## Discrete Mathematics: Types of Relations

In the field of mathematics, discrete mathematics deals with objects that can only take on distinct, separate values. One important concept in discrete mathematics is that of relations. Relations are used to describe the connections or associations between elements of sets. In this article, we will explore the different types of relations and provide examples to help illustrate their properties and usage.

## 1. Reflexive Relations

A reflexive relation is one in which every element is related to itself. In other words, for every element ‘a’ in a set, the relation R is true if (a, a) belongs to R. This means that every element is related to itself by the given relation.

For example, let’s consider the set A = {1, 2, 3}. The relation R = {(1, 1), (2, 2), (3, 3)} is reflexive, as each element in A is related to itself. However, the relation S = {(1, 1), (2, 2)} is not reflexive since element 3 is not related to itself.

## 2. Symmetric Relations

A symmetric relation is one in which if (a, b) belongs to the relation R, then (b, a) also belongs to R. In other words, if ‘a’ is related to ‘b’, then ‘b’ is also related to ‘a’.

For example, let’s consider the set B = {1, 2, 3}. The relation R = {(1, 2), (2, 1), (2, 3), (3, 2)} is symmetric because for every pair (a, b) in R, the pair (b, a) is also present in R. On the other hand, the relation S = {(1, 2), (2, 3)} is not symmetric since (2, 1) is not present in S.

## 3. Transitive Relations

A transitive relation is one in which if (a, b) and (b, c) belong to the relation R, then (a, c) also belongs to R. In other words, if ‘a’ is related to ‘b’ and ‘b’ is related to ‘c’, then ‘a’ is also related to ‘c’.

For example, let’s consider the set C = {1, 2, 3}. The relation R = {(1, 2), (2, 3), (1, 3)} is transitive because for every pair (a, b) and (b, c) in R, the pair (a, c) is also present in R. However, the relation S = {(1, 2), (2, 3), (1, 4)} is not transitive since (1, 4) is present in S but (2, 4) is not.

## 4. Equivalence Relations

An equivalence relation is a relation that is reflexive, symmetric, and transitive. It partitions a set into disjoint subsets called equivalence classes, where elements within each class are considered equivalent to each other.

For example, let’s consider the set D = {1, 2, 3, 4}. The relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)} is an equivalence relation. It is reflexive because every element is related to itself, symmetric because for every pair (a, b) in R, the pair (b, a) is also present in R, and transitive because for every pair (a, b) and (b, c) in R, the pair (a, c) is also present in R. The equivalence classes in this case would be {1, 2}, {3}, and {4}.

## 5. Partial Order Relations

A partial order relation is a relation that is reflexive, antisymmetric, and transitive. It provides a way to partially order elements in a set, indicating a partial ordering relationship between them.

For example, let’s consider the set E = {1, 2, 3, 4}. The relation R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 3), (1, 3)} is a partial order relation. It is reflexive because every element is related to itself, antisymmetric because for every pair (a, b) in R, if (a, b) and (b, a) both belong to R, then a = b, and transitive because for every pair (a, b) and (b, c) in R, the pair (a, c) is also present in R. This relation can be visualized as a directed acyclic graph.

## Conclusion

Relations are an important concept in discrete mathematics, allowing us to describe the connections and associations between elements of sets. Reflexive, symmetric, transitive, equivalence, and partial order relations are some of the key types of relations that are commonly used. Understanding these types of relations and their properties is essential for various applications in mathematics, computer science, and other fields.