Types of Relations: Definition, Classification and Examples

Types of Relations: One of the most important topics in set theory is relations and their various types of concepts. Sets, relations, and functions are all interconnected concepts. Sets are collections of ordered elements, whereas relations and functions are operations on sets. The relations establish the link between the two given sets. There are also different types of relations that describe the connections between the sets.

In mathematics, a relation is a relationship between two different sets of information. When two sets are considered, their relationship is established if there is a connection between the elements of two or more non-empty sets. As an example, Students in schools are expected to stand in a line in ascending order of their heights during the morning assembly. This establishes an ordered relationship between the students’ heights. In this post, we will learn more about the types of relations in maths.

Solve the Concept of Types of Relations

Definition of Relations

A relation describes the cartesian product of two sets. Cartesian product of two sets \(A\) and \(B,\) such that \(a \in A\) and \(b \in B,\) is given by the collection of all order pairs \(\left( {a,\,b} \right).\) Relation tells that every element of one set is mapped to one or more elements of the other set. Regarding relation, we can say that for every input, there are one or more outputs. Let’s know more about types of relations in maths.

The function is the special relation in which elements of one set are mapped to only one element of another set.

In mathematics, a set of order pairs is called the relation. For any two non-empty sets \(A\) and \(B,\) the relation \(R\) is the subset of the cartesian product of \(A \times B.\) The below figure shows the relation \(\left( {R:A \to B} \right)\) between set \(A\) and set \(B\) by an arrow diagram.

Domain and Range of Relations

There are several types of relations in maths. Relation defines the relationship between two sets. The relation of two sets is the subset of the cartesian product of the sets.
Let \(A\) and \(B\) be two non-empty sets of ordered pairs, the relation of set \(A\) with set \(B\) is written as \(R:A \to B.\) Here, set \(A\) is called the domain of the relation and set \(B\) is called the range of the relation.
The set of first elements in the ordered pair is a domain, and the set of second elements of the ordered pair is the range of the relation.

Example: In the ordered pair \(\left\{ {\left( {a,\,1} \right),\,\left( {b,\,2} \right),\,\left( {c,\,3} \right)} \right\},\) the elements \(\left\{ {a,\,b,\,c} \right\}\) are known as the domain of the relation, and the elements \(\left\{ {1,\,2,\,3} \right\}\) are known as the range of the relation.

Representation of Relations

Relation defines the relationship between two sets. There are three ways of representing the relations. They are:

Set-Builder Form

The relation can be written in set builder form using the variables \(x,\,y\) in the ordered pair with semi column followed by a relation between \(x\) and \(y.\)

The relation of two sets \(A = \left\{ {2,\,3,\,4} \right\}\) and \(B = \left\{ {4,\,9,\,16} \right\},\) in which elements of set A are the positive square root of elements of set B. The relation can be written in set builder form as follows.
\(R = \){\(\left( {x,\,y} \right):x\) is the positive square root of \(y,\,x \in A,\,y \in B\)}

Roaster Form

The relation can be represented in roaster form by writing all the possible ordered pairs of the two sets.
The relation of two sets \(A = \left\{ {2,\,3,\,4} \right\}\) and \(B = \left\{ {4,\,9,\,16} \right\},\) in which elements of set \(A\) are the square root of elements of set The \(B.\) relation can be written in roaster form as follows.
\(R = \left\{ {\left( {2,\,4} \right),\,\left( {3,\,9} \right),\,\left( {4,\,16} \right)} \right\}\)

Arrow Diagram

In this method, the relation between two sets is shown by using the arrow drawn from one set to another set.
The relation of two sets \(A = \left\{ {2,\,3,\,4} \right\}\) and \(B = \left\{ {4,\,9,\,16} \right\},\) in which elements of set \(A\) are the square root of elements of set \(B.\) The relation can be written by using an arrow diagram below. You can find more about type of relations in maths in this post.

Types of Relations

Relation defines the relationship between two sets. The relation of two sets is the subset of the cartesian product of the sets. There are various types of relations:

Empty Relation

A relation is said to be an empty relation if the element of any set is not mapped with another set or itself. Therefore, empty relation is also known as void relation.
Condition: No element of set \(P\) is mapped with another set \(Q\) or set \(P\) itself. The empty relation is shown by \(R = \emptyset .\)

Example: Consider set \(A\) consisting of \(10\) apples in the basket. Then finding the relation \(R\) of getting mangoes from the basket is not possible. Since this basket has only apples and not mangoes. So, the above relation is known as empty or void relation.

Universal Relation

A relation is universal if all the elements of any set are mapped to all the elements of another set or the set itself. Universal relation is also known as full relation, where all the elements are mapped.
Condition:
The relation \(R\) is universal; all the elements of set \(P\) is mapped with all the elements of set \(Q\) or set \(P\) itself.
\(R = P \times P\) or \(R = P \times Q.\)

Example: Consider set \(P\) consists of all whole numbers, and set \(Q\) consists of all integers. Then the relation \(R:P \to Q\) is universal because all the elements of set \(P\) are there in set \(Q.\) (We know that all whole numbers are integers).

