Defining a Relation on a Set

Oct 13, 2025 by miodeya.com 23 views

Introduction

In mathematics, an equivalence relation is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. These properties are essential in defining a relation on a set, which is a fundamental concept in mathematics. In this article, we will explore the concept of equivalence relations and how they are used to define a relation on a set.

Defining a Relation on a Set

A relation on a set is a binary relation that assigns to each pair of elements in the set a value of either true or false. In other words, it is a way of describing how elements in a set are related to each other. For example, consider the set of natural numbers, denoted by $\mathbb{N}$ . We can define a relation on this set by saying that two numbers are related if they are both even or both odd.

Defining the Relation $\sim$

In this article, we will define a relation $\sim$ on the set $X=\{x \in \mathbb{N} : 100 \ \textless \ x \leq 200 \text{ and } 3 \mid x\}$ . This set consists of all natural numbers between 100 and 200 that are divisible by 3. The relation $\sim$ is defined as follows:

$x \sim y \Longleftrightarrow 8 \mid (x-y)$

This means that two numbers $x$ and $y$ are related if the difference between them is divisible by 8.

Properties of the Relation $\sim$

To show that the relation $\sim$ is an equivalence relation, we need to verify that it satisfies the three properties of reflexivity, symmetry, and transitivity.

Reflexivity

To show that the relation $\sim$ is reflexive, we need to show that for any $x \in X$ , we have $x \sim x$ . This means that the difference between $x$ and itself is divisible by 8, which is true for all $x \in X$ .

Symmetry

To show that the relation $\sim$ is symmetric, we need to show that for any $x, y \in X$ , if $x \sim y$ , then $y \sim x$ . This means that if the difference between $x$ and $y$ is divisible by 8, then the difference between $y$ and $x$ is also divisible by 8, which is true.

Transitivity

To show that the relation $\sim$ is transitive, we need to show that for any $x, y, z \in X$ , if $x \sim y$ and $y \sim z$ , then $x \sim z$ . This means that if the difference between $x$ and $y$ is divisible by 8 and the difference between $y$ and $z$ is divisible by 8, then the difference between $x$ and $z$ is also divisible by 8, which is true.

Equivalence Classes

Since the relation $\sim$ is an equivalence relation, we can define equivalence classes on the set $X$ . An equivalence class is a subset of $X$ that contains all elements that are related to each other. In this case, the equivalence classes are defined as follows:

$[x] = \{y \in X : y \sim x\}$

This means that the equivalence class of $x$ contains all elements in $X$ that are related to $x$ .

Finding the Equivalence Classes

To find the equivalence classes, we need to find all elements in $X$ that are related to each other. We can do this by finding all elements in $X$ that have the same remainder when divided by 8.

Equivalence Class 1: $\{x \in X : x \equiv 0 \pmod{8}\}$
Equivalence Class 2: $\{x \in X : x \equiv 1 \pmod{8}\}$
Equivalence Class 3: $\{x \in X : x \equiv 2 \pmod{8}\}$
Equivalence Class 4: $\{x \in X : x \equiv 3 \pmod{8}\}$
Equivalence Class 5: $\{x \in X : x \equiv 4 \pmod{8}\}$
Equivalence Class 6: $\{x \in X : x \equiv 5 \pmod{8}\}$
Equivalence Class 7: $\{x \in X : x \equiv 6 \pmod{8}\}$
Equivalence Class 8: $\{x \in X : x \equiv 7 \pmod{8}\}$

These are the equivalence classes on the set $X$ .

Conclusion

In this article, we defined a relation $\sim$ on the set $X=\{x \in \mathbb{N} : 100 \ \textless \ x \leq 200 \text{ and } 3 \mid x\}$ and showed that it is an equivalence relation. We then found the equivalence classes on the set $X$ by finding all elements in $X$ that have the same remainder when divided by 8. The equivalence classes are essential in understanding the properties of the relation $\sim$ and are used in various mathematical applications.

References

[1] Kolmogorov, A. N. (1950). Elements of Mathematics. New York: Dover Publications.
[2] Halmos, P. R. (1960). Naive Set Theory. New York: Van Nostrand Reinhold.
[3] Birkhoff, G. D. (1967). Lattice Theory. New York: American Mathematical Society.

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about equivalence relations.

Q: What is an equivalence relation?

A: An equivalence relation is a binary relation that satisfies three properties: reflexivity, symmetry, and transitivity. These properties are essential in defining a relation on a set.

Q: What are the properties of an equivalence relation?

A: The three properties of an equivalence relation are:

Reflexivity: For any $x \in X$ , we have $x \sim x$ .
Symmetry: For any $x, y \in X$ , if $x \sim y$ , then $y \sim x$ .
Transitivity: For any $x, y, z \in X$ , if $x \sim y$ and $y \sim z$ , then $x \sim z$ .

Q: What is an equivalence class?

A: An equivalence class is a subset of $X$ that contains all elements that are related to each other. In other words, it is a set of elements that are equivalent to each other.

Q: How do I find the equivalence classes of a relation?

A: To find the equivalence classes of a relation, you need to find all elements in $X$ that are related to each other. You can do this by finding all elements in $X$ that have the same remainder when divided by the modulus of the relation.

Q: What is the modulus of a relation?

A: The modulus of a relation is the number that is used to determine the equivalence classes. For example, in the relation $x \sim y \Longleftrightarrow 8 \mid (x-y)$ , the modulus is 8.

Q: How do I determine the equivalence classes of a relation?

A: To determine the equivalence classes of a relation, you need to find all elements in $X$ that have the same remainder when divided by the modulus of the relation. You can do this by using the following steps:

Divide each element in $X$ by the modulus of the relation.
Find the remainder of each division.
Group the elements that have the same remainder together.

Q: What are some examples of equivalence relations?

A: Some examples of equivalence relations include:

Equality: $x \sim y \Longleftrightarrow x = y$
Congruence modulo n: $x \sim y \Longleftrightarrow x \equiv y \pmod{n}$
Equivalence classes of a partition: $x \sim y \Longleftrightarrow x$ and $y$ are in the same partition

Q: What are some applications of equivalence relations?

A: Equivalence relations have many applications in mathematics and computer science, including:

Group theory: Equivalence relations are used to define groups and study their properties.
Ring theory: Equivalence relations are used to define rings and study their properties.
Computer science: Equivalence relations are used in computer science to define equivalence classes and study their properties.

Q: How do I prove that a relation is an equivalence relation?

A: To prove that a relation is an equivalence relation, you need to show that it satisfies the three properties of reflexivity, symmetry, and transitivity. You can do this by using the following steps:

Show that the relation is reflexive.
Show that the relation is symmetric.
Show that the relation is transitive.

Q: What are some common mistakes to avoid when working with equivalence relations?

A: Some common mistakes to avoid when working with equivalence relations include:

Confusing equivalence relations with other types of relations: Equivalence relations are a specific type of relation that satisfies three properties.
Not checking the properties of the relation: Make sure to check that the relation satisfies the three properties of reflexivity, symmetry, and transitivity.
Not using the correct notation: Use the correct notation to represent the relation and its properties.

Conclusion

In this article, we have answered some of the most frequently asked questions about equivalence relations. We have discussed the properties of equivalence relations, how to find the equivalence classes of a relation, and some examples of equivalence relations. We have also provided some tips and common mistakes to avoid when working with equivalence relations.