relation & function | revision notes 2025 board exam | one one onto function #class12

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Relations and Functions
“Relations and Functions” are the most important topics in algebra. Relations and functions – these are the two different words having different meanings mathematically. You might get confused about their difference. Before we go deeper, let’s understand the difference between both with a simple example.

An ordered pair is represented as (INPUT, OUTPUT):

The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation which derives one OUTPUT for each given INPUT.

Note: All functions are relations, but not all relations are functions.

Relations and Functions

In this section, you will find the basics of the topic – definition of functions and relations, special functions, different types of relations and some of the solved examples.

What is a Function?
A function is a relation which describes that there should be only one output for each input (or) we can say that a special kind of relation (a set of ordered pairs), which follows a rule i.e., every X-value should be associated with only one y-value is called a function.

Domain It is a collection of the first values in the ordered pair (Set of all input (x) values).
Range It is a collection of the second values in the ordered pair (Set of all output (y) values).
Example:

In the relation, {(-2, 3), {4, 5), (6, -5), (-2, 3)},

The domain is {-2, 4, 6} and range is {-5, 3, 5}.

Note: Don’t consider duplicates while writing the domain and range and also write it in increasing order.

Types of Functions
In terms of relations, we can define the types of functions as:

One to one function or Injective function: A function f: P → Q is said to be one to one if for each element of P there is a distinct element of Q.
Many to one function: A function which maps two or more elements of P to the same element of set Q.
Onto Function or Surjective function: A function for which every element of set Q there is pre-image in set P
One-one correspondence or Bijective function: The function f matches with each element of P with a discrete element of Q and every element of Q has a pre-image in P.

One To One Function
Onto Function
Bijective Function
Special Functions in Algebra

Constant Function
Identity Function
Linear Function
Absolute Value Function
Inverse Functions
What is the Relation?
It is a subset of the Cartesian product. Or simply, a bunch of points (ordered pairs). In other words, the relation between the two sets is defined as the collection of the ordered pair, in which the ordered pair is formed by the object from each set.

Example: {(-2, 1), (4, 3), (7, -3)}, usually written in set notation form with curly brackets.

Different types of relations are as follows:

Empty Relations
Universal Relations
Identity Relations
Inverse Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Empty Relation
When there’s no element of set X is related or mapped to any element of X, then the relation R in A is an empty relation, and also called the void relation, i.e R= ∅.

For example, if there are 100 mangoes in the fruit basket. There’s no possibility of finding a relation R of getting any apple in the basket. So, R is Void as it has 100 mangoes and no apples.

Universal relation
R is a relation in a set, let’s say A is a universal relation because, in this full relation, every element of A is related to every element of A. i.e R = A × A.

It’s a full relation as every element of Set A is in Set B.

Identity Relation
If every element of set A is related to itself only, it is called Identity relation.

I={(A, A), ∈ a}.

For Example,

When we throw a dice, the total number of possible outcomes is 36. I.e (1, 1) (1, 2), (1, 3)…..(6, 6). From these, if we consider the relation (1, 1), (2, 2), (3, 3) (4, 4) (5, 5) (6, 6), it is an identity relation.

Inverse Relation
If R is a relation from set A to set B i.e., R ∈ A X B. The relation R-1= {(b,a):(a,b) ∈ R}.

For example,

If you throw two dice if R = {(1, 2) (2, 3)}, R-1 = {(2, 1) (3, 2)}. Here the domain is the range R-1 and vice versa.

Reflexive Relation
A relation is a reflexive relation iIf every element of set A maps to itself, i.e for every a ∈ A, (a, a) ∈ R.

Symmetric Relation
A symmetric relation is a relation R on a set A if (a, b) ∈ R then (b, a) ∈ R, for all a & b ∈ A.

Transitive Relation
If (a, b) ∈ R, (b, c) ∈ R, then (a, c) ∈ R, for all a,b,c ∈ A and this relation in set A is transitive.

Equivalence Relation
If a relation is reflexive, symmetric and transitive, then the relation is called an equivalence relation.
How to Convert a Relation into a Function?
A special kind of relation (a set of ordered pairs) which follows a rule i.e every X-value should be associated with only one y-value, then the relation is called a function

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