Relations and Functions: Definition, Types, and Examples

Comparing two numbers is finding the relation between them. Relations and Functions between any two entities gives us the link between them. Special relations in Mathematics obtain a precise correspondence between any two entities. These special relations are known as functions.

This article will discuss the concept of relations and functions in detail.

What is a Set?

A collection of well-defined or distinct elements is called a set. Such a set in mathematics is written enclosed in curly braces {}. The elements of a set can be anything ranging from numbers to alphabets to people. Two sets that contain the same elements irrespective of their positions are said to be similar.

What are Ordered Pair numbers?

Ordered pair numbers represent inputs and outputs and go hand in hand. They are represented in parenthesis as (INPUT, OUTPUT). 

What are Relations and Functions?

Relations and functions are used to portray the correspondence between two sets such that they are in the ordered pair form.

Consider an ordered pair (INPUT, OUTPUT) to understand the difference between relation and function. Then the relation is all about the relationship between the input and output, whereas the function determines one output for every input.

Definitions for Relations and Functions

Relations: Any set of ordered pair numbers in mathematics is a relation. Relation, in other words, is a bunch of ordered pairs. Relations and functions can be represented in various forms such as set builder, graphical, roster, and tabular. Let us represent a function f: A = {2,3,4} → B = {4,9,16} in various forms

Functions: A function is defined as a rule that relates every element of the domain to every element of the range. Here, the domain and range both are sets. For example, y = x+2 and y = 2x – 1 are functions as every input x gives an output y. 

Set builder form: {(x, y): f(x) = y2, x ∈ A, y ∈ B}

Roster form: {(2,4),(3,9),(4,16)}

Tabular form: 

X	Y
2	4
3	9
4	16

Terms associated with Relations and Functions

To better understand the concepts related to relations and functions, go through a few associated terms.

• Cartesian product: For two non-empty sets, P and Q, the cartesian product is the set of all ordered pairs of elements from P and Q.
• Domain: A set in which the elements are the first elements of the ordered pair in a relation R from a set A to B is known as the domain of the relation R. It is the set of inputs.
• Range: A set in which the elements are the second element of the ordered pair in a relation R from a set A to B is known as the range of the relation R. It is the set of outputs.
• Codomain: A codomain is the whole set Q related to R from set P to Q.

Types of Relations and Functions

The properties of relations and functions make them special. There are many such special functions and relations. Let us study these types of relations and functions now.

Relations

• Empty relation: A relation that has no elements or no element of set A is linked to any other element is known as an empty set. It is denoted as R = ∅.
• Universal relation: If every element of set A is linked with every element of set A, the relation R is called universal relation. It debuted as R = A×A.
• Identity relation: If each element of the relation R of the set is related to itself, it is termed an identity relation. It is denoted as R = {(a, a) : for all a ∈ A}
• Inverse relation: When relation R is defined from set P to Q, the inverse R^-1 is the relation from set Q to P. 
• Reflexive relation: If every element of set A maps to itself, it is called reflexive relation. It can be represented as a ∈ A, (a, a) ∈ R.
• Symmetric relation: A relation is said to be symmetric if and only if, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R.
• Transitive relation: If (a, b)∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R for a, b, c ∈ A, then the relation R is said to be transitive.
• Equivalence relation: If a relation R defined on set A is reflexive, symmetric, and transitive all at the same time, it is termed equivalence.

Types of functions

• One-to-one function: A function f: A →B is said to be one-to-one if all the elements in A can be mapped with the elements in B. The other name for this type is the injective function.
• Onto function: In a function f: A →B, if all the elements of B are images of some elements of A, the function is termed as an onto function.
• Many to one function: When more than one element of A is connected to the same element of B in a function f: A →B, it is said to be many to one function.
• Bijective function: A function that is both one-to-one and onto at the same time is called a bijective function.
• Constant function: A constant function f(x) = K, where K is a real number. For different values of the domain, the same range value for K is obtained.
• Identity function: A function where each element of set B gives the image of itself as the same element is called an identity function.
• There are a few algebraic functions such as linear function, objective function, quadratic function, cubic function and polynomial function. These functions are based on the degree of the algebraic functions.

What is the difference between a Relation and a Function?

Relations and functions are quite closely related to one another. A group of ordered pairs from one set of objects to another set of objects is a relation, whereas a relation that connects one set of inputs to another set of outputs is a function. Following are a few differences between relations and functions.

Consider a set (x, y), a collection of ordered pairs where x belongs to set A while y belongs to set B. Here, x is related to y. A group of such sets is called a relation. Only one x can be related to some y in a function. Here, x is from set A and y is from set B. 

