Types of Relations

Introduction

In mathematics, a relation is a set of ordered pairs. Each ordered pair consists of two elements, called the first element and the second element. The first element is often called the input or domain, while the second element is called the output or range.

Definition

A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product $A\times B$. In other words, a relation $R$ is a collection of ordered pairs $(a,b)$ where $a$ is in $A$ and $b$ is in $B$.

Example

Consider the set $A=\{1,2,3\}$ and the set $B=\{a,b,c\}$. The set of ordered pairs$\{(1,a),(2,b),(3,c)\}$ is a relation from $A$ to $B$.

Types of Relations

There are several different types of relations. Some of the most common types include:

• Reflexive: A relation $R$ is reflexive if for every element $a$ in $A$, the ordered pair $(a,a)$ is in $R$.
• Symmetric: A relation $R$ is symmetric if for every ordered pair $(a,b)$ in $R$, the ordered pair $(b,a)$ is also in $R$.
• Transitive: A relation $R$ is transitive if for every ordered pair $(a,b)$ in $R$ and every ordered pair $(b,c)$ in $R$, the ordered pair $(a,c)$ is also in $R$.

Mathematical Equations

There are several mathematical equations that can be used to describe relations. Some of the most common equations include:

• Domain: The domain of a relation $R$ is the set of all first elements in the ordered pairs of $R$. The domain of $R$ is denoted by $dom(R)$.
• Range: The range of a relation $R$ is the set of all second elements in the ordered pairs of $R$. The range of $R$ is denoted by $ran(R)$.
• Inverse: The inverse of a relation $R$ is the relation $R^{-1}$ that consists of the ordered pairs $(b,a)$ where $(a,b)$ is in $R$.
• Composition: The composition of two relations $R$ and $S$ is the relation $R\circ S$ that consists of the ordered pairs $(a,c)$ where there exists an element $b$ such that $(a,b)$ is in $R$ and$(b,c)$ is in $S$.

Graphs

Relations can also be represented graphically using Venn diagrams and directed graphs.

• Venn diagrams: A Venn diagram is a diagram that uses overlapping circles to represent sets. The ordered pairs in a relation can be represented by points inside the circles.
• Directed graphs: A directed graph is a graph in which the edges have arrows. The ordered pairs in a relation can be represented by edges in a directed graph, where the arrow points from the first element to the second element.

Applications of Relations

Relations have a wide variety of applications in mathematics, computer science, and other fields. For example, relations are used to represent relationships between people in a social network, to represent equations in a system of equations, and to represent data in a database.

Conclusion

Relations are a fundamental concept in mathematics that has a wide variety of applications. Understanding the concept of relations is essential for solving problems in a variety of fields.