive)
If a = b, then b = a (symmetric)
If a = b and b = c, then a = c (transitive)

Equality naturally partitions any set into equivalence classes, where each object forms its own class, since it is equal to itself only insofar as they are explicitly equal.

Conclusion

Understanding equivalence relations is important because they help simplify complex problems by classifying elements into groups or equivalence classes. Once a problem is limited to handling these classes, it often becomes easier to analyze and solve.

Equivalence relations exist not only in pure mathematical structures, but also have applications in a variety of fields, including computer science, engineering, logic, and other fields, providing a fundamental tool for understanding and organizing information.

Comments