Question

A student is to be taught basics of one-one functions. What would you teach? Make a informative notes

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!

Answer 2

Hey there!!
Down here ⏬

ONE - ONE FUNCTIONS

> First of all, we want to know what's meant by FUNCTIONS?
• It gives output for a given input.You gave a certain value/input to begin and do thier thing on value, and it gaves answer/output.
• e.g. : The function, f (x) = x-2 , subtracts
2 to any value you give/feed.
here, if we feed 2 as value , 2:f (2) = 2-2=0
• We cant have two different answers for a one value feeded. That is, A real function would give you one answer only.

> What are one-to-one functions?
• A one-to-one function, often written as 1-1, is a function that never repeat.
• A normal function can have two different input values that produce same answer, but a 1-1 function does not.
• e.g. : the function f (x) = x^2 is not a one-to-one function because it produces 4 as the answer when you input both a 2 and and a -2,
but, the function f (x) = x-3 is a 1-1 function because it produces a different answer for any input.

> SET :
• A group of numbers, variables, geometric figures, or just about anything.
• Sets are often written using set braces { }
• For e.g. : {1, 2,3} is the set containing the elements 1,2, 3

> ELEMENTS :
• A number, letter, point, line, or any other object contained in a set.
• For e.g. : The elements of the set {a,b,c} are the letters a,b, and c

> CO-ORDINATE PLANE :
•It is the plane formed by a horizontal axis (x-axis) and a vertical axis (y-axis) .

> GRAPH
• It is the picture obtained by plotting all the points of an equation or inequality.
• If there is only one variable, the graph is on a number line.
• If there are two variables,the graph is on the coordinate plane
• If there are 3 variables, the graph is in three-dimensional coordinates.
•In general, for n variables, the graph is in n dimensions.

> RELATION :
• It is a set if ordered pairs.
• For e.g. : {(1,2), (3,4), (1,a), (5, r)} ; this is a relation.

> VERTICAL LINES TEST :
• A test use to determine if a relation is a function. A relation is a function if there are no vertical lines that intersects the graph at more than one point.

> HORIZONTAL LINE TEST :
• A test use to determine if a function is one-to-one. If a horizontal line intersects a function's graph more than once, then the function is not one-to-one.

Hope it helps.
[ These are very basics ]