Relations in Math

Relations in math help to give a way of establishing a connection between any two objects or things. A relation describes the relationship between two objects that are usually represented as an ordered pair (input, output) or (x, y). Here, x and y are elements from sets.

Relations have several applications especially in the field of computer science to create relational database management systems (RDBMS). This article will elaborate on relations, their types, how to link elements from two sets using relations and associated examples.

Relations in math are used to describe a connection between the elements of two sets. They help to map the elements of one set (known as the domain) to elements of another set (called the range) such that the resulting ordered pairs are of the form (input, output). Furthermore, special types of relations that can be used to establish a correspondence between two quantities are known as functions. It can also be said that a function is a subset of a relation.

Relations Definition in Math

Relations in maths is a subset of the cartesian product of two sets. Suppose there are two sets given by X and Y. Let x ∈ X (x is an element of set X) and y ∈ Y. Then the cartesian product of X and Y, represented as X × Y, is given by the collection of all possible ordered pairs (x, y). In other words, a relation says that every input will produce one or more outputs.

Relations in Math Example

Suppose there are two sets X = {4, 36, 49, 50} and Y = {1, -2, -6, -7, 7, 6, 2}. A relation that states that "(x, y) is in the relation R if x is a square of y" can be represented using ordered pairs as R = {(4, -2), (4, 2), (36, -6), (36, 6), (49, -7), (49, 7)}.

Relations can be represented using different techniques. There are five main representations of relations. These are given as follows:

Set Builder Form: It is a mathematical notation where the rule that associates the two sets X and Y is clearly specified. If there are two sets X = {5, 6, 7} and Y = {25, 36, 49}. The rule is that the elements of X are the positive square root of the elements of Y. In set-builder form this relation can be written as R {(a, b): a is the positive square root of b, a ∈ X, b ∈ Y}.

Roster Form: In roster form, all the possible ordered pairs of the two sets that follow the given relation are written. Using the same example as mentioned above, the relation, the elements of set X are the positive square roots of the elements of set B is represented as R = {(5, 25), (6, 36), (7, 49)}.

Arrow Diagram: Such a diagram is used to visually represent the relation between the elements of the two given sets. The arrow diagram of the above-mentioned example is given as

Tabular Form: When the input and the output of a relation are expressed in the form of a table it is known as the tabular representation of a relation. In this, a table is drawn with two columns. The first one denotes the input and the second expresses the output. Using the relation that the elements of X = {5, 6, 7} are the positive square roots of the elements of Y = {25, 36, 49}, the table is given as follows:

The fifth representation that uses the graphing method will be covered in upcoming sections.

Two sets can have different types of connections thus, different kinds of relations are needed so as to classify these connections. The main types of relations are listed below:

Empty Relation

An empty relation is one where any element of a set is neither mapped to an element of another set nor to itself. This relation is denoted as R = ∅. For example, P = {3, 7, 9} and the relation on P, R = {(x, y) where x + y = 76}. This will be an empty relation as no two elements of P are added up to 76.

Universal Relation

If all the elements belonging to one set are mapped to all the elements of another set or to itself then such a relation is known as a universal relation. It is written as R = X × Y where each element of X is related to every element of Y. Example, P = {3, 7, 9}, Q = {12, 18, 20} and R = {(x, y) where x < y}.

Identity Relation

If all elements in a set are related to itself then it becomes an identity relation. It is written as I = {(x, x) : for all x ∈ X}. For example P = {3, 7, 9} then I = {(3, 3), (7, 7), (9, 9)}

Inverse Relation

If the elements of one set are the inverse pairs of another set then such a relation is known as an inverse relation. In other words, the inverse of a relation is an inverse relation. The inverse of a relation R is denoted as R^-1. i.e., R^-1 = {(y, x) : (x, y) ∈ R}

Reflexive Relation

In a set, if all the elements are mapped to themselves then it is a reflexive relation. Thus, if x ∈ X then a reflexive relation is defined as (x, x) ∈ R. For example, P = {7, 1} then R = {(7, 7), (1, 1)} is a reflexive relation.

Symmetric Relation

A relation is said to be a symmetric relation if one set, X, contains ordered pairs, (x, y) as well as the reverse of these pairs, (y, x). In other words, if (x, y) ∈ R then (y, x) ∈ R for the relation to be symmetric. Suppose P = {3, 4}, then a symmetric relation can be R = {(3, 4), (4, 3)}.