In a function, one input can connect with only one output, while this is not the case with a relation. Thus, all the functions are relations. However, all the relations are not functions. A function cannot have a one-many relationship. However, it can always have many one relationships.

How to recognise if a Relation is a Function?

We have seen various relations and functions examples. Now let us see how to determine if a relation is a function. First of all, examine the input values. Then observe the output values. If all the input values are unique, then the relation is a function, and if they are repeating, the relation is not a function.

Relations and Functions examples 

Relation example

In the below relation {(-2, 3), {4, 5), (6, -5), (-2, 3)}, find the domain and range. Here the domain is {-2,4,6} and the range is {-5,3,5}.

Function example

Consider the below examples of a function, A = {(2, 5), (2, 5), (3, -7), (4, -8), (4, -8)}. Generally, it is not a function if the input values or X- values are repeated. However, in this case, the input values are repeating along with the associated output values or Y-values. Therefore, it is a function.

Another Examples for Relations and Functions 

Example 1: Given three relations R, S, T from A = {x, y, z} to B = {u, v, w} defined as: 1) R = {(x, u), (z, v)}, 2) S =, 3) T = {(x, u), (x, v), (z, w)}. Using the definition of relations and functions, determine which of the given relations is/are function(s). 

Solution: allow us to check each part one by one. 

1) For R = {(x, u), (z, v)}, each element of A isn’t mapped to part of B which violates the definition of a function. Hence, R isn’t a function. 

2) For S = {(x, u), (y, v), (z, w)}, each element of A is mapped to a singular element of B which satisfies the definition of a function. Hence, S could be a function. 

3) For T = {(x, u), (x, v), (z, w)}, element x of A is mapped to 2 different elements of B which violates the definition of a function. Hence, T isn’t a function. 

Answer: S = {(x, u), (y, v), (z, w)} may be a function. 

Example 2: Define a relation R from A to A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y = x + 1}. Determine the domain, codomain and range of R. 

Solution: we are able to see that A = {1, 2, 3, 4, 5, 6} is that the domain and codomain of R. 

To determine the range, we determine the values of y for every value of x, that is, when x = 1, 2, 3, 4, 5, 6 

x = 1, y = 1 + 1 = 2; 
x = 2, y = 2 + 1 = 3; 
x = 3, y = 3 + 1 = 4; 
x = 4, y = 4 + 1 = 5; 
x = 5, y = 5 + 1 = 6; 
x = 6, y = 6 + 1 = 7. 
Since 7 doesn’t belong to A and therefore the relation R is defined on A, hence, x = 6 has no image in a very. 

Therefore, range of R = (2,3,4,5,6) 

Answer: Domain =Codomain = (1,2,3,4,5,6), Range = (2,3,4,5,6) 

Relations and functions worksheet

A binary relation between two sets such that every element from the first set called the domain associates with each element of the second set called the range is called a function. A relation is a bunch of ordered pairs from two sets containing objects from both sets. To understand the concepts better, it is necessary to go through various relations and function examples. 

Practicing ample relations and functions examples will give the students an idea about the key difference between relation and function. Solving Relations and functions worksheets are a great way to do so. 

You will find various relations and functions examples in day-to-day life, such as money being a function of time of temperature and location being a function of time. It is, therefore, necessary to practice the relations and functional worksheets to get a clear idea about these real-life applications.

Solving these worksheets will also help students understand the concepts related to variable functions, calculus, and probability. It will also help develop the reasoning capability. You can find a wide range of relations and functions worksheets on the internet which can be downloaded for free.

Make sure you practice the above-mentioned theory and concepts to enhance your understanding of the concept of functions and relations. 

Frequently Asked Questions

1. Does the relation define the function?

Ans. The relation between a function and its graph can be defined as the set of ordered pairs (x,y) where y is a real number and x is a variable that represents a point on the graph.

2. What are relations in Math?

Ans. Relations in Math are a way to describe how two or more numbers are related.For example, you might say that the relation between x and y is “x+y = 10” or that the relation between x and y is “x=y-1”.

3. How to determine if a relation is a function?

Ans. Determining whether a relation is a function can be done in two steps:

1. Determine whether the domain and range of the relation are finite, or countable
2. If they are finite, determine whether each member in the domain is paired with at least one member of the range

4. Why is it important the basic concept of relations and functions?

Ans. The basic concept of relations and functions is important because it helps us to understand the relationships between things. It allows us to see how objects are connected, and it helps us to predict what will happen in the future. Relationships can be used for predictions about the world around us.

5. Why all functions are relations but not all relations are functions?

Ans. functions are relations that always satisfy the property that their range is a subset of their domain. Relations are a broader category than functions and include many types of mathematical relationships that do not necessarily satisfy the above condition.