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Types Of Relations
Types of relations are based on the linking of the elements of one set with the elements of another set. The types of relations are based on the domain and range of the elements of the two sets. The different types of relations are empty relation, universal relation, reflexive relation, symmetric relation, and transitive relation.
An equivalence relation is a relation formed only if it is a reflexive relation, symmetric relation, and transitive relation. Let us learn more about the different types of relations, with the help of examples, FAQs.
1.  What Are The Types Of Relations? 
2.  Types Of Relations 
3.  Examples on Types Of Relations 
4.  Practice Questions 
5.  FAQs On Types Of Relations 
What Are The Types Of Relations?
Types of relations are based on the linking of elements of a set with its own elements, or with the elements of another set. Some of the important types of relations are as follows.
 Empty relation
 Universal relation
 Identity relation
 Reflexive relation
 Symmetric relation
 Transitive relation
 Equivalence relation
 Antisymmetric relation
 inverse relation
Let us recall what is a relation. A relation is a form of connection between the elements of one set and the elements of another set. The relation is governed by a function. A relation is between the elements of one set and the elements of another set.
A function defines the relation between two sets. For a function f(x) = x^{2}, the relation R between the two sets A and B is such that R: A → B = {(1, 1), (2, 4), (3, 9), (4, 16), (5, 25), (6, 36)}. The relation is generally the subset of the cartesian product between two sets. Further, a relation could be represented in a set builder form or roaster form as R = {(x, y):y = x^{2}, x ∈ A, and y ∈ B }. For the given sets, based on the rules which are termed as functions, the elements are related and are expressed as relations. For the cartesian product of two sets, the elements are paired and represented, and each element of one set is related to the element of another set.
Types Of Relations
There are basically 9 types of relations: empty relation, universal relation, identity relation, reflective relation, symmetric relation, transitive relation, equivalence relation, antisymmetric relation, and inverse relation. Each of these is defined (over a set A) as follows.
Empty Relation
A relation R on a set A is said to be an empty relation if no element of set A is related to any other element of set A. R = φ ⊂ A × A. A relation between two sets is defined such that no element of one set is linked with any element of another set.
Example: If A is the set of students of grade 8 of a boys school then the relation R = { (a, b)  a and b are sisters } is an empty relation.
Universal Relation
A relation R is said to be a universal relation if every element of set A is related to every other element of set A. i.e., R = A × A.
Example: If A is the set of students of grade 6 of a school then the relation R = { (a, b)  the difference in heights of a and b is less than 3 feet} is a universal relation.
Identity Relation
A relation R is said to be an identity relation if it contains only the ordered pairs where every element of set A is related to ONLY itself. i.e., R = { (a, a) }.
Example: If A = {1, 2, 3} then R = { (1, 1) (2, 2) (3, 3) } is the identity relationship.
Reflexive Relation
A relation R such that every element a ∈ A, is such that every element is related to itself. (a, a) ∈ R. The difference between reflexive relation and identity relation is that, in the identity relation, it should contain only the ordered pairs of the form (a, a), whereas, in reflexive relation, it may contain any other ordered pairs along with the ordered pairs of the form (a, a).
Example: N is the set of all natural numbers and the relation R = { (a, b)  a = b} is a reflexive relation.
Symmetric Relation
A relation R on a set A such that a_{1}, a_{2} are elements of A, and if (a_{1}, a_{2}) ∈ R, and also if (a_{2}, a_{1}) ∈ R, then the relation R is a symmetric relation. A reallife example of this is, if John is a brother of Sam, then Sam can also be said to be a brother of John. The following is a mathrelated example of a symmetric relation.
Example: N is the set of all natural numbers and the relation R = { (a, b)  a = b} is a symmetric relation because whenever a = b, then it obviously means that b = a.
Transitive Relation
A relation R on a set A such that the elements a_{1}, a_{2}, a_{3} belongs to set A, such that (a_{1}, a_{2}) ∈ R, (a_{2}, a_{3}) ∈ R, then it implies that (a_{1}, a_{3}) ∈ R, and the relation R is a transitive relation. This can also be understood with a simple example, If P, Q, R are three persons such that P is a friend of Q, Q is a friend of R, then P is said to be a friend of R. The following is a mathrelated example of a transitive relation.
Example: N is the set of all natural numbers and the relation R = { (a, b)  a = b} is a transitive relation because whenever a = b and b = c then it obviously means that a = c.
Equivalence Relation
A relation R on a set A, if it is a reflexive, symmetric, and transitive relation, then it is called an equivalence relation.
Example: We have already seen that the relation R = { (a, b)  a = b} on the set of natural numbers is reflexive, symmetric, and transitive and hence it is an equivalence relation.
Antisymmetric Relation
A relation R on a set A is said to be antisymmetric if the following condition is satisfied: "whenever (a, b) and (b, a) are elements of R then a = b".
Example: The relation R = { (a, b)  a ≤ b} on the set of whole numbers is antisymmetric because whenever a ≤ b and b ≤ a, then a = b.
inverse Relation
The inverse relation of a relation R is denoted by R^{1} and is obtained by interchanging the elements of each ordered pair of R.
Example: If R = { (1, 2) (3, 5) (5, 7) } then R^{1} = { (2, 1) (5, 3) (7, 5) }.
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Examples on Types Of Relations

