Equivalence Relation Definition
An equivalence relation on a set is a binary relation that satisfies three fundamental properties:
- Reflexivity: ∀ a ∈ S: a ~ a
- Symmetry: ∀ a, b ∈ S: a ~ b ⇒ b ~ a
- Transitivity: ∀ a, b, c ∈ S: (a ~ b) ∧ (b ~ c) ⇒ a ~ c
What is Reflexive Relation?
A relation R on a set A is called reflexive relation if
(a, a) ∈ R ∀ a ∈ A, i.e. aRa for all a ∈ A, where R is a subset of (A ✕ A), i.e. the cartesian product of set A with itself.
This means if element “a” is present in set A, then a relation “a” to “a” (aRa) should be present in relation R. If any such aRa is not present in R then R is not a reflexive relation.
What is Symmetric Relation?
A relation R on a set A is called symmetric relation if and only if
∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R and vice versa i.e., where R is a subset of (A x A), i.e. the cartesian product of set A with itself.
This means if an ordered pair of elements “a” to “b” (aRb) is present in relation R, then an ordered pair of elements “b” to “a” (bRa) should also be present in relation R. If any such bRa is not present for any aRb in R then R is not a symmetric relation.
What is Transitive Relation?
A relation R on a set A is called transitive relation if and only if
∀ a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, where R is a subset of (A x A), i.e. the cartesian product of set A with itself.
This means if an ordered pair of elements “a” to “b” (aRb) and “b” to “c” (bRc) is present in relation R, then an ordered pair of elements “a” to “c” (aRC) should also be present in the relation R. If any such aRc is not present for any aRb & bRc in R then R is not a transitive relation.
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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?