Verify Reflexive Relation
The process of identifying/verifying if any given relation is reflexive:
- Check for the existence of every aRa tuple in the relation for all a present in the set.
- If every tuple exists, only then the relation is reflexive. Otherwise, not reflexive.
Follow the example given below for better understanding.
Example for Reflexive Relation
Consider set A = {a, b} and a relation R = {{a, a}, {b, b}}.
For the element a in A:
⇒ The pair {a, a} is present in R.
⇒ Hence aRa is satisfied.For the element b in A:
⇒ The pair {b, b} is present in R.
⇒ Hence bRb is satisfied.As the condition for ‘a’, ‘b’ is satisfied, the relation is reflexive.
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?