Question

can anyone explain me functions??

Answer 1

*One-one functions* This can be made to be understood or explained by going through these basics. 
> Elements 
> Sets
> Types of sets 
> Relations
> Types of relations
> Domain 
> Function 
> One one function. 


Set* :- A collection of well defined objects. A set may have infinite or finite objects .Every object is called element of the set. 

Subset :- It is the set of few or all elements of a set. 

Power set :- It is the set of all subsets of the set including itself and the empty set. 

Union on sets :- Let A, B two sets . Then A union B represented by A∪B is the set of all elements of both A and B. 

Intersection of sets :- Intersection would contain the common elements of Sets .

Relation :- A connection between 2 sets with some kinda rule is known as Relation .
A × B defines a relationship with all the possible ordered pairs formed between A, B. 
Let every ordered be in the form of ( a, b) then a = xb .x can be sum ,difference or some kinda arithmetical variable or literals. 

Relationships exhibit in three types, qualifying the three gives the fourth one. 

Reflexive :-Consider a relation A × A, For every a ∈ A, there must exist (a, a) ∈ R. Then R is reflexive. 

Symmetric :- For a relationship A × B, for every ( a, b) ∈ R there must exist ( b, a) ∈ R then Relation R is symmetric. 

Antisymmetric :- For a relationship A × B, if (a, b) ∈ R and ( b, a) ∈ R it implies that a = b. 

Transitive :- if (a, b) ∈ R and also ( b, c) ∈ R then if ( a, c) is also present in R. Then R is transitive. 

Equivalence :- A relationship obeying transitive, reflexive and symmetric is known as equivalence. 

In relation, a element may be connected with one element or multiple elements .

In a relationship, the first set is called domain. The second set is co-domain .


Now what is a function . Let's dive in. 

`What is a function ?` 
A function has co-domain and domain as relation .What makes it different. In a relation, there is no compulsion for connection between elements in domain, co-domain. 
But..., In a function, A element of domain should be connected with a single element in its codomain .

The elements connected with a element in domain is called Image .The elements in domain are called pre Images of co-domain elements. 

The set of all images is called *Range* .

Now, What is a one-one function ? 

A function in which each single element in domain is associated with only single element in its co-domain .in other words , Every element has not more than one image, also if two elements have same images then they must be same elements. 

Let n(A) be the number of elements in Set A and n(B) be the number of elements in set B

A one -to-one function is possible only when n(A) ≤ n(B) .

If n(A) > n(B) then number of possible one to one functions are 0 .

If n(A) ≤ n(B) then number of possible one to one functions are P( n(B) , n(A) )

One to one functions are also called injective functions, or injections .

All linear functions are injections, as there would be only one possible value for each

Hope it helps you!