Equivalence Relation: FAQs

1. Define Equivalence Relation.

An equivalence relation is a binary relation on a set that satisfies three properties: reflexivity, symmetry, and transitivity. It is a way to partition a set into distinct subsets or “equivalence classes.“

2. What is Symmetry in an Equivalence Relation?

Symmetry means that if “a” is related to “b,” then “b” is also related to “a.” In formal terms, if (a, b) is in R, then (b, a) is also in R.

3. What is Transitivity in an Equivalence Relation?

Transitivity means that if “a” is related to “b” and “b” is related to “c,” then “a” is related to “c.” In formal terms, if (a, b) is in R and (b, c) is in R, then (a, c) is also in R.

4. Can you Give an Example of an Equivalence Relation?

The relation “is congruent modulo n” on the set of integers is an equivalence relation. Two integers are related if their difference is divisible by the integer “n.”

5. What is Reflexivity in an Equivalence Relation?

Reflexivity means that for any element “a” in the set, it is related to itself. In formal terms, if R is an equivalence relation on a set A, then for all “a” in A, (a, a) is in R.

6. What is an Equivalence Relation in Real Life?

Here are some example of equivalence relation in real life:

• In mathematics, congruence relations are used to classify numbers based on their remainders when divided by a particular number (e.g., congruence modulo 5).
• Consider the relation “is a sibling of.” If person A is a sibling of person B, this relation is reflexive (a person is a sibling of themselves), symmetric (if A is a sibling of B, then B is a sibling of A), and transitive (if A is a sibling of B and B is a sibling of C, then A is a sibling of C).

7. What is the Smallest Equivalence Relation?

The smallest equivalence relation on a set is known as the trivial equivalence relation or the equality relation. This relation is formed by pairing each element in the set with itself, making sure that every element is related to itself but not necessarily to any other element in the set.

9. Is an Empty Relation an Equivalence Relation?

Yes, empty relation is an Equivalence Relation.

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.