Mathematical correspondence

---

Given two sets: X and Y, and a function f, which determines some binary relation between some element of X with some element of Y, we will say that this function: f, defines a correspondence between X and Y, which we will represent:

$f:X\to Y$

when at least one element in X is related to at least one element in Y. Functions are sometimes called mappings or transformations.

An example

Sets X, Y and the relations between them.

Imagine 4 friends are going to have lunch and there are 4 available dishes in the menu. If we associate every person to the food they are ordering we have a relationship between both sets. Let's call our sets X and Y, X being the friends and Y being the dishes, the relationships between sets are represented by arrows. Of course, no one is forced to order anything and more than one person can order the same, so there might be elements in both sets that have no relation, ie: one of them is not hungry or no one likes one of the dishes in the menu.

In this case Dany and Michael ordered fries, Louis ordered tuna and Rebecca ordered a sandwich. All of them ordered something, so arrows come from all of them, but no one ordered icecream, so no arrow goes in that element.

As we said before, at least one element in X must be related with at least one element in Y to be called a correspondence, otherwise there is no relation between both sets.

There is no need for these correspondences to be strictly numerical, and still they are mathematical correspondences.

Definitions

In a correspondence there are different sets and elements:

Initial set

First set in the correspondence, we can represent it as bg (beginning), in this example it will be:

X = bg(f) = {Dany, Michael, Louis, Rebecca}

Final set

Second set in the correspondence, represented as fin (final), let it be represented as:

Y = fin(f) = {Fries, Sandwich, Tuna, Icecream}

Origin set

It is the set consisting of all elements in the initial set that are related to any element of the final set, let it be represented as:

or(f) = {Dany, Michael, Louis, Rebecca} (All of them ordered something so all of them are related to Y, therefore all of them belong to the origin set)

Image set

It is the set consisting of all the elements in the final set that are related (pointed by arrows) to elements in the origin set, let it be represented as Im (image):

Im(f) = {Fries, Sandwich, Tuna} (Icecream wasn’t ordered so it is not related to any person, therefore is not in the image set)

Homologous element

Two elements, one from the origin set and one from the image set, if they are related by the correspondence f they are homologous.

These pairs of elements are homologous: (Dany, Fries), (Michael, Fries), (Louis, Tuna).

Image element

Given an element x from the origin set and another element y from the image set, y is image of x, it is represented like:

f(x) = y

If element x is related to the element y following the correspondence f, using the example we have:

f(Dany) = Fries

f(Rebecca) = Sandwich

Inverted correspondence

Given a correspondence between the sets A and B, represented as:

$f:A\to B$

it is defined as inverse correspondence of f, which is called $f^{-1}$:

$f^{-1}:B\to A$

the function that relates f with its origin.

Example:

$f:X\to Y$

(Dany, Fries), (Rebecca, Sandwich)

The inversed correspondence relates the set of foods Y to the persons in X

$f^{-1}:Y\to X$

(Fries, Dany), (Sandwich, Rebecca)

Mathematical application

Given a mathematical correspondence between all of the elements in X with the elements in Y, this correspondence is called an application between X and Y when each element of X is related with a single element of Y. It’s usually called function between X and Y. To express this in a more mathematical way, let's assign numbers to the people we had in our previous examples and letters to the available dishes, as it goes:

$Dany\to 1.$ Ice-cream icon $Fries\to D$

$Michael\to 2.$ Ice-cream icon$Sandwich\to B$

$Louis\to 3.$ Ice-cream icon$Tuna\to C$

$Rebecca\to 4.$ Ice-cream icon$Icecream\to A$

Injective application

Kinds of mathematical applications

Given 2 sets X,Y and all of the possible applications A that can be generated from these 2 sets:

• If every image has only one correspondence in the origin, it is an injective application.

Less formally, "Every element in the final origin related to an element in the origin set has only one incoming arrow"

• If the application applies to the whole final set, it is a surjective application.

Colloquially: "Every element in the final element has an arrow pointing at it, at least"

• There is a third kind which is being injective and surjective at the same time, these are bijective applications

Again, in a nutshell: "All origin elements point to only one final element, and all final elements are pointed by only one arrow, even shorter, all of the elements are related 1 to 1"

Injective non surjective application

Surjective application

Since it is an injective application every image element will have just one origin element and being non surjective implies that at least one element in the final set has no origin element preceding it. In this kind of applications the cardinality of X is always smaller than that of Y, since at least one element in the final set has no preceding element in the origin and image elements have no more than one origin element.

Example

Everyone in the group orders something from the menu, but there are more dishes than people in the group and no one will repeat dish since they want to share, hence it is injective, but not surjective.

Surjective non injective application

A non injective application has at least one image element with more than one origin element and being surjective means all elements in the set have at least one origin element. In this kind of application the cardinality of X is always bigger than that of Y, since Y has at least one origin for each final element and at least one has two, X has at least one more element.

Example

Everyone in the group orders, but they won’t share, therefore it can happen that two people want to order the same dish for themselves.

Injective and surjective (Biyective)

File:Bijection.svg.png

Bijective application

If an application is surjective and bijective at the same time, it is bijective. By being injective every element with origin has only one, and by being surjective every element in the final set has an origin, therefore both have the same number of elements, same cardinality and the correspondence between elements is always 1 to 1.

Example

Everyone in the groups orders one dish, and the group is so big that they match the number of dishes in the menu. This case could also be considered as if someone was delivering free gifts in the streets and there is the same number of people as gifts, if it is just one gift per person, then that would be a bijective application too.

Non injective and non surjective

This kind of application has at least one image element with two or more origins and at least one element in the final set has no origin. It doesn´t have a specific name. X and Y sets cannot be compared according to their cardinality, or their number of elements.

Example

Everyone orders, some people order the same, but they don´t order the whole menu.

References

• Hurtado, F. (1997). Atlas de matematicas. Idea books, S.A 1st Edition. ISBN 978-84-8236-049-2. Search this book on
• Campos, Neila (2003). "ALGEBRA LINEAL" (PDF).
• Zotes, F. (2009). "Cardinalidad de conjuntos".
• Thomas Ara, Luis (1974). Algebra Lineal. ISBN 978-84-400-7995-4. Search this book on

This article "Mathematical correspondence" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Mathematical correspondence. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Burma Economic Watch
• City Centre Plaza
• GeeksForGeeks
• RFL Reserve Championship
• Thiebou Yapp
• Shibu Chacko
• Manoranjan TV

---