Relation and Its Types

The relationship between the elements of different pairs of sets is termed as relations between the sets. 

The set consisting of all the x-values is called the domain while the set consisting of all the y-values is called the range.

For example- The following table represents the x-value and y-value in separate columns.

After pairing each pair in the same order as they are listed.

Ordered Pair = {(1 , 6) , (-3 , 2) , (5 , 0) , (-1 , -5) , (4 , 2)}

Domain= {1 , -3 , 5 , -1 , 4}

Range= {6 , 2 , 0 , -5 , 2}

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

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.

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-

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 asymmetric 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.

Representation of Relations

There are two ways by which a relation can be represented-

1. Roster method 

2. Set-builder method

The roster form and set-builder for for a set integers lying between -2 and 3 will be-

Roster form

I= {-1,0,1,2}

Set-builder form

I= {x:x∈I,-2<x<3}

Types of Relations and Relationships

The different types of relations are as follows-

1. Empty Relation- When there are no relations between any elements of a set, the relation is said to be an empty relation. R=

2. Universal Relation- When all the elements in the set are related to each other, the relation is said to be a universal relation. R=A x A

3. Identity Relation- When all the elements of set A are related to itself only, then the relation is said to be an identity relation. I={(a,a),∈A}

4. Inverse Relation- When ordered pairs are obtained by interchanging each ordered pair, the relation is said to be an inverse relation. R-1={(b,a):(a,b)∈R}

5. Reflexive Relation- When all the elements of set A map to themselves, then the relation is said to be a reflexive relation. If A={a,b}, then R{(a,a),(b,b)}

6. Symmetric Relation- When (a,b)R,(b,a)∈R, for all a & b∈A, then the relation is said to be a symmetric relation.

7. Transitive Relation- When (a,b)∈R,(b,c)R, for all a,b,c ∈A, then (a,c)∈R

8. Equivalence Relation- When the relation is reflexive, symmetric and transitive at the same time, then the relation is said to be an equivalence relation.