Relations in mathematics - definition, formulas, graphs, examples (2023)

Relationships in mathematicsthey help create a connection between two objects or things. A relationship describes a relationship between two objects, usually represented as an ordered pair (input, output) or (x, y). Here x and y are elements of sets.

Relations have many uses, especially in the field of computer science, for building relational database management systems (RDBMS). This article will discuss relationships, their types, how to combine elements from two sets using relationships and related examples.

What is a relation in mathematics?

Relationships in mathematicsare used to describe the relationship between the elements of two sets. They help map the elements of one set (called the domain) to the elements of another set (called the range) in such a way that the resulting ordered pairs have the form (input, output). In addition, special types of relationships that can be used to create correspondences between two quantities are called functions. You can also say that a function is a subset of a relation.

Definition of relationship in mathematics

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

Relations in a mathematical example

Suppose there are two sets X = {4, 36, 49, 50} and Y = {1, -2, -6, -7, 7, 6, 2}. A relation that says "(x,y) is in the relation R if x is asquares y' can be represented byI ordered a pairstrongly R = {(4, -2), (4, 2), (36, -6), (36, 6), (49, -7), (49, 7)}.

Representation of relationships

Relationships can be shown using a variety of techniques. There are five main representations of relationships. They are listed as follows:

Set up a form to create:It is a mathematical notation that clearly states the rule that connects two sets X and Y. If there are two sets X = {5, 6, 7} and Y = {25, 36, 49}. As a rule, the elements of X are the positive square root of the elements of Y. Inset builder formthis relation can be written as R {(a, b): a is the positive square root of b, a ∈ X, b ∈ Y}.

Form of composition:Wlist form, write all possible ordered pairs of two sets resulting from the given relation. Using the same example as mentioned above, the relation that the elements of X are the positive square roots of the elements of B is represented as R = {(5, 25), (6, 36), (7, 49)}.

Arrow diagram:Such a diagram is used to visually represent the relationship between the elements of two sets of data. The arrow diagram of the above example is given as

Table format:When the inputs and outputs of a relation are expressed in matrix form, it is called a matrix representation of the relation. In this case, a table with two columns is drawn. The first is the input and the second is the output. Using the relation that the elements of X = {5, 6, 7} are positivesquare rootsbetween elements Y = {25, 36, 49}, the matrix is ​​as follows:

We will discuss the fifth representation using the graphical method in the following sections.

Types of relationships

Two sets can have different types of relationships, so different types of relationships are needed to classify these relationships. Mrtypes of relationshipsare listed below:

Empty reference

An empty relation is one in which no element of a set is mapped either to an element of another set or to itself. We denote this relationship as R = ∅. For example, P = {3, 7, 9} and the relation in P, R = {(x, y) where x + y = 76}. This will be an empty relation because the two elements of P will not be added to 76.

Universal attitude

If all elements of a set map to all elements of another set or to each other, then such a relationship is called a universal relationship. This is written as R = X × Y, where each element of X is associated with each element of Y. Example: P = {3, 7, 9}, Q = {12, 18, 20} and R = {(x , y ) ) where x < y}.

Identity relationship

If all elements of a set are related to each other, then it becomes an identity relation. This 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 relationship

If the elements of one set are inverse pairs of another set, then this relation is called anreverse relationship. In other words, an inverse relationship is an inverse relationship. We denote the inverse of R as R^-1. i.e. R^-1= {(y, x): (x, y) ∈ R}

Reflex attitude

In a set, if all the elements are in harmony with each other, it is areflective relationship. So, if x ∈ X, we define a reflexive relation as (x, x) ∈ R. For example, P = {7, 1} then R = {(7, 7), (1, 1)} is a reflexive relation.

A symmetrical relationship

The relationship is said to be asymmetrical relationshipif the set X contains ordered pairs (x, y), as well as the inverse of these pairs (y, x). In other words, if (x, y) ∈ R then (y, x) ∈ R for the relation to be symmetric. Let P = {3, 4}, then the symmetric relation can be R = {(3, 4), (4, 3)}.