Identity Relation

A relation is said to be an identity relation if all the elements are related to itself. \(^{”}{I^{”}}\) generally denotes identity relation.
Condition:
If the relation \(\left( I \right)\) is identity, then all the elements of set \(P\) are related with itself, such that for every \(x \in P,\) it is \(\left( {x,\,x} \right) \in I.\)

Example: In the set \(P = \left\{ {1,\,2,\,3,\,4} \right\},\) then the identity relation is given by \(I = \left\{ {\left( {1,\,1} \right),\,\left( {2,\,2} \right),\,\left( {3,\,3} \right),\,\left( {4,\,4} \right)} \right\}\)

Inverse Relation

If a set with elements has the inverse pairs of another set, then the relation is called inverse relation.
Condition:
Consider \(R = \left\{ {\left( {a,\,b} \right):a \in P,\,b \in Q} \right\}\) be the relation from set \(P\) to set \(Q,\) then the relation from set \(Q\) to set \(P\) is known as inverse relation, such that \({R^{ – 1}}:Q \to P = \left\{ {\left( {b,\,a} \right):\left( {a,\,b} \right) \in R} \right\}\)
The range of relation R and domain of the inverse relation \({R^{ – 1}}\) are the same.

Example: Let the relation \(R = \left\{ {\left( {a,\,x} \right),\,\left( {b,\,y} \right)} \right\},\) then the inverse relation \({R^{ – 1}} = \left\{ {\left( {x,\,a} \right),\,\left( {y,\,b} \right)} \right\}\)

Reflexive Relation

A relation is said to be reflexive if all the elements of any set are mapped to itself.
Condition:
If the relation \(\left( R \right)\) is reflexive, then all the elements of set \(P\) are mapped with itself, such that for every \(x \in P,\) then \(\left( {x,\,x} \right) \in R.\)

Example: If the set \(A = \left\{ {1,\,2,\,3} \right\},\) then the relation \(\left\{ {\left( {1,\,1} \right),\,\left( {2,\,2} \right),\,\left( {3,\,3} \right)} \right\}\) is reflexive relation.
Here, every element of set \(A\) is mapped with itself, such that \(1 \in A,\,\left( {1,\,1} \right) \in R.\)

Symmetric Relation

A relation is said to be symmetric, in which the ordered pair of a set and the reverse ordered pair are present in the relation.
Condition:
The relation \(\left( R \right)\) is symmetric on set \(P,\) if \(\left( {x,\,y} \right) \in R,\) then, \(\left( {y,\,x} \right) \in R,\) such that \(a,\,b \in P.\)
If \(x = y\) is true, then \(y = x\) also true in the symmetric relation, where \(x,\,y \in P.\)

Example: For the set \(P = \left\{ {a,\,b} \right\},\) the relation \(R = \left\{ {\left( {a,\,b} \right),\,\left( {b,\,a} \right)} \right\}\) is called symmetric relation, where \(a,\,b \in P.\)

Transitive Relation

The relation \(R\) on set \(P,\) if \(\left( {x,\,y} \right) \in R\) and \(\left( {y,\,z} \right) \in R,\) then \(\left( {x,\,z} \right) \in R,\) for all \(a,\,b,\,c \in R\) is called transitive relation.

Example: For the set \(P = \left\{ {a,\,b,\,c} \right\},\) the relation \(R = \left\{ {\left( {a,\,b} \right),\,\left( {b,\,c} \right),\,\left( {a,\,c} \right)} \right\}\) is called transitive relation, where \(a,\,b,\,c \in P.\)

Equivalence Relation

A relation is said to be equivalence if it is reflexive, transitive and symmetric. Universal and identity relations are equivalence relations.
Conditions:
1. If the relation \(\left( R \right)\) is reflexive, then all the elements of set \(P\) are mapped with itself, such that for every \(x \in P,\) then \(\left( {x,\,x} \right) \in R.\)
2. The relation \(\left( R \right)\) is symmetric on set \(P,\) if \(\left( {x,\,y} \right) \in R,\) then \(\left( {y,\,x} \right) \in R,\) such that \(a,\,b \in P.\)
3. The relation \(R\) on set \(P,\) if \(\left( {x,\,y} \right) \in R\) and \(\left( {y,\,z} \right) \in R,\) then \(\left( {x,\,z} \right) \in R,\) for all \(a,\,b,\,c \in R\) is called transitive relation

Other Types of Relations

The other types of relations based on the mapping of two sets are given as follows:

One to One Relations

A relation is said to be a One to One relation if all the distinct elements of one set are mapped to distinct elements of another set.

One to Many Relations

A relation is said to be One to Many relations if the same element of one set is mapped to another set’s more than one element.

Many to One Relation

A relation is said to be Many to One relation if all the distinct elements of one set are mapped to the same element of another set.

