User:Rafael Cabanillas Murillo/sandbox

Gallery of mathematical correspondences

Main concepts

Here, in this section, we will define some main concepts related with mathematical correspondences:

• Cardinality of a set: Suppose we have a set with a finite number of elements. Then, the cardinality of the set is the number of elements that is has got.

For example: Let $A$ be a set with some elements. Let these elements belower case letters: $A\equiv \{a,b,d,c,l,m,n\}$. Then, the cardinality of $A$ is 7 because we can count seven letters. That is, using mathematical language: $|A|=7$

• Image of a correspondence: For example if we have an element $a$ that belongs to a set $A$ (In mathematical language: $a\in A$), and an element $b$ ∈ to another set $B$ (In mathematical language: $b\in B$) and there is a mathematical relationship from $a$ to $b$, using other words, $a$ is related with $b$. Then $a$ has an image: $b$.

• Mathematical Relation: Let suppose that we have an element $p$ and an element $t$. Let's suppose there are a relation from $p$ to $t$. In mathematical language, that is represented using an arrow from $p$ to $t$:

$p\rightarrow t$.

A good example of that is:

Let's suppose that we know a person called Peter. It has 20 years old. Then, we can define a relation from Peter to 20:

$Peter\rightarrow 20$.

Definition of a mathematical correspondence

A mathematical correspondence is a mathematical relationship that involves some mathematical sets. It tells us the relation between these sets.

For example, if we have 2 sets, denoted with $X$ and $Y$, then imagine that $X$ has the same cardinality as $Y$ (that is, the same number of elements).

• We are going to call the elements in $X$: $\{x_{1},x_{2},x_{3},...,x_{i}\}$.

• We are going to do the same with $Y$: $\{y_{1},y_{2},y_{3},...,y_{i}\}$.

In mathematics, we define a correspondence from $X$ to $Y$ using that kind of language:

• $F:X\rightarrow Y$

And we define the behaviour of the correspondence too: F is a correspondence that relates every $x_{i}$ with every $Y_{i}$ where $x_{i}\in X$ and $y_{i}\in Y$.

That is, explained in a more graphical way:

Bijective generic application

In short, $F:X\rightarrow Y$ is a correspondence because it relates elements $x_{i}\in X$ with elements $y_{i}\in Y$.

Let's explain now a simpler, more trivial example. But probably it will be more bearable for you to understand:

Imagine that three people want to travel from Alaska to EEUU. Their names are: Maria, Peter and Astrid. There are three means of transportation: By train, by car and by plane.

Maria likes travelling by car, and train. We can relate the element Maria with the elements "car" and "train".

Peter likes travelling by plane. We can relate the element "Peter" with the element "plane".

Astrid likes travelling by train. Finally, we make a mathematical relationship that goes from the element "Astrid" to the element "train".

Graphically, that is:

Representative example of a mathematical correspondence

In the next section, we will describe many types of correspondences.

Types of mathematical correspondences

The inverse of a correspondence

An inverse correspondence is the contrary relation of a correspondence. If we denote a correspondence by $S$, them, its inverse is denoted $S'$.

For example, we have two sets:

$T=\{\,$

,

,

,

$\}\,$

and

$P=\{\,$

,

,

,

$\}\,$

Suposse that it exist a relation (Let's call it S) between the elements in P and the elements in T. To construct the inverse of S (Let's call it $S'$), all we have to do is invert the relations (ie, if in $S$ is the image of , then in $S'$ is the image of .

This is the correspondence $S$

This is the inverse of the correspondence $S$, $S'$

Univocal correspondence

An univocal correspondence is a relation where every element in the set $P$ is related with one or with no elements in the set $T$. An element in the set $P$ cannot be related with two or more elemments in $T$.

Example 1

Example 2

Example 3

Example 4

Non Univocal Correspondence

A non univocal correspondence is a relation where at least one element in the set $P$ is related with two or more elements in the set $T$.

Example 1

Example 2

Example 3

Example 4

Biunivocal correspondence

A biunivocal correspondence $F$ is a relation which is univocal and whose inverse $F'$ is univocal too. That is, every element in $A$ is related with one or none elements in $B$ and every element in $B$ is related with one or none elements in $A$.

