Example of Equivalence Relation
A classic example of an equivalence relation is the relation of “equality” on the set of real numbers. Given any two real numbers “a” and “b”:
- Reflexivity: “a = a” is always true for any real number “a.”
- Symmetry: If “a = b,” then “b = a.”
- Transitivity: If “a = b” and “b = c,” then “a = c.”
Some other examples include:
- Congruence (in modular arithmetic)
- Congruence of Geometric Shapes
- Equivalence of Parallel Lines
Equivalence Relations
Equivalence Relation is a type of relation that satisfies three fundamental properties: reflexivity, symmetry, and transitivity. These properties ensure that it defines a partition on a set, where elements are grouped into equivalence classes based on their similarity or equality. Equivalence relations are essential in various mathematical and theoretical contexts, including algebra, set theory, and graph theory, as they provide a structured way to compare and classify elements within a set.
In this article, we will learn about the key properties of equivalence relations, how to identify any relation to be an equivalence relation, and their practical applications in fields such as abstract algebra, discrete mathematics, and data analysis. We’ll explore examples and exercises to deepen our understanding of Equivalence Relation.
Table of Content
- What is an Equivalence Relation?
- Equivalence Relation Definition
- Example of Equivalence Relation
- Properties of Equivalence Relation
- How to Verify an Equivalence Relation?