Transitive Relation

Suppose (x, y) ∈ R and (y, z) ∈ R then R is a transitive relation if and only if (x, z) ∈ R. For example, P = {p, q, r}, then a transitive relationcan be R = {(p, q), (q, r), (p, r)}

Equivalence Relation

An equivalence relation is a type of relation that is symmetric, transitive, and reflexive.

One to One Relation

In a one-to-one relation each element of one set will be mapped to a distinct element in another set. For example, suppose there are two sets P = {1, 2, 3} and Q = {a, b, c}. Then a one to one relation can be R = {(1, a), (2, b), (3, c)}

One to Many Relation

In a one-to-many relation, a single element of one set will be mapped to more than one element in another set. For example, given two sets P = {1, 2, 3} and Q = {a, b, c}, a one to many relation is written as R = {(2, a), (2, b), (2, c)}

Many to One Relation

If more than one element of one set are mapped to a single distinct element of another set then such a relation is referred to as many to one relation. For example, P = {1, 2, 3} and Q = {a, b, c}, then R = {(1, a), (2, a), (3, a)} is a many to one relation.

Many to Many Relation

In a many-to-many relation, one or more elements of one set will be mapped to the same or a different element of another set. If P = {1, 2, 3} and Q = {a, b, c}, then R = {(2, a), (3, a), (2, c)} is an example of a many to many relation.

Relations also be represented graphically using the cartesian coordinate system. An element of a relationship can either be expressed in the form of an ordered pair, (x, y) or it can be given in the form of an equation (or inequality). The ordered pair represents the position of points in a coordinate plane. Suppose a relation is given as y = x - 2 on the set of all real numbers, then the steps to plot the graph are as follows:

• Substitute x with numerical values; x = -1, 0, 2 (some random numbers)
• Find the corresponding values of y using the given relation; y = -3, -2, 0.
• Write these test points as ordered pairs; {(-1, -3), (0, -2), (2, 0)}.
• Plot these points on a cartesian plane. If the relation is already given in the form of ordered pairs then plot them on the plane.
• Join these points to get the graph of the given relation. For the given example the graph will be a straight line.

Important Notes on Relations in Math:

• A relation is used to establish a connection between the elements of the same or different sets.
• An ordered pair, of the form (input, output), is used to denote an element of relation.
• The cartesian product of two sets can be described using relations.
• Relations can be represented using the set-builder form, roster form, arrow diagram, graphical form, and tabular form.
• There are many different types of relations such as empty relation, universal relation, many to one relation, etc.

☛Related Articles:

• Relations and Functions
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• x and y axes

What is Relation Definition in Math?

A relation in math gives the relationship between two sets (say A and B). Every element of a relationship is in the form of ordered pair (x, y) where x is in A and y is in B. In other words, a relation is a subset of the cartesian product of A and B.

What are Functions and Relations in Math?

A relation helps to establish a connection between the elements of two sets such that the input and output form an ordered pair (input, output). A function is a subset of a relation that determines the output given a specific input. All functions are relations but all relations are not functions. For example, R = { (1, 2), (1, 3), (2, 3)} is a relation but not a function as 1 is mapped twice (to both 2 and 3).

What are the Different Types of Relations in Math?

There are nine different types of relations in math. These are given as follows:

• Empty Relation
• Universal Relation
• Identity Relation
• Inverse Relation
• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation

There are four other types of relations based on mapping.

• One to One Relation
• One to Many Relation
• Many to One Relation
• Many to Many Relation

What is a Relation Equation?

When a relation is expressed in the form of an equation it is known as a relation equation. y = x² is an example of a relation equation. The graph of this relation will be a parabola.

How is a Relationship in Math Represented?

There are 5 commonly used ways to represent a relation. These are the set-builder form, the roster form, the tabular form, the arrow diagram, and by using a graph.

How Do You Write a Relation in a Graph?

If there exists an ordered pair (x, y) such that x is related to y then such a relation can be plotted on a graph. To represent a relation on a graph, simply mark the ordered points on it. The x coordinate represents the distance of the point from the y-axis and the y coordinate denotes the distance from the x-axis.

What are Symmetric Relations in Math?

A symmetric relation in math can be defined as a relation that contains the ordered pair (x, y) as well as the reverse of this pair (y, x). Thus, for a symmetric relation if (x, y) ∈ R then (y, x) ∈ R.

Are all Functions Relations in Math?

All functions are relations. A function is a relation where each input will have only one output. Thus, a one to one relation and a many to one relation will form a function.