Relation- Domain of a Relation

In simple terms, relations are described as connections between different things. Every day, we see hundreds of relationships such as your relationship with your mother, the relationship between husband and his wife, student and teacher, and so on. Also, we regularly build new relationships and sometimes break the old ones; however, every relationship needs to have at least two members. 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.

This article talks about the domain of a relation. You will find brief information on the concept of domain of a relation in Maths, co-domain of a relation with examples, the difference between co-domain, domain and range, and so on. So, let’s start by describing the domain of a relation in the Maths study material. 

Explain relation in the mathematical context 

The relation is studied to find the connection between two or more sets of elements. Mathematically, it is represented as “A relation R from set P to set Q is a subset 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 the second component is a member of Q.

What is a domain and codomain of a relation 

As mentioned earlier, the relation will always have ordered pairs where the second element is referred to as an image, whereas the first element is called a preimage. It also means that among the two sets, one will be the image, and the second one will be the preimage. The set with all the first elements of the ordered pairs of relation R is called the domain of a relation. However, the codomain of the relation can be described as 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 ={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.

Here is the solution –

We simply need to find set B to find the codomain.

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

B = { 2, 4 , 6 , 8}

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

Example 2:

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.

Here is the solution –

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

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

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

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

Domain of R = { a | (a,b) ∈ R}

The 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.

Conclusion 

Just like in life, relations in mathematics plays a significant role as they are one of the most significant topics of maths. Relations are described as connections between different things, and domains are described as numbers that you offer to functions. Both domain and range are usually denoted using interval notation. The set which has all the first elements of the ordered pairs of relation R is called the domain. 

In this article describing the domain of a relation, we studied the concept of the domain of relation in mathematics in length. We covered several other topics, such as a relation in a mathematical context, domain and codomain of relation along with examples, and other related topics. We hope this study material must have helped you better understand the domain of a relationship.