Domain and Range of Relation: A relation is a rule that connects elements in one set to those in another. \(A\) and \(B\) If are non-empty sets, then the relationship is a subset of Cartesian Product \(A \times B\).

The domain is the set of initial members of all ordered pairs. On the other hand, the range is the collection of all ordered pairs’ second components and only comprises the items utilised by the function. In this article, we will discuss the domain and range of relations in detail.

Introduction

We necessarily know questions such as, “How is he connected to you?” in our daily lives. The below are some appropriate options:

(i) He is my father.
(ii) He is my teacher.
(iii) He is not related to me.

Domain and Range of Functions

As a result, we may conclude that the term relation refers to a connection between two people. In mathematics, extending this concept, a relation is defined as a link between two or more mathematical objects. Examples,

(i) A number \(m\) is related to a number \(n\) if \(m\) divides \(n\) in the set of \(N\)
(ii) A real number \(x\) is related to a real number \(x \le y.\)
(iii) A point \(p\) is related to a line \(L\) if \(p\) lies on \(L\).
(iv) A student \(X\) is related to a school \(S\) if \(X\) is a student of \(S\).

What is a Relation?

The ordered pair collection is defined as the relationship between two sets in which the object from each set forms the ordered pair. A relation \(R\) is the subset of the cartesian product of any two non-empty sets \(A\) and \(B\). According to the relation, every element of one set is mapped to one or more elements of the other set.

In terms of relationships, we may say that every input has one or more outputs. If \((a,\,b) \in \,R\) we say that \(a\) is related to \(b\) under the relation \(R\) and is written as a \(R\,b\). The function is a particular connection in which elements from one set are mapped to just one set’s element.

A set of ordered pairs is referred to as a relation in mathematics. An arrow diagram shows the relationship \((R\,:A \to B)\) between sets \(A\) and \(B\) in the image below.

There is a relationship between sets \(A\) and \(B\) in the diagram above. Ordered pairs can be finite or infinite in a relation. The term “relation on \(A”\)  refers to a relationship between sets \(A\) and \(A\). \({2^{mn}}\) is the maximum number of relations defined from sets \(A\)  (with \(m\)  elements) and (with \(n\) elements).

Domain and Range of a Relation

Now it’s clear that, like a Cartesian product, a relationship will have ordered pairs. The second element in these ordered pairings is the image of the first element, while the first element is the preimage of the second element.

This implies that one set will have all the preimages, while the other will contain all the images. The relation domain is the set that contains all the initial elements of all the ordered pairs of relation  The domain set may or may not be identical to set \(R\) The domain set may or may not be identical to set \(A\). The range of the relation, on the other hand, is the set that contains all the second items.  The second set \(B\) might be the same size as or more significant than the relation’s range.

This is because elements in set \(B\) might be unrelated to any element in set \(A\). As a result, the entire set \(B\)  is known by a new name, i.e. the relation’s codomain. Relations can be expressed in the same manner as the sets are represented, i.e., using the Roster technique or the Set-builder approach. The use of arrow diagrams provides another more precise depiction.

For example, consider the set \(N = \left\{ {9,10,\,11,\,12,\,13} \right\}\). Let us now create a relation \(A\) from \(N\) to \(N\) such that \(y\) is two greater than \(x\) in the ordered pair \((x,\,y)\) in \(A\). This can be expressed in one of three ways:

1. Set-builder method: \(A = \left\{ {(x,\,y):y = x + 2;\,x,\,y \in N} \right\}\)
2. Roster method: \(A = \left\{ {(9,\,11),\,(10,\,12),\,(11,\,13)} \right\}\)
3. Arrow diagram:

Since the domain of \(A\) is not \(N\)  It is equal to \(\left\{ {9,\,10,\,11} \right\}\) Because for other values from \(N\) the output doesn’t lie in set \(N\) Also, the codomain of \(A\) is \(N\) But the range of \(A\)  is \(\left\{ {11,\,12,\,13} \right\}\) , which is a subset of the codomain.

Relation Representation

Apart from set notation, there are various ways to express the relation, such as using tables, plotting it on an \(XY – \) axis, or using a mapping diagram.

Types of Relations

Following are the different types of relations:

1. Empty Relation

The relation \(R\) in \(A\) is an empty relation, also known as the void relation when no element of set \(A\) is connected or mapped to any other member of \(A\).

