OpenLLM-France/wikimedia huggingface dataset:
# Tuples, vectors, matrices/R/Sets/Introduction/Section

Important product sets are $\mathbb {R} ^{2}=\mathbb {R} \times \mathbb {R}$ and $\mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R}$. The ordering of the elements is essential. In general, for a set $M$ and some $n\in \mathbb {N}$, we denote the $n$-th fold product set of $M$ with itself as
    $M^{n}=\underbrace {M\times \cdots \times M} _{n{\text{ times}}}\,.$
The elements have the form
    $\left(x_{1},x_{2},\ldots ,x_{n}\right),$
where every $x_{i}$ is from $M$. Such an ordered finite sequence of $n$ elements is also called an $n$-tuple over $M$. For $n=2$, it is called a pair, for $n=3$, it is called a triple. For
    $x=\left(x_{1},x_{2},\ldots ,x_{n}\right)\,,$
the element $x_{i}$ is called the $i$-th component or the $i$-th entry of the tuple. In this context, the $i$ is called the index of the tuple, and $\{1,\ldots ,n\}$ is called the index set of the tuple.
More generally, for every index set $I$, there exist $I$-tuples. In such an $I$-tuple, to every index $i\in I$ some mathematical object is assigned; the tuple is often written as $x_{i}$, $i\in I$. If all $x_{i}$ are from one set $M$, then we call this an $I$-tuple from $M$. For $I=\mathbb {N}$, we call this a sequence in $M$.
A finite index set can always be replaced by a set of the form $\{1,\ldots ,n\}$ (this procedure is called a numbering of the index set), but this is not always useful. If we start with the index set
    $I=\{1,\ldots ,n\}\,$
and if we are interested in a certain subset $J\subseteq I$, then it is natural to stick to the original notation from $I$ instead of introducing a new numbering $\{1,\ldots ,m\}$ for $J$. Quite often, there is a "natural“ index set for a certain problem that represents a part of the structure of the problem (and is easier to remember).
An $n$-tuple over a set $M$ of the form
    $\left(a_{1},\,\ldots ,\,a_{n}\right)$
is also called a row tuple (of length $n$), and an $n$-tuple of the form
    ${\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}$
is called a column tuple. These are just two different ways to represent the tuple, but if the tuple represents some structure (like a vector, to which a matrix (see below) shall be applied), then this difference is relevant.
When $I$ and $J$ are two sets and $I\times J$ is their product set, then we can express an $I\times J$-tuple in $M$ as a "table“, that assigns to every pair $(i,j)$ an element $a_{ij}\in M$. In particular, for $I=\{1,\ldots ,m\}$ and $J=\{1,\ldots ,n\}$, we call an $I\times J$-tuple also an $m\times n$-matrix, and write this as
    ${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}.$
The row tuple
    $\left(a_{i1},\,a_{i2},\,\ldots ,\,a_{in}\right)$
is called the $i$-th row of the matrix, and
    ${\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{mj}\end{pmatrix}}$
is called the $j$-th column of the matrix.
# Set family/Intersection and union/Definition

Let $M_{i}$, $i\in I$, be a family of subsets of a set $G$. Then
    $\bigcap _{i\in I}M_{i}={\left\{x\in G\mid x\in M_{i}{\text{ for all }}i\in I\right\}}\,$
is called the intersection of the sets, and
    $\bigcup _{i\in I}M_{i}={\left\{x\in G\mid {\text{ there exists some }}i\in I{\text{ with }}x\in M_{i}\right\}}\,$
is called the union of the sets.
# Set families/Introduction/Section

Not only elements but also sets can be indexed by an index set. This is called a family of sets.

## Definition

Let $I$ be a set and let, for every $i\in I$, a set $M_{i}$ be given. Such a situation is called a family of sets
    $M_{i}{\text{ , }}i\in I.$
$I$
Here, the sets might be independent of each other, but they can also be subsets of a certain set.

## Definition

Let $M_{i}$, $i\in I$, be a family of subsets of a set $G$. Then
    $\bigcap _{i\in I}M_{i}={\left\{x\in G\mid x\in M_{i}{\text{ for all }}i\in I\right\}}\,$
is called the intersection of the sets, and
    $\bigcup _{i\in I}M_{i}={\left\{x\in G\mid {\text{ there exists some }}i\in I{\text{ with }}x\in M_{i}\right\}}\,$

## Definition

Let $I$ be a set and let, for every $i\in I$, a set $M_{i}$ be given. Then the set
    $M=\prod _{i\in I}M_{i}=\{(x_{i})_{i\in I}:\,x_{i}\in M_{i}{\text{ for all }}i\in I\}\,$
$M_{i}$
As soon as one of the sets $M_{i}$ is empty, then also the product is empty, because then there is no possible value for the $i$-th component. However, if all sets $M_{i}$ are not empty, then also their product is not empty, as for every index $i$, there exists at least one element $x_{i}\in M_{i}$. In a formal-axiomatic introduction of set theory, one has to postulate that such a choice is possible. This is the content of the axiom of choice.

