Verify Transitive Relation
To verify transitive relation:
- Firstly, find the tuples of form aRb & bRc in the relation.
- For every such pair check if aRc is also present in R.
- If any of the tuples does not exist then the relation is not transitive else it is transitive.
Follow the below illustration for a better understanding
Example for Transitive Relation
Consider set R = {(1, 2), (1, 3), (2, 3), (3, 4), (1,4)}
For the pairs (1, 2) and (2, 3):
⇒ The relation (1, 3) exists
⇒ This satisfies the condition.For the pairs (1, 3) and (3, 4):
⇒ The relation (1, 4) exists.
⇒ This satisfies the condition.So the relation is transitive.
Hence the transitive property is proved.
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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?