## Introduction to Relation and Function

Relation and function both are closely related to each other, and to have a clear understanding of them, one must take proper knowledge from the maths experts on our website.

### Relation

A relation is nothing but the connection of two sets by any means. In mathematics, it is a collection of ordered pairs that contain elements from one set to the other set.

A relation is a collection of ordered pairs, which contains an object from one set to the other set. For instance, X and Y are two sets, and ‘a’ is the object from set X and b is the object from set Y, then we can say that the objects are related to each other if the order pairs of (a, b) are in relation.

Consider two arbitrary sets X and Y. The set of all ordered pairs (x,y) where x∈X and y∈Y is called the Cartesian product of X and Y. The product is designated as, read as X cross Y. By deﬁnition,

X х Y = {(x, y)} I x ∈ X and y ∈ Y}

X х Y ≠ Y х X . The Cartesian product deals with ordered pairs, so the order in which the sets are considered is important. Using n(A) for the number of elements in a set A, we have:

n(X х Y) = n(X) х n(Y)

### Function

It is a relation that defines the set of inputs to the set of outputs.

Note that all functions are relations, but not all relations are functions.

The relation that defines the set of input elements to the set of output elements is called a function. Each input element in the set X has exactly one output element in the set Y in a function. A function requires two conditions to be satisfied to qualify as a function:

Every x∈X must be related to y∈Y, i.e., the domain of f must be X and not a subset of X.

There is a requirement of uniqueness, which can be expressed as:

(x,y) ∈f and (x,z) ∈f ⇒ y = z

Sometimes we represent the function with a diagram: f : A⟶B or Af⟶B

Functions are sometimes also called mappings or transformations.

### Difference between Function and Relation in Maths

A relation from a set X to a set Y is any subset of the Cartesian product X×Y.

A relation X to Y is a subset of X Y.

Let there be an X set and a Y set. An ordered pair (x,y) is called a relation in x and y. The first element in an ordered pair is called the domain, and the set of second elements is called the range of the relation.

Let us consider R as a relation from X to Y. Then R is a set of ordered pairs where each ﬁrst element is taken from X and each second element is taken from Y. That is, for each x ∈ X and y ∈ Y, follows exactly one of the following:

x, y R; then“x is R-related to y”, written as xRy.

x, y ∉ R; then “x is not R-related to y”, written as xRy

If R is a relation from a set X to itself, that is, if R is a subset of X2 =X X, we say that R is a relation on X.

### The Difference between a Relation and a Function

To understand the difference between a relationship that is a function and a relation that is not a function. All functions are relations, but not all relations are functions. The difference between a relation and a function is that a relationship can have many outputs for a single input, but a function has a single input for a single output. This is the basic factor to differentiate between relation and function. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

A Relation is a group of ordered pairs of elements. It can be a subset of the Cartesian product. It is a dyadic relation or a two-place relation. Relations are used, so those model concepts are formed. Relations give a sense of meaning like “greater than,” “is equal to,” or even “divides.”

Function pertains to an ordered triple set consisting of X, Y, F. X, where X is the domain, Y is the co-domain, and F is the set of ordered pairs in both “a” and “b.” Each ordered pair contains a primary element from the “A” set. The second element comes from the co-domain, and it goes along with the necessary condition.

In a set B, it pertains to the image of the function. The domain and co-domain are both sets of real numbers. It doesn’t have to be the entire co-domain. It can be known as the range. Relations show the properties of items. In a way, some things can be linked in some way, so that’s why it’s called “relation.” It doesn’t imply that there are no in-betweens that can distinguish between relation and function.

One thing good about it is the binary relation. It has all three sets. It includes the X, Y, and G. X and Y are arbitrary classes, and the “G” would have to be the subset of the Cartesian product, X x Y. They are known as the domain set of departure or even co-domain. G would be understood as a graph.

The function can be an item that takes a mixture of two-argument values that can give a single outcome. There is another difference between relation and function. The function should have a domain that results from the Cartesian product of two or more sets but is not necessary for relations.

### Similarities between Logarithmic and Exponential Functions

Log functions can be written as exponential functions.

Logs of products involve addition and products of exponentials involve addition.

### Benefits of Knowing the Difference

Other than learning the topics, students have to understand the difference between these topics. When you know the difference, it becomes easy to break down the seeds of knowledge and gain the consciousness of tiny topics related to it.

Our subject matter experts offer you a detailed explanation of the topic, Relation and Function, in the online maths class. You can join the maths online class to know more about the relation and function.

## FAQs on Know The Difference Between Relation and Function

**1. What is the importance of Relation and Function?**

Relation and Function plays an important role in mathematics. It helps students maintain a link between any other two entities. In maths, you often come across the relation between numbers. For instance, ‘s’ is greater than ‘d’. Relation and function map the objects of one set to the objects of another set.

**2. Where can I find relevant resources for maths online?**

Vedantu has provided you with different resources to help you ace your exam. Each resource has a definite name and is available to download as per the particular class. You can easily find maths resources in the study material section available on the website, select the class or maths as a subject from the list and enjoy the well of healthy resources that benefits you in achieving your dreams. Download all these resources for free and start preparing.

**3. How to know if a relation is a function?**

It is easy to find if you know the concepts. Our maths experts have already pointed out that a relation is a function only when each element in a domain is with the unique elements of another domain or a set. This is how you identify whether a relation is a function or not.

Always have a note in mind, a function is always a relation, but vice versa is not necessarily true. To know more about the topic, download the detailed notes of the chapter from the Vedantu or use the mobile app to get it directly on the phone.

**4. What are the best textbooks for mathematics on relation and function?**

NCERT textbooks are the best source to study maths, as well as various topics including relations and function. NCERT books cover the CBSE syllabus with thorough explanation, and these textbooks have included various illustrations to explain topics in a better and more fun way. We have provided these textbooks to download for free. NCERT textbooks also help you prepare for competitive exams like engineering entrance exams. Our maths teachers prefer these books because of the easy explanation of complex topics.

**5. What is the Basic Difference Between Relation and Function in Math?**

A relation represents the relationship between the input and output elements of two sets whereas a function represents just one output for each input of two given sets.

**6. What is Domain and Range in Function?**

Domain is a set of all input elements of a set and range is a set of all output elements of a set.

**7. What are the Different Types of Functions in Maths?**

One-to-One or Injective

Onto or Surjective

One-To-One Correspondence or Bijective

Inverse functions

Logarithmic and exponential functions are two special types of functions.

Linear function

Quadratic function

Polynomial function