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

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

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.

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.

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.

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.

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 .

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.

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.

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

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.

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