Often in maths, we come across with the word ‘relation’. Generally speaking, by relation we usually understand some connection between the two living or non-living things. Like the relations of mother-daughter, brother-sister, teacher-student etc. We are quite familiar with these relations. Today, we will learn about a new concept of “relations” in maths. Mathematically, we can also define a relationship between the two elements of a set. Let’s begin!

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## Relation

Given any two non-empty sets A and B, A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B.

Consider an example of two sets, Â A = {2, 5, 7, 8, 9, 10, 13} and B = {1, 2, 3, 4, 5}. The Cartesian product A Ã— BÂ has 30 ordered pairs such asÂ A Ã— B =Â {(2, 3), (2, 5)â€¦(10, 12)}. From this, we can obtain a subset ofÂ A Ã— B, by introducing a relation R between the first element and the second element of the ordered pair (x, y) as

R = {(x, y): x = 4y – 3, x âˆˆ A and yÂ âˆˆ B}

Then, R = {(5, 2), (9, 3), (13, 4)}.

Â Â Â (Arrow representation of the Relation R)

## Representation of Relation

A relation is represented either by Roster method or by Set-builder method. Consider an example of two sets A = {9, 16, 25} and B = {5, 4, 3, -3, -4, -5}. The relation is that the elements of A are the square of the elements of B.

- In set-builder form, R = {(x, y): x is the square of y,Â x âˆˆ A and yÂ âˆˆ B}.
- In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.

**Â Download Relations Cheat Sheet PDF by clicking on Download button below**

## Terminologies

- Before getting into details, let us get familiar with a few terms:
- Image: Suppose we are looking in a mirror. What do we see? An image or reflection. Similarly, for any ordered pairs, in any Cartesian product (say AÂ Ã— B), the second element is called the image of the first element.
- Domain: The set of all first elements of the ordered pairs in a relation R from a set A to a set B.
- Range: The set of all second elements in a relation R from a set A to a set B.
- Codomain: The whole set B. RangeÂ âŠ†Â Codomain.

## Total Number of Relations

For two non-empty set, A and B. If the number of elements in A is h i.e., n(A) = h & that of B is k i.e., n(B) = k, thenÂ the number of ordered pair in the Cartesian product will be n(AÂ Ã— B) = hk.Â The total number of relations is 2^{hk}.

## Solved Examples for You

**Question 1: Let A = {5, 6, 7, 8, 9, 10} and B = {7, 8, 9, 10, 11, 13}. Define a relation R from A to B by**

**R = {(x, y): y = x + 2}. Write down the domain, codomain and range of R.**

**Answer :**Here, R**Â **= {(5, 7), (6, 8), (7, 9), (8, 10), (9, 11)}.

Domain = {5, 6, 7, 8, 9}

Range = {7, 8, 9, 10, 11}

Co-domain = {7, 8, 9, 10, 11, 13}.

**Question 2: What is the relation in math?**

**Answer: **It refers to a relationship between sets of values. In addition, in mathematics, the relationship is between the x-values and y-values of ordered pairs. Moreover, the set of x-values is known as the domain, and the set of all y-values is known as the range. Furthermore, the brackets show that the values form a set.

**Question 3: What makes a relation a function?**

**Answer:** Generally, relationship refers to a set X to a set Y is called a function of each element of X is related to exactly one element in Y. As given an element x in X, there is only one element in Y that is related to then this is a function as each element from X is related to only one element in Y.

**Question 4: What are the properties of relation?**

**Answer: **The properties of relation are reflexivity, transitivity, symmetry, and connectedness. Besides, we consider some properties of binary relations.

**Question 5: Are relations sets?**

**Answer:** A binary relation over two sets X and Y is a set of ordered pairs (x, y) that consist of elements x in X and y in Y. As if it is a subset of the Cartesian product X Ã— Y. Most importantly, it encodes the information of relation.

The example for onto function doesn’t qualify as a function in the first place. Does it??

It is a relation but not a function because a single element in the domain has been mapped to two elements in the co domain. Isn’t it??

Please tell me if I’m correct or not.

It is really confusing.

An onto function exists if and only the co-domain is equal to the range that is every element in set A (the domain) is mapped to every element in set B (the range/codomain) i.e without leaving out any element. Irrespective of whether it is a one to one mapping or not. Therefore it is a function. Put simply, take set A as a set of sons and set B as a set of fathers, a function requires that every son has one father (which is normal) yet every father can have more than one son(which is also normal) so if one element in set A maps to more than one element in set B it is not a function (and we will need DNA test to know who really is the father) but if more than one element in set A maps to one element in set B it is still a function (the elements are just brothers). What and onto function requires is that every father has a son. Sorry if I made it a bit complicated I feel that if I continue I might make it worse just it study a bit more and from different sources, videos or books and you will understand it better.