Example 1: For a set A representing the boys in a school, a relation R_{1} exists such that R_{1} = {(a, b): a is the sister of b}, and another relation R_{2} = {(a, b): the difference between the heights of two students is less than 2 meters}. Find the types of relations in each of the given cases.
Solution:
The given relation is R_{1} = {(a, b): a is the sister of b}. Since it is a boy's school and we do not have any girls, this relation does not exist. Hence it is an empty relation and R_{1} = φ.
Now, the second relation is R_{2} = {(a, b): the difference between the heights of two students is less than 2 meters}. Since the difference between every two students is always less than 2 meters, this relation exists, and it contains all ordered pairs of A × A. Hence R is a universal relation.
Answer: R_{1} is empty relation and R_{2} is universal relation.

Example 2: For A to be the set of triangles in a plane having a relation R such that R = {(A_{1}, A_{2})  A_{1} is congruent to A_{2}}. Show that the type of relation R is an equivalence relation.
Solution:
Here R is a reflexive relation since each triangle is congruent to itself. Further whenever (A_{1}, A_{2}) ∈ R (means A_{1} is congruent to A_{2}), we can say that (A_{2}, A_{1}) ∈ R (means A_{2} is congruent to A_{1}). From this, we can say that the relation R is symmetric. Finally, when (A_{1}, A_{2}) ∈ R, (A_{2}, A_{3}) ∈ R, (i.e., when A_{1} is congruent to A_{2}, and A_{2} is congruent to A_{3}), we have A_{1} is congruent to A_{3}, i.e., (A_{1}, A_{3}) ∈ R. Therefore R is a transitive relation.
Further, since it is a reflexive relation, symmetric relation, and transitive relation, it can be called an equivalence relation.
Answer: R is an equivalence relation.

Example 3: If R = { (1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} is a relation on the set A = {1, 2, 3} then what type of relation R is? a) Reflexive b) Symmetric c)Transitive. Select all options that apply.
Solution:
Since (a, a) ∈ R for every a ∈ A, R is reflexive.
(1, 2) ∈ R but (2, 1) does not belong to R. So R is NOT symmetric.
(1, 2) and (2, 3) ∈ A but (1, 3) does not belong to R. So R is NOT transitive.
Thus, the type of relation of R is just reflexive but neither symmetric nor transitive.
Answer: a) Reflexive.
FAQs on Types Of Relations
What Are 6 Types Of Relations?
The different types of relations are based on the domain and range of the relations. The 6 important types of relations are as follows.
 Empty relation
 Universal relation
 Reflexive relation
 Symmetric relation
 Transitive relation
 Equivalence relation
What Are Relations?
Relations are used to link the elements of one set with the elements of another set. The functions define how the relations are formed between the elements of the two sets. The relations can be termed as different relations, based on the domain and range of the relation.
What is An Equivalence Relation in Types Of Relations?
An equivalence relation is a type of relationship which satisfies the conditions of reflexive relation, symmetric relation, and transitive relation.
How To Write Types Of Relations?
The different types of relations are written in set builder form or in roster form. in setbuilder form a mathematical or English statement is given to represent the elements of a relation, and in roster form, the elements are written individually in brackets. The example of setbuilder form is R = {(x, y), x = 2y, and x, y, ∈ R}, and an example of roster form is R = {(1, 3), (1, 5), (2, 6), (2, 8)}.
What is Identity Relation in Types Of Relations?
The relation in which every element of a set is related to itself is called an identity relation. The relation R across a set such that every element a ∈ A, is related to itself. and (a, a) ∈ R. The identity relationship is also sometimes referred to as a reflexive relationship.
What is Universal Relation in Types Of Relations?
The type of relations in which every element of a given set is related to every element of another set is called a universal relation. If there are n elements in set A, and if there are m elements in set B, then the number of elements in the universal relation across A and B is m × n.
What is Void Type Of Relations?
The void type of relations is also called an empty relation. In this, none of the elements of a given set is related to any other element of another set. The cardinal number of a void relation is zero.
What is The Difference Between Relations And Functions?
The relations relates every element of a given set to every other element of another set. The function is a rule which helps to connect the elements of the two sets. The function defines the relations across the sets, and the function is a subset of a relation.
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