Relations and their Representation: Relation is one of the crucial topics in the set theory. Relation describes the way of connection between any two objects or things. A relation tells the cartesian product of two sets. 

Relations are generally represented in three methods: general statement form, roaster or tabular form, and set-builder form. Relations between two sets are shown even by using arrow diagrams also. In this article, we will discuss the methods of representation of the relations in detail.

Relations

A relation tells the cartesian product of two sets. Relation is one of the crucial topics in the set theory. Relation describes the way of connection between any two objects or things. The cartesian product of any two sets \(A\) and \(B,\) such that \(a ∈A\) and \(b∈B,\) is given by the collection of all order pairs \((a, b).\)

Relation describes that every element of one set is mapped to one or more elements of another set. We can say that relation has one or more outputs for every input value. The function is the special relation, in which elements of one set is mapped to only one element of another set.

In mathematics, a set of ordered pairs is called the relation. For any two non-empty sets \(A\) and \(B,\) the relation \(R\) is the subset of the cartesian product of \(A×B.\) The below figure shows the relation \((R: A⟶B)\) between set \(A\) and set \(B\) by an arrow diagram.

Terminologies Used in Relations

Let us discuss the terminologies used in the relation in detail below:

1. Domain:
   The set of all first elements of the given ordered pairs in a relation \(R\) from a set \(A\) to a set \(B\) is known as the domain of the relation \(R.\)
2. Range:
   The set of all second elements of the given ordered pairs in a relation \(R\) from a set \(A\) to a set \(B\) is known as the range of the relation \(R.\)
3. Co-domain:
   The whole set \(B\) (second set) in a relation \(R\) from a set \(A\) to a set \(B\) is known as the co-domain of the relation \(R.\)

Total Number of Relations

A relation tells the cartesian product of two sets. The cartesian product of any two sets \(A\) and \(B,\) such that \(a ∈A\) and \(b∈B,\) is given by the collection of all order pairs \((a, b).\) For two non-empty sets \(A\) and \(B,\) the number of elements in set \(A\) is \(p,\) and the number of elements in set \(B\) is \(q.\)

1. The number of ordered pairs in the cartesian product is \(p×q.\)
2. The total number of relations from set \(A\) to \(B\) is \({2^{pq}}.\)

Types of Relations

A relation tells the cartesian product of two sets. There are various types of relations; some of them are listed below:

1. Empty Relation:
   A relation is said to be an empty relation if the element of any set is not mapped with another set or itself.
2. Universal Relation:
   A relation is universal if all the elements of any set are mapped to all the elements of another set or the set itself.
3. Identity Relation:
   A relation is said to be an identity relation if all the elements are related to itself.
4. Inverse Relation:
   A set with elements as the inverse pairs of another set, then the relation is called inverse relation. 
5. Reflexive Relation:
   A relation is said to be reflexive if all the elements of any set are mapped to itself.
6. Symmetric Relation:
   A relation is said to be symmetric, in which the ordered pair of a set and the reverse ordered pair are present in the relation.
7. Transitive Relation:
   The relation \(R\) on set \(P,\) if \((x, y)∈R\) and \((y, z)∈R,\) then \((x, z)∈R,\) for all \(x, y, z∈R\) is called transitive relation.
8. Equivalence Relation:
   A relation is said to be equivalence if it is reflexive, transitive and symmetric—Universal and identity relations equivalence relations.

Representation of Relations

There are four ways of representing the relations, they are:

1. Statement or listing method
2. Set builder representation of a relation
3. Roaster or tabular method of representation
4. Representation by arrow diagrams

Statement or Listing Method

In this method of representing the relations, the well-defined property among the given elements is listed in the statement form, enclosed in the curly brackets.For example, the relation of the sets is given by \(\{ (1,1),(2,4),(3,9),(4,16),(5,25)\} \) can be written in statement form as \({\rm{\{ perfect}}\,{\rm{squares}}\,{\rm{of}}\,{\rm{the}}\,{\rm{numbers}}\,{\rm{1,2,3,4\} }}\)

Set-Builder Form

The relation can be written in set builder form using the variables \(x, y\) in the ordered pair with semi column followed by a relation between \(x\) and \(y.\) In the set builder representations of relations, instead of listing all the elements of the relations, they will show with the property among the given elements.

The variables and the set rule are separated by a colon \(( :)\) or a vertical line \((|).\) The set builder notation is widely used for the representation of the relations.

The relation of two sets \(A = \{ 2,3,4\} \) and \(B = \{ 4,9,16\} ,\) in which elements of set A are the square root of elements of set \(B\). The relation can be written in set builder form as follows.\(R = \left( {x,y} \right):x\) is the square root of \(y,x \in A,y \in B\} \)

Roaster Form

The relation can be represented in roaster form by writing all the possible ordered pairs of the two sets. 

The relation of two sets \(A = \{ 2,3,4\} \) and \(B = \{ 4,9,16\} ,\) in which elements of set \(A\) are the square root of elements of set \(B.\) The relation can be written in roaster form as follows. \(R = \{ (2,4),(3,9),(4,16)\} \)

Arrow diagram

In this method, the relation between two sets is showing by using the arrow drawn from one set to another set.The relation of two sets \(A = \{ 2,3,4\} \) and \(B = \{ 4,9,16\} ,\) in which elements of set \(A\) are the square root of elements of set \(B.\) The relation can be written by using an arrow diagram as follows.

