1. What an Equivalence Class Means
An equivalence class is the group of all elements that are connected to a given element under an equivalence relation. If a relation treats certain elements as “equivalent”, then each element sits inside a group with all the elements related to it.
So an equivalence class collects everything that behaves alike according to the relation.
2. Formal Definition
If R is an equivalence relation on a set A, then the equivalence class of an element a is:
2.1. Definition
[a] = \{ x \in A \mid (a,x) \in R \}
In words: all elements that relate to a.
3. Understanding the Idea
Each element has its own class, but many elements may share the same class if they relate to each other. These classes form a partition of the whole set — every element belongs to exactly one class.
3.1. Key Point
If two elements are equivalent, they fall into the same class.
4. Examples of Equivalence Classes
Here are some simple examples showing how classes look:
4.1. Example 1 (Same Colour Relation)
Suppose objects are grouped by colour.
If an object is red, its equivalence class is the set of all red objects. Same idea for blue, green, etc.
4.2. Example 2 (Equality)
Under equality, each element is equivalent only to itself.
[a] = \{a\}
4.3. Example 3 (Same Remainder Modulo n)
On integers, define a relation where:
a \sim b \; \text{if} \; a \equiv b \pmod{3}
The equivalence classes are:
- [0] → {…, -6, -3, 0, 3, 6, …}
- [1] → {…, -5, -2, 1, 4, 7, …}
- [2] → {…, -4, -1, 2, 5, 8, …}
5. How to Find an Equivalence Class
To find the equivalence class of an element:
- Use the definition of the relation.
- List all elements that are related to the chosen element.
- The collection you get is the class.
5.1. Illustration
If R groups numbers with the same remainder mod 4, then:
[2] = \{2,6,10,14,\ldots\}
6. Properties of Equivalence Classes
Equivalence classes have neat and predictable behaviour:
6.1. Key Properties
- Each element belongs to exactly one equivalence class.
- Two classes are either identical or completely disjoint.
- The collection of all classes divides the set into non-overlapping groups.
7. Why Equivalence Classes Matter
Equivalence classes help organise a set into natural groups based on a shared property. They are used in modular arithmetic, geometry (similar shapes), algebra (congruence), and many classification problems.