For instance, suppose the fruit basket contains \(100\) mangoes. There is no way to find a mangoes. There is no way to find a relation \(R\) that will result in an apple being placed in the basket. is Void because it has \(100\) mangoes but no apples.

2. Universal Relation

A set \(A\) is a universal relation if every element of \(A\) is connected to every element of \(A\) in this entire relation. For example, \(R\, = \,A \times A\) is a universal relation

3. Identity Relation

Identity relation occurs when each element of set \(A\) is only related to itself.

\({I_A} = \,\{ (a,\,a),\,a \in R\} \)

For example,

When we roll the dice, there are a total of \(36\) possible outcomes, i.e.,

\(\left\{ {(1,\,\,1),\,(1,\,\,2),\,(1,\,\,3)……(6,\,6)} \right\}\)  From these, if we consider the relation
\(\left\{ {(1,\,\,1),\,(2,\,\,2),\,(3,\,\,3),\,(4,\,\,4),\,(5,\,\,5),(6,\,\,6)} \right\}\) , it is an identity relation.

4. Inverse Relation

If \(R\)  is a relation from \(A\) set to set \(B\) such that \(R \in \,\,A \times B\) then relation \({R^{ – 1}} = \,\{ (b,\,\,a);(a,\,b) \in R\} \)
For example,

If we roll two dice and if \(R = \,\{ (1,\,\,2),(2,\,3)\} \) then \({R^{ – 1}} = \,\{ (2,\,\,1),(3,\,2)\} \) Here the domain of is the range of \(R\) is the range of \({R^{ – 1}}\) and vice versa.

5. Reflexive Relation

If every element of set \(A\) maps to itself, then the relation is reflexive. i.e for every \(a \in \,A,\,(a,\,a) \in R\)

6. Symmetric Relation

A relation \(R\) on a set \(A\) is symmetric if \((a,\,\,b) \in R\) then \((b,\,\,a) \in R\)  for all \(a,\,\,b \in A\)

7. Transitive Relation

A relation \(R\) on a set \(A\) is said to be transitive if \((a,\,\,b) \in R,\,(b,\,c) \in R\)  then \((a,\,\,c) \in R,\) for all \(a,\,b,\,c \in A\)

8. Equivalence Relation

A relation \(R\) on a set \(A\) is an equivalence relation if it is reflexive, symmetric and transitive.

Solved Examples – Domain and Range of Relations

Q.1. Write the domain and range of the given relation: (eye colour, student’s name).
\(a,b,\,c \in A = \{ \left( {{\rm{blue}},\,{\rm{John}}} \right),\,\left( {{\rm{green}},\,{\rm{William}}} \right),\,\left( {{\rm{brown}},\,{\rm{Wilson}}} \right),\,\left( {{\rm{blue}},\,{\rm{Moy}}} \right)\)
\(\left( {{\rm{brown}},\,{\rm{Abraham}}} \right),\,\left( {{\rm{green}},\,{\rm{Dutt}}} \right)\} \)
State whether the relation is a function.
Ans: Domain: \(\left\{ {{\rm{blue}},\,{\rm{green}},\,{\rm{brown}}} \right\}\) Range:
\(\left\{ {{\rm{John}},\,{\rm{William}},\,{\rm{Wilson}},\,{\rm{Moy}},\,{\rm{Abraham}},\,{\rm{Dutt}}} \right\}\)
Because the eye colours are repeated, the relationship is not a function.

Q.2. Write the domain and range of the given relation:
\(\{ (4,\,3),\,( – 1,\,7),\,(2,\, – 3),\,(7,\,\,5),\,(6,\, – 2)\} \)
Ans: The ordered pairs’ domain is the first component. Range, on the other hand, is the ordered pairs’ Second Component. If you find any duplicates, remove them.
Domain \( = \{ 4,\,\, – 1,\,\,2,\,\,7,\,\,6\} \) Range \( = \{ 3,\,\,7,\,\, – 3,\,\,5,\,\, – 2\} \)

Q.3. From the following arrow diagram, find the domain and range and depict the relation between them.