Transitional relationship

Let (x, y) ∈ R and (y, z) ∈ R, then R is onetransitional relationshipif and only if (x, z) ∈ R. For example, P = {p, q, r}, then the transitive relation can be R = {(p, q), (q, r), (p, ) }

Equivalence relation

Someequivalence relationit is a type of relationship that is symmetrical, transitive and reflexive.

One to one relationship

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

One-to-many relationship

In a one-to-many relationship, one element in 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}, the one-to-many relationship is written as R = {(2, a), (2, b) , ( 2, c)}

A many-to-one relationship

If more than one element of one set maps to a single, distinct element of another set, the relationship is called a many-to-one relationship. For example, P = {1, 2, 3} and Q = {a, b, c}, then R = {(1, a), (2, a), (3, a)} is a multiple -Connection.

A more-to-more relationship

In a many-to-many relationship, one or more members of a set will be mapped to the same or different members 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 many relation.

Biblical relationships

Relationships can also be shown graphically usingCartesian coordinate system. An element of a relation can be expressed as an ordered pair (x, y) or given as an equation (orinequality). The ordered pair represents the position of points in acoordinate plane. Suppose the relation is given as y = x - 2 in the set of allreal numbers, then the steps to create a graph are as follows:

• Replace x with numeric values. x = -1, 0, 2 (some random numbers)
• Find the corresponding y-values ​​using the given relation. y = -3, -2, 0.
• Record these test points as ordered pairs. {(-1, -3), (0, -2), (2, 0)}.
• Plot these points in aCartesian plane. If the relationship is already given in the form of ordered pairs, draw them on the plane.
• Connect these points to get a graph of the given relationship. For this example, the graph would be aa straight line.

Important notes about relationships in mathematics:

• A relation is used to create a connection between elements of the same or different sets.
• An ordered pair of characters (input, output) is used to denote a relation element.
• The Cartesian product of two sets can be described by a relation.
• Relationships can be displayed using Set form, Chart form, Arrow diagram, Chart form, and Table form.
• There are many different types of relationships such as empty relationship, global relationship, many-to-one relationship, etc.

☛Relevant articles:

• Relationships and features
• Coordinate geometry
• x and y axes

Frequently asked questions about relationships in mathematics

What is the definition of relation in mathematics?

ONErelationship in mathematicsgives the relationship between two sets (say A and B). Each element of the relation is an ordered pair (x, y) where x is in A and y is in B. In other words, the relation is a subset of the Cartesian product of A and B.

What are functions and relations in mathematics?

The relationship helps create a bond between the elements of the twolandscapeso that input and output form an ordered pair (input, output). The function is aa subsetthe relationship that determines the output given the input. All functions are relations, but not all relations are functions. For example, R = {(1, 2), (1, 3), (2, 3)} is a relation but not a function because 1 is assigned twice (both 2 and 3).

What are the different types of relationships in mathematics?

In mathematics, there are nine different types of relations. They are listed as follows:

• Empty reference
• Universal attitude
• Identity relationship
• Inverse relationship
• Reflex attitude
• A symmetrical relationship
• Transitional relationship
• Equivalence relation

There are four other types of mapping relationships.

• One to one relationship
• One-to-many relationship
• A many-to-one relationship
• A more-to-more relationship

What is a relationship equation?

When the relation is expressed in the form anequationthis is known as the ratio equation. y = x²is an example of a relationship equation. The graph of this relationship will be aparabola.

How is the relationship represented in mathematics?

There are 5 common ways to represent relationships. These are the Create Sets form, the Composition form, the Table form, the Arrow chart, and the Chart form.

How to save a relationship in a chart?

If there is an ordered pair (x, y) in which x is related to y, then this relationship can be represented graphically. To display relationships on a graph, simply select the ordered points on the graph. The x-coordinate represents the distance of the point from the y-axis, and the y-coordinate representsdistancefrom the x-axis.

What are symmetrical relations in mathematics?

A symmetric relation in mathematics can be defined as a relation that contains an ordered pair (x, y) and the inverse of that pair (y, x). So, for a symmetric relation if (x, y) ∈ R then (y, x) ∈ R.

Are all functions relations in mathematics?

allOperationsthere are relationships. A function is a relationship in which each input will have only one output. So a one-to-one relationship and a many-to-one relationship will form a function.