Equivalence relation

Introduction

In mathematics, an equivalence relation defined on a set is a reflexive, symmetric, and transitive binary relation. For example, a subset of the cartesian product A B is a binary relation over the sets A and B consisting of components of the form (a, b) such that an a ∈ A and b ∈ B. The ‘equal to (=)’ relation, reflexive, symmetric, and transitive, is a highly frequent and easy-to-understand example of an equivalence relation. As the name implies, two components of a set are said to be equivalent if and only if they belong to the same equivalence class. We will learn about equivalence relations definition, classes, and partitions in this article, complete with proofs and examples.

What is Equivalence Relation?

A binary equivalence relation is reflexive, symmetric, and transitive and is defined on a set X. The relation cannot be an equivalence relation if any of the three conditions, namely transitive, reflexive, and symmetric, are not met. The equivalence relation separates the set into equivalence classes that are distinct. If two elements of the set belong to the same equivalence class, they are said to be equivalent. The sign ‘~’ is commonly used to represent an equivalence relation.

Examples of Equivalence Relation

After discussing what equivalence relation is in detail, let’s look at the examples.

• ‘Is equal to (=)’ is an equivalence relation on any set of integers A, as we have a = a, a = b b = a, and a = b, b = c a = c for all elements a, b, c A. It means that (=) is transitive, symmetric, and reflexive.
• On the set of triangles, ‘is comparable to (~)’ is defined: It’s transitive, symmetric, and reflexive.
• The expression ‘has the same birthdate’ is reflexive, symmetrical, and transitive when applied to a group of persons.
• The equivalence relation ‘Is congruent to’ defined on the triangles is reflexive, symmetric, and transitive.
• On the set of integers, ‘congruence modulo n (≡)’ is defined: Transitive, symmetric, and reflexive.
• The equivalence relation ‘Has the same absolute value’ defined on the set of real numbers is reflexive, symmetric, and transitive.

Proof of Equivalence Relation

Let’s look at an example to verify that a connection is an equivalence relation. If a is equal to b, describe a relation R on the set of natural numbers N as (a, b) ∈ R if a is equal to b. We’ll now demonstrate that R is reflexive, symmetric, and transitive.

• Reflexive Property – 

Because each natural number is equal to itself, a = a for all a ∈ N ⇒ (a, a) ∈ R for all a ∈ N. As a result, R is reflexive.

• Symmetric Property –

Let (For a, b ∈ N, let (a, b) ∈ R ⇒ a = b ⇒ b = a ⇒ (b, a) ∈ R (a, R is symmetric because a and b are arbitrary.

• Transitive Property – 

a, b, c ∈ N, let (a, b) ∈ R and (b, c) ∈ R ⇒ a = b and b = c ⇒ a = c (a, c) R a = b and b = c a = c (a, c) R. R is transitive because a, b, and c are arbitrary.

R is an equivalence relation since it is thrice of the above when defined on the set of natural numbers N.

Connection of Equivalence Relation to other relations

• A reciprocal, a system that can be classified, and a linear relationship are all examples of incomplete orders.
• Equality is both a complete order and a relationship of equivalence.
• The only inductive, symmetric, and antisymmetric connection on a set is equality.
• In algebraic expressions, equal variables can be substituted for one another, a feature unavailable for equivalence-related variables.
• Individuals within the equivalence classes of an equivalence relation can replace each other, but not within a class.
• An asymmetric, irreflexive, and incomplete bidirectional order is asymmetric, irreflexive, and bidirectional.
• The ternary comparability relation is the ternary equivalent of the standard equivalence relation.
• A reliance relationship, also known as a tolerance relationship, is mutual and symmetrical.
• A sequence can be inductive as well as bidirectional.

Important Notes on Equivalence Relation

• An equivalence relation definition is a binary relation that is reflexive, symmetric, and transitive and is defined on a set X.
• The equivalence relation separates the set into distinct equivalence classes.
• Equivalent items are those that belong to the same equivalence class.

Conclusion

An equivalence relation on it induces a partition on a set. Any partition, on the other hand, produces an equivalence relation. Equivalence relations are essential because each equivalence class can often be ‘converted’ into another set (quotient space) by treating it as a single unit. The key focus is conceptual understanding, and anyone who has mastered this skill will be successful. To better understand the issue, practice sums after going over the concept. If you haven’t mastered equivalence relations enough, they can be challenging.