Relation – Range of a Relation

Range of a Relation is an important concept studied in mathematics to get an insight of output elements in relation to R for a given set,pair of sets or a function. Relation is studied to know the connection between the elements of two or more sets. Mathematically, it is represented as “A relation R from set A to set B is a subset of the Cartesian product A*B”. The subset is derived by describing a relationship between elements of A and B. Given two non empty sets A and B, the Cartesian product A*B is the set of all ordered pairs whose first component is a member of A and second component is a member of B.

Range of a Relation:

Range can be defined as a set of all the output members(Second element) of ordered pairs in Relation R. Range is the set of all permissible outputs satisfying the criterion of the given Relation (R ). The elements in the first set are referred to as inputs and the elements in the second set are called the outputs.

Range of R = { b | (a,b) ∈ R}

Let A and B be two sets, then Relation R from A to B is a subset of A*B

i.e. R ⊆ A*B

Also, if a ∈ A and b ∈ B, then

R = {(a, b) | a ∈ A and b ∈ B }

Where a is the first member of the ordered pair and b is the second member of the ordered pair.

Below are some examples to get a better understanding of the concept of Range of a relation.

Example 1:

If R is a relation from Set A ={2,4,5} to Set B ={1,2,3,4,6,8} defined by xRy implies that x completely divides y, such that x∈A and y∈B. Find Range.

Solution:

A = {2,4,5}

B = {1,2,3,4,6,8}

R = {(2,2), (2,4), (2,6), (2,8),(4,4),(4,8)}

(Cartesian product of given two sets A*B, in relation to R)

Therefore, Range = {2, 4, 6, 8}

(All the second elements of ordered pairs in relation to R will be the elements of Range.)

 Example 2:

If R is a relation from Set A = {2, 3, 5} to Set B = {all positive even numbers less than 10} defined by xRy implies that the product of x and y is less than 25, and then Find Co-domain.

Solution:

A = {2, 3, 5}

B = {2, 4, 6, 8}

R = {(2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (5,2), (5,4)}

(Cartesian Product of given sets A*B in relation to R)

Therefore, Range = {2, 4, 6, 8}

(Set of second elements in ordered pairs in relation to R)

Till now, we learned to find the range for any set or given pair of sets. Moving further, in the example below, we will learn how to find the range for a given function.

Example 3:

Given a function: f = 5x – 1, Find the Range if 1<x<5.

Solution :

The values of x will be from 2 to 4. 

So, the domain is {2, 3, 4}. 

The values of f obtained by putting domain elements in the function will be range. 

if x=2 then f = 5(2)-1 = 9, 

if x=3 then f = 5(3)-1 =14, 

if x=4 then f = 5(4)-1 = 19, 

So, the range is {9,14,19}.

Things to Remember :

Note that, Set of all permissible inputs or the inputs which satisfy the criterion of a relation is called Domain. Domain can also be defined as a set of all the first elements of ordered pairs in Relation R.

Domain of R = {a | (a, b) ∈ R}

And a set of all second elements whether in relation to R or not, are called Co-domain.

Conclusion:

In this article, we discussed the meaning of the term Range of a relation and how it is useful to find the connection between a given pair of sets or for a given function.We discussed a few examples to get deep insight of the concept learned here. Further, we discussed the relationship of Range with Co-domain and domain of a relation. We saw notations and mathematical representation of terms and their use in relation and function. To find the relation between two sets, Range of a relation is one important property that helps to find the association of properties of a given pair of sets or a given function.