Answer: We know that the domain is the set of the first element of the ordered pair.
So, Domain \( = \{ 3,\,\,4,\,\,5\} \)
And the range of a relation is the set of second elements of the ordered pair.
Range \( = \{ 3,\,\,4,\,\,5,\,6\} \)
From the given arrow diagram, we can write a relation \(R = \{ (3,\,\,4),\,(\,4,\,\,6),\,(5,\,3),(5,5)\} \)

Q.4. Determine the domain and range of the relation R defined by
\(R = \{ x,\,\, – 2,\,\,2x + 3\} :\,x \in \{ 0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5\,\} \)
Ans: Given \(x = \{ 0,\,\,1,\,\,2,\,\,3,\,\,4,\,\,5\} \)
\(x = 0 \Rightarrow x – 2 = 0 – \,\,2 = \,\, – 2\) and \(2x + 3 = 2 \times 0 + 3 = 3\)
\(x = 1 \Rightarrow x – 2 = 1 – \,\,2 = \,\, – 1\) and \(2x + 3 = 2 \times 1 + 3 = 5\)
\(x = 2 \Rightarrow x – 2 = 2 – 2 = 0\) and \(2x + 3\, = 2 \times 2\, + 3\, = \,7\)
\(x\, = 3\,\, \Rightarrow \,x\, – \,2\,\, = \,3 – \,2\, = 1\) and \(2x + 3 = \,2 \times 3 + 3\,\, = \,9\)
\(x = \,4\, \Rightarrow \,x – 2\, = \,4 – 2\, = 2\) and \(2x + 3\, = 2 \times 4 + \,3\, = 11\)
\(x\,\, = \,\,5\, \Rightarrow \,x – 2\, = \,5 – \,2\, = \,3\) and \(2x – \,3\, = 2 \times 5\, + 3\, = \,13\)
Hence \(R = \,\{  – 2,\,\,3),\,( – 1,\,\,5)\,,\,(0,\,\,7)\,\,(1,\,\,9)\,(2,11)\,,\,(3,\,\,1\,3)\)
Domain of \(R = \,\{  – 2,\, – 1,\,\,0,\,\,1,\,\,2,\,\,3\} \)
Range of \(R\, = \,\{ 3,\,\,5,\,\,7\,,\,\,9,\,\,11,\,\,13\,\} \)

Q.5. A relationship between set \(X\) and set \(Y\) is depicted in the diagram below. Write the same thing in the roster and set builder Forms, and then figure out the domain and range.

Ans: In the set builder form \(R = \left\{ {\left( {x,\,y} \right):x\,{\rm{is}}\,{\rm{the}}\,{\rm{square}}\,{\rm{of}}\,y;\,x \in X,\,y \in Y} \right\}\)
In roster form \(R = \{ (2,1)(4,\,\,2)\} \)
Domain \( = \,\{ 2,\,4\} \)
Range \( = \,\{ 1,\,\,2\} \)

Summary

In this article, we have discussed a relation and how to represent relations in three different forms, types of relations. Also, we discussed the domain and range of relations in detail, along with the solved examples and frequently asked questions.

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Frequently Asked Questions: Domain and Range of Relation

Q.1. What is the domain for the relation?
Ans: The domain of a relation is the set of the initial elements of all the ordered pairs of the relation.

Q.2. What is the range of a relation?
Ans: The set of the second elements of all the ordered pairs of a relation is the range of a relation.

Q.3. What is the difference between relation and function?
Ans: A relation is defined as a connection between two or more sets of values. It may also be thought of as a subset of the Cartesian product. A function is defined as a relationship in which each input has just one output. Every function is a relation, but not every relation is a function.

Q.4. What is the difference between domain and range?
Ans: The domain of a function is all the values that go into it, and the range is all the values that come out of it in its simplest form. For a relation from \(A \to B\) all the elements of set \(A\) which are related to the elements of set \(B\) are the elements of the domain and all related elements of set \(B\) form the range.

Q.5. If a set \(A\) has n elements, and has m elements, what is the number of relations from \(A\,to\,B\)
Ans: A relation from \(A\,to\,B\) is a subset of the Cartesian product \((A\, \times \,B)\) of two sets \(A\) and \(B\). There will be \((n\, \times \,m)\) elements \((A\, \times \,B)\) in if the set \(A\) has \(n\) elements and the set \(B\) has \(m\) elements. As a result, there will be \({2^{n\, \times \,m}}\)  subsets \((A\, \times \,B)\) of and therefore \({2^{n\, \times \,m}}\) relations from \(A\,to\,B\) may be defined.

We hope this detailed article on the domain and range of relations helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!