Domain and Range of a Relation

A relation in mathematics is a connection or association between elements of two sets. Domain and range of relations help in signifies the behaviour of relations at different values. The number of possible relations between two sets A and B depends on the sizes of these sets. Domain and range of relations help to analyze it more deeply. The main applications of relation are in database systems, graph theory, and social networks.

In this article, we will cover the concepts of domain, range, and co-domain of relation. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of seven questions have been asked on this concept, including one in 2018, two in 2020, and four in 2023.

Introduction to relations

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B.

The subset is derived by describing the relationship between the first element and the second element of the ordered pairs in A × B.

The second element is called the image of the first element.

If the element (a,b) belongs to R, (here a belongs to A and b belongs to B), then the relation is represented as a R b.

Example:

Set A and Set B: Let A={1,2} and B={x,y}. A possible relation R could be: R={(1,x),(2,y)}. This indicates that 1 is related to x and 2 is related to y.

Domain of relation

The domain of a relation is the set of all first elements (or components) of the ordered pairs in the relation. In other words, it is the set of all possible inputs with at least one associated output in the relation.

The domain of a relation R is the set of all first elements of the ordered pairs in a relation R.

eg. R={(a,b),(c,d)}, then the domain is {a,c}

The image given below represents the domain of a relation.

Range of relation

The range of any Relation is the set of output values of the relation. For example, if we take two sets A and B, and define a relation R: {(a,b): a ∈ A, b ∈ B} then the set of values of B is called the domain of the function.

The range of a relation R is the set of all second elements of the ordered pairs in a relation R.

eg. R={(a,b),(c,d)}. Then Range is {b,d}

The image given below represents the range of a relation.

Co-domain of relation

The codomain of a relation is the set into which all the second elements (or components) of the ordered pairs in the relation are constrained to lie. It is the set of all possible outputs, not just the actual outputs associated with the domain's elements.

Co-domain in a relation from A to B is the set B itself.

Codomain(R)=B

The distinction between Codomain and Range

• Codomain: The set B in A×B, which includes all possible second elements of the ordered pairs, whether they are used or not.
• Range (or Image): The set of all actual second elements of the ordered pairs in the relation. In other words, it is the set of all outputs that are produced by the relation.

Solved Examples Based On the Domain, Range, and Co-domain of Relations:

Example 1: What is the domain of the relation? $\left\{(a, b): b \in \mathbb{Z} \text { and } a=\frac{b}{2}\right\}_{\text {(Here } Z \text { is the set of integers) }}$
1) $R$
2) $z$
3) N
4) None of these

Solution:
Since, $a=\frac{b}{2}$ and $b \in \mathbb{Z}$
So, values that a will take will be of the form (integer/2), which will include all the integers and half of all the integers.
Domain $=\{\ldots-1.5,-1,-0.5,0.0 .5,1.1 .5, \ldots$.
Hence, the answer is the option 4.
Example 2: If relation R is such that $R:\{(a, b): a=b-3$ and $b \in \mathbb{N}\}$, then the domain of R is
1) N
2) $\{-1,0,1,2,3, \ldots$.
3) $\{-2,-1,0,1,2,3, \ldots$.

4) W

Solution:

Since a = b - 3 and

So, b = {1,2,3,4,.....}

Thus, a = -2, -1, 0, 1, 2, .....

Hence, the answer is the option 3.

Example 3: Let N denote the set of all natural numbers. Define two binary relations on N as $R_1=\{(x, y) \in \mathbb{N} \times \mathbb{N}: 2 x+y=10\}$ and $R_2=\{(x, y) \in \mathbb{N} \times \mathbb{N}: x+2 y=10\}$ Then :
1) Range of $R_1 s\{2,4,8\}$.
2) Range of $R_2$ is $\{1,2,3,4\}$.
3)Both $R_1$ and $R_2$ are symmetric relations
4) Both $\mathrm{R}_1$ and $R_2$ are transitive relations

Solution:
As we have learned
Range -
The range of the relation $R$ is the set of all second elements of the ordered pairs in a relation $R$.
- wherein
eg. $R=\{(a, b),(c, d)\}$. Then Range is $\{b, d\}$

$\begin{aligned}
& R_1:\{(x, y) \in N \times N: 2 x+y\}=10 \\
& R_2=\{(x, y) \in \mathbb{N} \times \mathbb{N}: x+2 y=10\} \\
& \quad R_1 \quad x=\frac{10-y}{2} \\
& \text { So for } \\
& \text { i.e } y=2,4,6,8
\end{aligned}$

lend\{aligned\}\$
For $R_1$,ordered pairs are $(4,2)(3,4)(2,6)(1,8)$
For $R_2, \quad x+2 y=10 ; y=\frac{10-x}{2}$
ordered pairs are $(2,4)(4,3)(6,2)(8,1)$
range of $R_2$ is $\{1,2,3,4\}$

Hence, the answer is the option 2.

Example 4: If R is a relation such that it is defined from A to A, then what is the range of R? Here A = {1,2,3,4}.

1) {1,2,3}

2) {2,3,4}

3) {1,2,3,4}

4) {2,3}

Solution:

As we have learned,

R is defined from .

Range is {1,2,3,4}.

Hence, the answer is the option 3.

Example 5: What is the domain of relation? {(1,3); (2,9); (5,7); (3,8)}

1) {1,2,3,5}

2) {1,2,3}

3) {3,7,9,8}

4) {1,2,3,5,7,8,9}

Solution:

As we have learned,

In this question, the domain is {1,2,3,5} as these elements are the first elements of ordered pairs in R.

Hence, the answer is the option 1.

Summary

We concluded that the domain of a relation is crucial in various fields, as it helps determine the scope and applicability of the relation or function in question. The codomain of a relation is crucial for correctly interpreting the relation's scope and for ensuring proper definitions in mathematical, computational, and practical contexts.