Many to Many Relation

A relation is said to be Many to Many relations if one or more than one element is mapped with the same element of another set.

Solved Examples – Types of Relations in Maths

Q.1. Let \(A\) be the set of two male persons in a family. \(R\) be a relation defined onset \(A\) is “is a brother of”, check whether \(R\) is symmetric or not?
Ans:
Let \(a,\,b\) are two persons in a family, then \(a,\,b \in A.\)
The given relation on set “is a brother of”.
If \(a\) is the brother of \(b,\) then \(b\) is also the brother of \(a.\)
\(R = \left\{ {\left( {a,\,b} \right),\,\left( {b,\,a} \right)} \right\}\)
Hence, the relation \(R\) is symmetric.

Q.2. Find the inverse relation of \(R = \left\{ {\left( {1,\,2} \right),\,\left( {3,\,4} \right),\,\left( {5,\,6} \right)} \right\}.\)
Ans:
Given relation is \(R = \left\{ {\left( {1,\,2} \right),\,\left( {3,\,4} \right),\,\left( {5,\,6} \right)} \right\}.\)
Consider \(R = \left\{ {\left( {a,\,b} \right):\,a \in P,\,b \in Q} \right\}.\) be the relation from set \(P\) to set \(Q,\) then the relation from set \(Q\) to set \(P\) is known as inverse relation, such that \({R^{ – 1}}:Q \to P = \left\{ {\left( {a,\,b} \right):\left( {a,\,b} \right) \in R} \right\}.\)
So, inverse relation is obtained by taking the reverse of the given ordered pairs.
\({R^{ – 1}}:\left\{ {\left( {2,\,1} \right),\,\left( {4,\,3} \right),\,\left( {6,\,5} \right)} \right\}\)

Q.3. Find the identity relation on the set \(P = \left\{ {x,\,y,\,z} \right\}.\)
Ans:
We know that the relation \(\left( I \right)\) is identity, then all the elements of set \(P\) are related with itself, such that for every \(a \in P,\) then \(\left( {a,\,a} \right) \in I.\)
Given set is \(P = \left\{ {x,\,y,\,z} \right\}\)
Then the identity relation is given by \(I = \left\{ {\left( {x,\,x} \right),\,\left( {y,\,y} \right),\,\left( {z,\,z} \right)} \right\}.\)

Q.4. Let \(P = \left\{ {1,\,2,\,3} \right\},\,R\) be a relation defined on set \(P\) as “is greater than” and \(R = \left\{ {\left( {2,\,1} \right),\,\left( {3,\,2} \right),\,\left( {3,\,1} \right)} \right\}.\) Verify \(R\) is transitive.
Ans:
Given set \(P = \left\{ {1,\,2,\,3} \right\},\)
Let \(P = \left\{ {1,\,2,\,3} \right\},\) then we have
\(\left( {b,\,a} \right) = \left( {2,\,1} \right) \to 2\) is greater than \(1.\)
\(\left( {c,\,b} \right) = \left( {3,\,2} \right) \to 3\) is greater than \(2.\)
\(\left( {c,\,a} \right) = \left( {3,\,1} \right) \to 3\) is greater than \(1.\)
Thus in a transitive relation, if \(\left( {x,\,y} \right) \in R,\,\left( {y,\,z} \right) \in R,\) then \(\left( {x,\,z} \right) \in R.\)
So, the relation \(R = \left\{ {\left( {2,\,1} \right),\,\left( {3,\,2} \right),\,\left( {3,\,1} \right)} \right\}\) is transitive.

Q.5. Let set \(A\) contain the bag of red balls. Then the relation \(R\) of getting white balls from the bag of red balls is what type of relation?
Ans:
Given set \(A\) contains the bag contains the red balls. From the bag of red balls, relation \(R\) of getting the white ball is not possible. So, the given relation is an empty relation.

Summary

In this article, we have studied the relation and its definition, which gives the relationship between two sets. We studied the different ways of representing the relation, such as roaster, set-builder form, and arrow diagram method. We discussed the domain and range of the relation.

We also discussed the different types of relations, such as universal relation, empty or void relation, identity relation, inverse relation, reflexive relation, transitive relation, symmetric relation, and equivalence relation, along with the solved examples. In this article, we also studied other types of relations such as one to one relations, one to many relations, many to one relations, and many to many relations.

Frequently Asked Questions (FAQs)

Q.1: What are the four types of relations?
Ans: The four types of relations are:
1. One to one
2. One to many
3. Many to one
4. Many to many

Q.2: What is equivalence relation?
Ans: A relation is said to be equivalence if it is reflexive, transitive, and symmetric.

Q.3: What are the different types of relations?
Ans: The different types of relations are universal relation, empty relation, identity relation, inverse relation, reflexive, transitive, symmetric, and equivalence relation.

Q.4: What is the relation?
Ans: A relation describes the cartesian product of two sets. Relations define the operations performed on the sets.

Q.5: Are all functions are relations?
Ans: Yes. All functions are relations. Functions are special kinds of relations.

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