Relations and Functions


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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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,  A = {2, 5, 7, 8, 9, 10, 13} and B = {1, 2, 3, 4, 5}. The Cartesian product A × B has 30 ordered pairs such as A × B = {(2, 3), (2, 5)…(10, 12)}. From this, we can obtain a subset of A × 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 ∈ A and y ∈ B}

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

relations      (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, x ∈ A and y ∈ B}.
  • In roster form, R = {(9, 3), (9, -3), (16, 4), (16, -4), (25, 5), (25, -5)}.


 Download Relations Cheat Sheet PDF by clicking on Download button below

relation cheat sheet

relation cheat sheet


  • 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 A × 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. Range ⊆ 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, then the number of ordered pair in the Cartesian product will be n(A × B) = hk. The total number of relations is 2hk.

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 = {(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 × Y. Most importantly, it encodes the information of relation.

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2 responses to “Relations”

  1. Al quba says:

    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.

    • Eli Simeon says:

      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.

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