Notations of Various Symbols used in the Representation of Relations

Below are some of the important symbols or notations used in the representation of relations.

1. \(N-\) Natural numbers or positive  integers
2. \(W-\) Whole numbers
3. \(Z-\) Integers
4. \(Q-\) Rational numbers
5. \(R-\) Real numbers
6. \(I-\) Imaginary numbers
7. \(C-\) Complex numbers
8. \(|\) or \(:-\) Such that
9. \(∈-\) Belongs to
10. \(∉-\) Not belongs to
11. \(⊂-\) Subset
12. \(∀-\) Such that

Solved Examples – Representation of Relations

Q.1. Let \(P = \{ 1,2,3\} ,R\) be a relation defined on set

\(P\) as “is greater than”, then represent the relation on the given set in the roasted method.
Ans: Given set \(P = \{ 1,2,3\} ,\)
The cartesian product of sets is \(\{ (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3)\} \)
Given the relation defined on set \(P\) as “is greater than”.
From the ordered pairs of the cartesian product, we have
\((2, 1)⟶2\) is greater than \(1.\)
\((3, 2)⟶3\) is greater than \(2.\)
\((3, 1)⟶3\) is greater than \(1.\)
Therefore, the set of ordered pairs \(\{ (2,1),(3,2),(3,1)\} \) gives the relation to the given set.

Q.2. The relation of two sets \(A = \{ 5,6,7\} \) and \(B = \left\{ {23,36,49} \right\},\) in which elements of set \(A\) are the square root of elements of set \(B.\) Represent the above relation in the roaster method.
Ans: Given the relation between two sets \(A = \{ 5,6,7\} \) and \(B = \left\{ {23,36,49} \right\},\) is elements of set \(A\) are the square root of elements of set \(B.\)
The property used between the elements in a given relation is “square root”.
In the roaster method, we can write all the possible ordered pairs that follow the given property.
So, relation \(R = \{ (5,25),(6,36),(7,49)\} .\)

Q.3. Observe the relation shown in the given figure and write the relation in the roaster form between the sets \(P\) and \(Q.\)

Ans: From the above arrow diagram, we see that the elements of set \(P = \{ x,y,z\} \) and the elements of set \(Q = \{ a,b,c\} \)
The relation shown in the arrow diagram describes that each element of set \(P\) is mapped with the elements of set \(Q.\)
The ordered pairs of the relation between the sets \(P\) and \(Q\) are given by
\(R = \{ (x,a),(y,b),(z,c)\} \)
The above shown ordered pairs of the relation shown in the roaster method.

Q.4. The relation of two sets \(A = \{ 2,3,4\} \) and \(B = \{ 8,27,64\} ,\) in which elements of set \(A\) are the cube root of elements of set \(B.\) Represent the above relation in the arrow diagram.
Ans: The relation of two sets \(A = \{ 2,3,4\} \) and \(B = \{ 8,27,64\} ,\) in which elements of set \(A\) are the cube root of elements of set \(B.\) The relation can be written by using an arrow diagram as follows.

Q.5. Observe the given relation of ordered pairs given, \(R = \{ (1,2),(1,3),(2,3)\} \) of two sets \(A = \{ 1,2,3\} \) and \(B = \{ 1,2,3\} \) in the set builder method.
Ans: Given relation is \(R = \{ (1,2),(1,3),(2,3)\} \)
From the given relation, the property used among the elements is “less than.”
As \(1\) is less than \(2\)
\(2\) is less than \(3\)
\(1\) is less than \(3\)
The above relation can be shown in the set-builder form by using the variables and conditions used among the elements.
The set-builder form of given relation is \(R = \{ (x,y):x\) is less than \(y,x \in A,y \in B\} \)

Summary

In this article, we have studied the definitions of relations and terminologies used in relations. We have studied the types of relations and the formula used to find the relations. 

Relation is one of the crucial topics in the set theory that describes the way of connection between any two objects or things. Relation describes that every element of one set is mapped to one or more elements of another set. We can say that relation has one or more outputs for every input value. The function is the special relation, in which elements of one set is mapped to only one element of another set.

There are namely four types of Representation of Relations:

• – Statement or listing method
• – Set builder representation of a relation
• – Roaster or tabular method of representation
• – Representation by arrow diagrams

This article also gives the representation of relations such as the roaster method, set-builder method and arrow method by using examples.

Frequently Asked Questions (FAQ) – Relations and Representation

Q.1. What are forms of representations of relations?
Ans: The methods of representations of relations are
1. Statement method
2. Roster form
3. Set-builder form
4. Arrow method

Q.2. What is a relation in set theory?
Ans: A relation tells the cartesian product of two sets.

Q.3. How do you represent the relation in set-builder form?
Ans: The relation can be written in set builder form using the variables \(x, y\) in the ordered pair with semi column followed by a relation between \(x\) and \(y.\)

Q.4. How do you represent the relation in roaster form?
Ans: The relation can be represented in roaster form by writing all the possible ordered pairs of the two sets.

Q.5. Give any three symbols used in the set-builder form?
Ans: The symbols used are
\(∈-\)Belongs to
\(∉-\)Not belongs to
\(⊂-\)Subset

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