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Relations and Functions: Definition, Types, and Examples

relations and functions

Comparing two numbers is finding the relation between them. Relation 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 relation and function

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: 

XY
24
39
416

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 functions and relations

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 example, 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.

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. 

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