 # Difference Between Relations and Functions

Relations and functions are both closely related to each other. One needs to have a clear knowledge to understand the concept of relations and functions to be able to differentiate them. In this article we are going to distinguish between relations and functions.

## What is Relation?

• Two or more sets can be related to each other by any means is known as Relation.

• Let us consider an example two set A and set B having m elements and n elements respectively, we can easily have a relation with any ordered pair which shows a relation between the two sets A and B.

### What is Function?

• A function can have the same range mapped as that of in relation, such that a set of inputs is related with exactly one output.

Let us consider an example Set A & Set B are related in a manner that all the elements of Set A are related to exactly one element of Set B or many elements of the given set A are related to one element of given  Set B. Thus this type of relation is known as a function.

We see that a given function cannot have one to Many Relation between the set A and set B.

### What Is A Relation?

• Relation in Mathematics can be defined as a connection between the elements of two or more sets, the sets must be non-empty.

• A relation R is formed by a Cartesian product of subsets.

• For example, let us say that we have two sets then if there is a connection between the elements of two or more non-empty sets then the only relation is established between the elements.

### Relations and Functions Notes:

There are three ways to represent a relation in mathematics.

1. Roster Form - Roster form is basically a representation of a set which lists down all of the elements present in the set and are separated by commas and enclosed within braces.

2. Set- builder Form - A shorthand method is used to write sets and is often used for sets with an infinite number of elements. It is used with different types of numbers, such as integers, real numbers, and so on. The set –builder form is also used to express sets with an interval or an equation.

3. By Arrow Diagram -In the by arrow diagram method, the relation between sets is denoted by drawing arrows from first components to the second components of all the pairs which belong to the relation.

### Different Types of Relations In Mathematics-

There are different types of relations in math which define the connection between the sets.

There are eight types of relations.

1. Empty Relation -We can write an empty Relation R = ∅.

1. Universal Relation - Universal Relation can also be known as a Full relation as every element of set A is related to every element in B. An empty and universal relation can also be named as a trivial relation.

1. Identity Relation - A relation is called an identity relation if every element of set A is related to itself only.

1. Inverse Relation - Suppose we have a relation R from set A to set B, R∈ A×B. Then the inverse relation of R can be written as R -1 = {(b, a) : (a, b) ∈ R}.

1. Reflexive Relation - If every element of set A maps for itself, then set A is known as a reflexive relation.

1. Symmetric Relation - A relation R on a set A is known as a symmetric relation if (a, b) ∈R then (b, a) ∈R, such that for all a and b ∈A.

1. Transitive Relation - A relation R in a set A is said to be transitive if (a, b) ∈R , (b, c) ∈R , then (a, c) ∈R such that for all a, b, c ∈A.

1. Equivalence Relation -A relation is said to be an equivalence relation if (if and only if) it is Transitive, Symmetric, and Reflexive.

NOTE: All functions are relations, but not vice versa.

What is a Function?

• A function is a relation that says that there should be only one output for each input.

• In simpler words, we can say that it is a special kind of relation or a set of ordered pairs that follow a rule that every value of x should be associated to only one value of y. This is known as a Function.

### Types of Functions-

In terms of relations, the types of functions can be defined as:

Here are Relations and Functions class 12 notes.

1. One to One Function:

• Let there be a function f: A → B is said to be One to One if for each value of A there is a distinct value of B.

• The one to one function is also known as the Injective function.

2. Many to One Function:

• A many to one function is one which maps two or more elements of A to the same element of set B.

3. Onto Function :

• A function for which every element of set B there is pre- image in set A is known as Onto Function

• The onto function is also known as Subjective function.

4. One-one and Onto Function:

• The function f matches with each element of A with a discrete element of B and every element of B has a pre- image in A.

• The one-one and onto function is also known as Bijective function.

### Some Special Functions in Algebra-

1. Constant Function- We can write a constant function as f(x) = c.

2. Identity Function- We can write a constant function as f(x) =x.

3. Absolute Value Function- We can write an absolute value function as f(x) =|x|.

4. Inverse Functions- We can write an inverse function as f -1 (x).

5. Linear Functions- We can write a linear function as f(x ) = mx+c.

Let’s know the difference between Relations and Functions are –

## Difference between Relations and Functions:

 Relations Functions A relation is defined as a relationship between sets of values. Or, it is a subset of the Cartesian product. A function is defined as a relation in which there is only one output for each input. A relation is generally denoted by “R” A function is generally denoted by “F” or “f”. We can say that every relation is not a function. We can say that every function in Mathematics is a relation.

The above table shows the difference between relations and functions in Mathematics.

### Questions to be solved:

Question 1) What is the sum of two functions given below:

f(x) = 2x+3 , g(x) = 4x+4

Answer) From the Relations and functions class 12 explanation,

(f+g)(x) = 2x+3+4x+4

=6x+7