## Example

For $n\in \mathbb {N}$, let
    $M_{n}={\left\{x\in \mathbb {N} \mid x\geq n\right\}}\,$
be the set of all natural numbers that are at least as large as $n$. This is a family of subsets of $\mathbb {N}$ indexed by the natural numbers. We have the inclusions
    $M_{n}\subseteq M_{m}{\text{ for }}n\geq m.$
The intersection
    $\bigcap _{n\in \mathbb {N} }M_{n}$
is empty, because there is no natural number which is above every other natural number.

## Example

For $n\in \mathbb {N} _{+}$, let
    $\mathbb {N} n={\left\{x\in \mathbb {N} _{+}\mid x{\text{ is a multiple of }}n\right\}}\,$
be set of all positive natural numbers that are multiples of $n$. This is a family of subsets of $\mathbb {N}$ indexed by the positive natural numbers. We have the inclusions
    $\mathbb {N} n\subseteq \mathbb {N} m{\text{ for }}m{\text{ divides }}n.$
The intersection
    $\bigcap _{n\in \mathbb {N} _{+}}\mathbb {N} n$
is empty, because no positive natural number is a multiple of every positive natural number ($0$ is such a multiple).

## Example

Let $x$ be a real number and let $x_{n}$ denote the rational number that consists of the digits of $x$ in the decimal system up to the $n$th digit after the point. We define the intervals
    $M_{n}=\left[x_{n},x_{n}+{\left({\frac {1}{10}}\right)}^{n}\right]\subset \mathbb {R} \,.$
These are intervals of length ${\left({\frac {1}{10}}\right)}^{n}$, and we have
    $\{x\}=\bigcap _{n\in \mathbb {N} }M_{n}\,.$
The family $M_{n}$, $n\in \mathbb {N}$, is a family of nested intervals for $x$.
# Set theory/Indexed family of sets/Definition

Let $I$ be a set, and let, for every $i\in I$, a set $M_{i}$ be given. Such a situation is called a family of sets
    $M_{i}{\text{ , }}i\in I.$
The set $I$ is called the index set of the family.
# Product set/Arbitrary/Definition

Let $I$ be a set, and let, for every $i\in I$, a set $M_{i}$ be given. Then the set
    $M=\prod _{i\in I}M_{i}=\{(x_{i})_{i\in I}:\,x_{i}\in M_{i}{\text{ for all }}i\in I\}\,$
is called the product set of the $M_{i}$.
# Indexed set family/Natural numbers starting with n/Example

For $n\in \mathbb {N}$, let
    $M_{n}={\left\{x\in \mathbb {N} \mid x\geq n\right\}}\,$
be the set of all natural numbers that are at least as large as $n$. This is a family of subsets of $\mathbb {N}$ indexed by the natural numbers. We have the inclusions
    $M_{n}\subseteq M_{m}{\text{ for }}n\geq m.$
The intersection
    $\bigcap _{n\in \mathbb {N} }M_{n}$
is empty because there is no natural number that is above every other natural number.
# Indexed set family/Multiples of n/Example

For $n\in \mathbb {N} _{+}$, let
    $\mathbb {N} n={\left\{x\in \mathbb {N} _{+}\mid x{\text{ is a multiple of }}n\right\}}\,$
be the set of all positive natural numbers that are multiples of $n$. This is a family of subsets of $\mathbb {N}$ indexed by the positive natural numbers. We have the inclusions
    $\mathbb {N} n\subseteq \mathbb {N} m{\text{ for }}m{\text{ divides }}n.$
The intersection
    $\bigcap _{n\in \mathbb {N} _{+}}\mathbb {N} n$
is empty because no positive natural number is a multiple of every positive natural number ($0$ is such a multiple).
# Indexed set family/Nested intervals/Example

Let $x$ be a real number, and let $x_{n}$ denote the rational number that consists of the digits of $x$ in the decimal system up to the $n$th digit after the point. We define the intervals
    $M_{n}=\left[x_{n},x_{n}+{\left({\frac {1}{10}}\right)}^{n}\right]\subset \mathbb {R} \,.$
These are intervals of length ${\left({\frac {1}{10}}\right)}^{n}$, and we have
    $\{x\}=\bigcap _{n\in \mathbb {N} }M_{n}\,.$
The family $M_{n}$, $n\in \mathbb {N}$, is a family of nested intervals for $x$.
# Mapping/Bijective/Definition

