Relation and Its Types

Types of Relation in Maths

We often speak of relations between any two or more objects.

What are Relations?

  • Relations in Mathematics are one of the most important topics of set theory.

  • Relations and functions generally define the different operations performed on sets.

  • Relation in Mathematics can be defined as a connection between the elements of two or more sets, the sets must be non-empty.

  • A relation R is formed by a Cartesian product of subsets.

  • For example, let us say that we have two sets then if there is a connection between the elements of two or more non-empty sets then only a relation is established between the elements.


Representation of a Relation Math –

There are three ways to represent a relation in mathematics. (image will be updated soon)


Here’s a Little Description of all the Three Ways of Representation of a Relation Math.


1. Roster Form

Roster form is basically a representation of a set which lists down all of the elements present in the set and are separated by commas and enclosed within braces. For example, 

Let us take set A = {1,2,3,4,5} and another set B= {1,2,3,4,5…………20}

And let us assume R be a relation ‘has its square, from set A to set B, then 

R= {(1,1), (2,4), (3,9), (4,16)}


2. Set- builder form 

A shorthand method which is used to write sets and is often used for sets with an infinite number of elements. It is used with different types of numbers, such as integers, real numbers and so on. The set - builder form is also used to express sets with an interval or an equation.

Suppose we have a given set:

{3,6,9,12}

 

Let’s write the given set in set-builder form,

3×1= 3

3×2= 6

3×3= 9

3×4= 12

{x: x = 3n, n∈N and 1n≤4}



  3. By arrow diagram

In the by arrow diagram method, the relation between sets is denoted by drawing arrows from first components to the second components of all the pairs which belong to the relation.(image will be updated soon)



Different Types of Relations in Mathematics-

There are different types of relations in math which define the connection between the sets. There are eight types of relations in mathematics,

Here are the Types of Relations in Mathematics-

Empty Relation

Reflexive Relation

Transitive Relation

Anti-symmetric Relation

Universal Relation

Inverse Relation

Equivalence Relation


1. What is an Empty Relation?

  • If no element of set X is related or mapped to any of the elements of set Y, then the relation is known as an Empty Relation.

  • An empty relation is also known as a void relation.

  • We can write an empty Relation R = ø.

  • Let us take an example if suppose we have a set X consisting of exactly 200 elephants in a farm. Are there any chances of finding a relation of getting a rabbit in the poultry farm? No! The relation R is a void or empty relation since there are only 200 elephants and no rabbits.

2. What is a Universal Relation?

  • A relation R in a set, let’s say we have a universal Relation A because, in this relation, each element of A is related to every element of A, such that the Relation R = A×A.

  • Universal Relation can also be known as a Full relation as every element of set A is related to every element in B.

  • Let’s take an example, suppose we have set A which consists of all the natural numbers and set B which consists of all whole numbers. Then the relation between set A and set B is universal since every element of set A is in set B.

  • An empty and universal relation can also be known as a trivial relation.

3. What is an Identity Relation?

  • A relation is called an identity relation if every element of set A is related to itself only.

  • It is represented as I = {(A, A), ∈ a}.

  • For example, when we throw two dice, the number of possible outcomes we get is equal to 36: (1,1), (1,2) ……. (6,6). Now, let’s define a function R: {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}, such a relation is known as an identity relation.

4. What is an Inverse Relation?

  • Suppose we have a relation R from set A to set B, R∈ A×B. Then the inverse relation of R can be written as R-1 = {(b, a) :(a, b) ∈ R}.

  • Let us take an example of throwing two dice if relation R = {(1,2), (2,3)} then the inverse relation R-1 can be written as R-1= {(2,1), (3,2)}.

  • Here, the domain of R is the range of the inverse function R-1 and vice versa.

5. What is a Reflexive Relation?

  • If every element of set A maps for itself, then set A is known as a reflexive relation.

  • It is represented as a∈ A, (a,a) ∈ R . 

6. What is a Symmetric Relation?

  • A relation R on a set A is known as a symmetric relation if (a, b) ∈R then (b, a) ∈R , such that for all a and b ∈A.

7. What is a Transitive Relation?

  • A relation R in a set A is said to be transitive if (a, b) ∈R , (b, c) ∈R , then (a, c) ∈R such that for all a, b, c ∈A.

8. What is an Equivalence Relation?

  • A relation is said to be an equivalence relation if (if and only if) it is Transitive, Symmetric, and Reflexive.


How to Convert a Relation into a Function?

A special kind of relation (a set of ordered pairs) which follows a rule that every value of X must be associated with only one value of Y is known as a Function.

 

Questions to be Solved-

Question 1) Three friends X, Y, and Z live in the same society close to each other at a distance of 4 km from each other. If we define a relation R between the distances of each of their houses. Can R be known as an equivalence relation?

Solution) We know that for an equivalence Relation, R must be reflexive, symmetric, and transitive.

R is not reflexive as X cannot be at a distance of 4 km away from itself. The relation, R can be said as symmetric as the distance between X and Y is 4 km which is the same as the distance between Y and X. R is said to be transitive as the distance between X and Y is 4 km, the distance between Y and Z is also 4 km and the distance between X and Z is also 4 km.

Therefore, this relation is not an equivalence relation.

FAQ (Frequently Asked Questions)

Question 1) What are the Types of Relations Math?

Answer) There are many different types of relations in mathematics, 

Here they are- 

  • Empty Relation

  • Reflexive Relation

  • Symmetric Relation

  • Transitive Relation

  • Anti-symmetric Relation

  • Universal Relation

  • Inverse Relation

  • Equivalence Relation

Question 2) What are the Different Types of Relation and Functions?

Answer) The different types of relation and functions are-

Relations Maths – 

  • Empty Relation

  • Reflexive Relation

  • Symmetric Relation

  • Transitive Relation

  • Anti-symmetric Relation

  • Universal Relation

  • Inverse Relation

  • Equivalence Relation 

Functions –

  • One to one function

  • Many to one function

  • Many to Many function

  • One to many function