Please make notes on Relation and functions for class 10 and be marked the brainliest
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Answer:
Relations
Relation: Meaning
Let A and B be two sets, then a relation R from A to B is a subset of A ×B.
Thus, R is a relation from A to B . ⇔ R ⊆ A × B
Recall that A × B is a set of all ordered pairs whose first member is from the set A and second member is from B i.e.,
A × B = {(x,y) : x ∈ A and y ∈ B }
If R is a relation from a non-empty set A to a non-empty set B and if (a, b) ∈ R, then we write aRb which is read as a is related to b by the relation R. If (a, b) ∉ R then we say that a is not related to b by the relation R. In other words, a relation is a set of inputs and outputs, often written as ordered pairs (input, output). We can also represent a relation as a mapping or a grap.
types of relations
(i) universal relation : if a be any set, then r = a × a ⊆ a × a and so it is a relation on a. this relation is called the universal relation on a.
(ii) empty (or void) relation : if a be any set, then φ ⊆ a × a and so it is a relation on a. this relation is called empty (or void) relation.
(iii) reflexive relation : a relation r is said to be reflexive, if every element of a is related to itself.
thus, r is reflexive ⇔ (a,a) ∈ r, for all a ∈ a .
(iv) symmetric relation : a relation r on a set a is said to be a symmetric if (a,b) ∈ r ⇔ (b,a) ∈ r for all a,b ∈ a
(v) transitive relation : a relation r on any set a is said to be transitive if (a,b) ∈ r, (b,c)∈ r ⇒(a,c)∈ r,for all a,b,c ∈ a.
i.e., arb and brc ⇒ arc for all a,b,c ∈ arc
equivalence relation : a relation r on a set a is said to be an equivalence relation on a if
(a) it is reflexive
(b) it is symmetric
(c) it is transitive
Functions
Function : Meaning
Let A and B be two non-empty sets, then a rule f which associates each element of A with a unique element of B is called a function or mapping from A to B.
If f is a mapping from A to B, we write f : A → B (read as f is a mapping from A to B).
If f associates x ∈ A to y ∈ B then we say that y is the image of the element x under the map (or function) f and we write y = f (x). And, the element x is called the pre-image of y.
Types of Functions
(1) One-One Function (Injective) : A function f : A → B is said to be a one-one function or an injective if different elements of A have different images in B.
Thus, f : A → B is one-one.
⇔ a≠b ⇒ f(a) ≠ f(b) for all a,b ∈ A or f(a) = f(b)⇔ a=b for all a,b ∈ A
(2) Onto-Function (Surjective) : A function f : A → B is said to be an onto function or surjective if and only if each element of B is the image of some element of A i.e. for every element y ∈ B there exists some x ∈ A such that y = f (x). Thus f is onto if range of
f = co-domain of f.
(3) One-one and onto Function (Bijective Function) : A function is a bijective if it is one-one as well as onto
Hope it helps you
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.

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 asetoforderedpairs, which follows a rule i.e every X-value should be associated with only one y-value is called a function.
For example:
DomainRange-1-31339
Let us also look at the definition of Domain and Range of a function.
DomainIt is a collection of the first values in the ordered pair Setofallinput(x values).RangeIt is a collection of the second values in the ordered pair Setofalloutput(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.
Read here:
One To One Function
Onto Function
Bijective Function
Special Functions in Algebra
Some of the important functions are as follow:
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 orderedpairs. 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.
Relation Representation
There are other ways too to write the relation, apart from set notation such as through tables, plotting it on XY- axis or through mapping diagram.

Types of Relations
Different types of relations are as follows:
Empty Relations
Universal Relations
Identity Relations
Inverse Relations
Reflexive Relations
Symmetric Relations
Transitive Relations
Let us discuss all the types one by one.
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\Undefined control sequence \):a,b ∈ R}.
For Example,
If you throw two dice if R = {