Verify Symmetric Relation
To verify a symmetric relation do the following:
- Manually check for the existence of every bRa tuple for every aRb tuple in the relation.
- If any of the tuples does not exist then the relation is not symmetric else it is symmetric.
Follow the example given below for better understanding.
Example for Symmetric Relation
Consider set A = { 1, 2, 3, 4 } and a relation R = { (1, 2), (1, 3), (2, 1), (3, 4), (3, 1),(4.3) }
For the pair (1, 2) in R:
⇒ The reversed pair (2, 1) is present in the relation.
⇒ This pair satisfies the conditionFor the pair (1, 3) in R:
⇒ The reversed pair (3, 1) is present in the relation.
⇒ This pair satisfies the conditionFor the pair (2, 1) in R:
⇒ The reversed pair (1, 2) is present in the relation.
⇒ This pair satisfies the conditionFor the pair (3, 4) in R:
⇒ The reversed pair (4, 3) is present in the relation.
⇒ This pair satisfy the conditionFor the pair (3, 1) in R:
⇒ The reversed pair (1, 3) is present in the relation
⇒ This pair satisfies the conditionAs the set satisfy the condition, the relation is symmetric.
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?