Let $M$ and $L$ denote sets, and suppose that
    $F\colon M\longrightarrow L,x\longmapsto F(x),$
is a mapping. Then $F$ is called bijective if $F$ is injective as well as surjective.
# Mapping/Bijective/Inverse mapping/Definition

Let $F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping
    $G\colon M\longrightarrow L$
that sends every element $y\in M$ to the uniquely determined element $x\in L$ with $F(x)=y$, is called the inverse mapping of $F$.
# Mapping/Composition/Associativity/Fact

Let $L,M,N$ and $P$ be sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
    $G\colon M\longrightarrow N,y\longmapsto G(y),$
and
    $H\colon N\longrightarrow P,z\longmapsto H(z),$
be mappings.
    $H\circ (G\circ F)=(H\circ G)\circ F\,$
# Mapping/Graph/Definition

Let $L$ and $M$ be sets, and let
    $F\colon L\longrightarrow M$
be a mapping. Then the set
    $\Gamma =\Gamma _{F}={\left\{(x,F(x))\mid x\in L\right\}}\subseteq L\times M\,$
is called the graph of the mapping $F$.
# Mapping/Image under a mapping/Definition

Let $L$ and $M$ be sets, and let
    $F\colon L\longrightarrow M$
be a mapping. For a subset $S\subseteq L$, we call
    $F(S)={\left\{y\in M\mid {\text{there exists an }}x\in S{\text{ such that }}F(x)=y\right\}}\,$
the image of $S$ under $F$. For $S=L$,
    $F(L)=\operatorname {Im} F\,$
is called the image of the mapping.
# Mapping/Preimage under a mapping/Definition

Let $L$ and $M$ be sets, and let
    $F\colon L\longrightarrow M$
be a mapping. For a subset $T\subseteq M$, we call
    $F^{-1}(T)={\left\{x\in L\mid F(x)\in T\right\}}\,$
the preimage of $T$ under $F$. For a subset $T=\{y\}$ with one element, we call
    $F^{-1}(\{y\})$
also the preimage of $y$.
# Mapping/Definition

Let $L$ and $M$ denote sets. A mapping $F$ from $L$ to $M$ is given by assigning to every element of the set $L$ exactly one element of the set $M$. The unique element that is assigned to $x\in L$ is denoted by $F(x)$. For the mapping as a whole, we write
    $F\colon L\longrightarrow M,x\longmapsto F(x).$
# Mapping/Introduction/Section
## Definition

Let $L$ and $M$ denote sets. A mapping $F$ from $L$ to $M$ is given by assigning to every element of the set $L$ exactly one element of the set $M$. The unique element that is assigned to $x\in L$ is denoted by $F(x)$. For the mapping as a whole, we write
    $F\colon L\longrightarrow M,x\longmapsto F(x).$
If a mapping $F\colon L\rightarrow M$ is given, then $L$ is called the domain (or domain of definition) of the map, and $M$ is called the codomain (or target range) of the map. For an element $x\in L$, the element
    $F(x)\in M\,$
is called the value of $F$ at the place (or argument) $x$.
Two mappings $F\colon L_{1}\rightarrow M_{1}$ and $G\colon L_{2}\rightarrow M_{2}$ are equal if and only if their domains coincide, their codomains coincide, and if for all $x\in L_{1}=L_{2}$ the equality $F(x)=G(x)$ in $M_{1}=M_{2}$ holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called functions. However, we will usually reserve the term function for mappings where the codomain is a number set like the real numbers $\mathbb {R}$.
For every set $L$, the mapping
    $L\longrightarrow L,x\longmapsto x,$
which sends every element to itself, is called the identity (on $L$). We denote it by $\operatorname {Id} _{L}$. For another set $M$ and a fixed element $c\in M$, the mapping
    $L\longrightarrow M,x\longmapsto c$
that sends every element $x\in L$ to the constant value $c$ is called the constant mapping (with value $c$). It is usually again denoted by $c$.
There are several ways to describe a mapping, like a value table, a bar chart, a pie chart, an arrow diagram, or the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule that allows computing the values of the mapping for every argument. Such rules are, e.g., (from $\mathbb {R}$ to $\mathbb {R}$) $x\mapsto x^{2}$, $x\mapsto x^{3}-e^{x}+\sin x$, etc. In the sciences and in sociology, also empirical functions are important that describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximately) in a mathematical manner.
* ↑ Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things.
# Mapping/Injective/Definition

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
be a mapping. Then $F$ is called injective if for two different elements $x,x'\in L$, also $F(x)$ and $F(x')$ are different.
# Mapping/Surjective/Definition

