Relation – Co-domain of a Relation

Relation is a connection between things. We can see relations in our daily life such as the relationship of a husband and wife, mother and daughter, student and teacher etc. What we can notice from these examples is that there should be a minimum two numbers of elements for any relation to exist. Every relation has a pattern or property. Similarly we can see relations in the world of Mathematics as well, such as P >Q, n =6, line m is perpendicular to line y, line a is parallel to line b, set P is a subset of set Q and so on.

Relation is studied to find the connection between the elements of two or more sets. Mathematically, it is represented as “A relation R from set P to set Q is a subset of the Cartesian product P*Q”. The subset is derived by describing a relationship between elements of P and Q. Given two non empty sets P and Q, the Cartesian product P*Q is the set of all ordered pairs whose first component is a member of P and second component is a member of Q.

Co-domain of a Relation:

For any set A, Co-domain is the set of output elements/members. It does not depend on the given relation(R). 

Given two sets A and B, Co-domain can be defined as the set of all elements of set B. In layman’s language,  it is the Set of all possible outputs whether satisfying the given relation or not. It is also called the destination set or output set of a function.

Let us understand the concept in detail with some practical examples below:

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 divides y, such that x∈A and y∈B. Find Co-domain.

Solution:

Co-domain = Set B = {1, 2, 3, 4, 6, 8}

All the elements of set B will be the elements of Co-domain.

Example 2:

If R is a relation from Set A ={2,4,5} to Set B ={all positive even numbers less than 10} defined by xRy implies that the sum of x and y is less than 20, then Find Co-domain.

Solution:

We simply need to find set B to find the co-domain.

Set B = {all positive even numbers less than 10}

B = { 2, 4 , 6 , 8}

Therefore, Co-domain = { 2, 4 , 6 , 8}

Understanding the difference between Co-domain, Domain and Range:

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}

Range can be defined as a set of all the second elements of ordered pairs in Relation R. Range is the set of all permissible outputs satisfying the criterion of the given range. 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 represents the first element of the ordered pair and b represents the second element of the ordered pair.

Example 3:

Given Two Sets, A = {all positive even numbers less than 10) and B = {all positive odd numbers less than 10}

Relation: R = {(a,b) | a+b < 10}

Find Domain, Range and Codomain.

Solution:

Here A represents the set of Inputs (First element of ordered pair) and B represents the set of all possible Outputs (Second Element of ordered pair).

A = {2, 4, 6, 8}

B = {3, 5, 7, 9}

R = {(2,3), (2,5), (2,7), (4,3), (4,5), (6,3)}

Domain of R = {2, 4, 6}

(Set of all the first elements of ordered pair in relation R)

Co-domain = {3, 5, 7, 9}

(Set of all the second elements whether in relation to R or not)

Range = {3, 5, 7}

(Set of all the second elements in relation to R)

 Example 2:

Let P = {1, 2, 3, …….., 10}. Define a relation R from P to P by R = {(x, y): 3x – y = 0, where x, y ∈ P}. Write down its co-domain.

 Solution:

 The relation R from P to P is given as R = {(x, y): 3x – y = 0, where x, y ∈ P}

 i.e., R = {(x, y): 3x = y, where x, y ∈ P}

∴The roster form is given by R = {(1, 3), (2, 6), (3, 9)}

The whole set P is the co-domain of the relation R.

∴ Co-domain of R = P = {1, 2, 3… 10}

Conclusion

Codomain is the set of all the second elements in the ordered pair and it does not change in relation to R. Here, in this article, we discussed the meaning of the term co-domain of a relation and how it is useful to find the connection between a given pair of sets or for a given function. We saw notations and mathematical representation of the co-domain of relation, Range and Domain of a relation or function. We understood the concepts involved here with the help of practical examples. Further we discussed the difference between Co-domain and Range/Domain as the two look quite similar and are generally confused for the meaning. Meaning of the terms along with their practical problems will help to create a better understanding of the concepts.