Example of the correspondence $F$

Example of the correspondence $F'$

Non biunivocal correspondence

An non biunivocal correspondence $F$ is a relation which is univocal and whose inverse $F'$ is not univocal. That is, every element in $P$ is related with one or none elements in $T$ and at least one element in $T$ is related with two or more elements in $P$.

Example of the correspondence $F$

Example of the correspondence $F'$

Definition of function or application

An application or a function is a special type of correspondence. I has some characteristics. To ilustrate them, we are going to define an application $F:P\rightarrow T$.

♠ All of the elements in $P$ have an image. That is, every element in $P$ is related with any element in $T$

♠ From the latter characteristic, we can define that $F$ is univocal. It is univocal because each element in $P$ has one image (nor two or more). And because of that, $F$ satisfies the neccesary conditionnecessary condition for being univocal: "each element in A has at most 1 image".

File:FUNCTION.

Not a function

A correspondence that it is not a function violates the two conditions explained in the previous section, when we defined what is a function.

Here you have some examples of what is not a function or an application and why:

• When we have two sets, $A$ and $B$ and one or more elements of the set $A$ are not related with the elements in $B$, this is not a correspondence.

• When there are two sets, $P$ and $T$ and there is at least one element in $P$ that it is related with two or more elements in $T$.

In this correspondence, the blue element in $P$, has got two images.

In this correspondence, the yellow element in $P$ hasn't got any image.

In this example, the green element in $P$, has got two images. Also, the red element in $P$ hasn't got any image.

Types of functions

In this part of the article, we are going to see many types of functions or applications. We will explain them briefly using a triple point of view:

First, a brief explanation using the concept of image.

Then, the mathematical one. It is more suitable to define theorems using these types of explanations.

Finally, the grafical one.It is easier to understand and for those who are new to mathematical correspondences.

Injective application

♦ Brief Explanation: An injective application is a function in which all the elements in $T$ hasn't to be the image of an element in $P$. An element in $T$ cannot be the image of two or more elements in $P$.

♦ Mathematical definition: Let $f:X\rightarrow Y$ be a function. Let $x_{i}$ be the elements that belong to the set $X$ and $y_{i}$ the elements that belong to the set $Y$.

Then, f is injective if each distinct element $x_{i}\in X$ are related to at most one distinct element $y_{i}\in Y$. That is, every element $x_{i}\in X$ has, at most, one image. Then, the cardinality of $X$ must be less or equal than the cardinality of $Y$ if and only if X and Y are finite sets. In mathematical language, that is: $|X|\leq |Y|$

♦ Graphical examples:

Particular injective application

Injective generic application

Surjective application

♦ Brief Explanation: A surjective application is a function where an element in $T$ can be image of two or more elements in the set $P$. All the elements in $T$ must be the image of at least one element in the set $P$.

♦ Mathematical definition: Let $f:X\rightarrow Y$ be a function. Let $x_{i}$ be the elements that belong to the set $X$ and $y_{i}$ the elements that belong to the set $Y$.

Then, $f$ is surjective if every element $x_{i}\in X$ has an image $y_{i}\in Y$. It doesnt matter if an element $x_{1}$ has the same image as an element $x_{2}$. This means that the cardinality of $X$ is greater or equal than the cardinality of $Y$ if and only if X and Y are finite sets. That is, making use of symbols: $|X|\geq |Y|$

♦ Graphical examples:

Particular surjective application

Surjective generic application

Bijective application

♦ Brief Explanation: A biyective application is a function that satisfies that one in $P$ is paired with one and only one element in $T$. This condition is fullfilled for all the elements in $P$ and in $T$.

♦ Mathematical definition: Let $f:X\rightarrow Y$ be a function. Let $x_{i}$ be the elements that belong to the set $X$ and $y_{i}$ the elements that belong to the set $Y$.

Then $f$ is bijective if it is injective and surjective at the same time. That is, every element $x_{i}\in X$ has exactly one image $y_{i}\in Y$. It is worth noting that the cardinality of the set $X$ must be equal to the cardinality of the set $Y$ if and only if X and Y are finite sets. that is, in mathematical language: $|X|=|Y|$

♦ Graphical examples:

Particular bijective application

Bijective generic application