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
be a mapping. Then $F$ is called surjective if, for every $y\in M$, there exists at least one element $x\in L$, such that
    $F(x)=y\,.$
# Mapping/Quantifier/Interpretation as solution/Remark

The question, whether a mapping $F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation
    $F(x)=y\,$
(in the two variables $x$ and $y$). The surjectivity means that for every $y\in M$ there exists at least one solution
    $x\in L\,$
for this equation; the injectivity means that for every $y\in M$ there exists at most one solution $x\in L$ for this equation. The bijectivity means that for every $y\in M$ there exists exactly one solution $x\in L$ for this equation. Hence, surjectivity means the existence of solutions, and injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics, and they can also be interpreted as surjectivity or injectivity of suitable mappings.
# Mapping/Injective/Surjective/Bijective/Introduction/Section
## Definition

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
be a mapping. Then $F$ is called injective if for two different elements $x,x'\in L$, also $F(x)$ and $F(x')$

## Definition

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
be a mapping. Then $F$ is called surjective if, for every $y\in M$, there exists at least one element $x\in L$, such that
    $F(x)=y\,.$

## Definition

Let $M$ and $L$ denote sets, and suppose that
    $F\colon M\longrightarrow L,x\longmapsto F(x),$
is a mapping. Then $F$ is called bijective if $F$ is injective as well as
These concepts are fundamental!
The question, whether a mapping $F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation
    $F(x)=y\,$
(in the two variables $x$ and $y$). The surjectivity means that for every $y\in M$ there exists at least one solution
    $x\in L\,$
for this equation; the injectivity means that for every $y\in M$ there exists at most one solution $x\in L$ for this equation. The bijectivity means that for every $y\in M$ there exists exactly one solution $x\in L$ for this equation. Hence, surjectivity means the existence of solutions, and injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics, and they can also be interpreted as surjectivity or injectivity of suitable mappings.
In order to show that a certain mapping is injective, we often use the following strategy: One shows for any two given elements $x$ and $x'$ using the condition $F(x)=F(x')$ that $x=x'$ holds. This method is often easier than showing that $x\neq x'$ implies $F(x)\neq F(x')$.

## Example

The mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$
is neither injective nor surjective. It is not injective, because the different numbers $2$ and $-2$ are both sent to $4$. It is not surjective, because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping
    $\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$
is injective, but not surjective. The injectivity can be seen as follows: If $x\neq y$, then one number is larger, say
    $x>y\geq 0\,.$
But then also $x^{2}>y^{2}$, and in particular $x^{2}\neq y^{2}$. The mapping
    $\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$
is not injective, but surjective, since every nonnegative real number has a square root. The mapping
    $\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$
is injective and surjective.

## Definition

Let $F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping
    $G\colon M\longrightarrow L$
that sends every element $y\in M$ to the uniquely determined element $x\in L$ with $F(x)=y$,
$F$
The inverse mapping is usually denoted by $F^{-1}$.
# Mapping/Squaring/Injective and surjective/Example

The mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$
is neither injective nor surjective. It is not injective because the different numbers $2$ and $-2$ are both sent to $4$. It is not surjective because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping
    $\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$
is injective, but not surjective. The injectivity can be seen as follows: If $x\neq y$, then one number is larger, say
    $x>y\geq 0\,.$
But then also $x^{2}>y^{2}$, and in particular $x^{2}\neq y^{2}$. The mapping
    $\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$
is not injective, but surjective, since every nonnegative real number has a square root. The mapping
    $\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$
is injective and surjective.
# R to R/Linear/Mapping/Example

Let $a\in \mathbb {R}$ be fixed. This real number defines a mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax.$
For $a=0$, this is the constant zero mapping. For $a\neq 0$, we have a bijective mapping; the inverse mapping is
    $y\longrightarrow {\frac {1}{a}}y.$
Here, the inverse mapping has a similar form as the mapping itself.
# Mapping/Linear algebra/Section

We discuss two classes of mappings that are in the framework of linear algebra very important. They are both so-called linear mappings.

## Example

Let $a\in \mathbb {R}$ be fixed. This real number defines a mapping
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto ax.$
For $a=0$, this is the constant zero mapping. For $a\neq 0$, we have a bijective mapping; the inverse mapping is
    $y\longrightarrow {\frac {1}{a}}y.$
Here, the inverse mapping has a similar form as the mapping itself.

## Example

Let an $m\times n$-matrix
    ${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}$
be given, where the entries $a_{ij}$ are real numbers. Such a matrix defines a mapping
    $\varphi \colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m},$
by sending an $n$-tuple $x={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}\in \mathbb {R} ^{n}$ to the $m$-tuple
    $\varphi (x)={\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}={\begin{pmatrix}a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n}\\a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n}\\\vdots \\a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{j=1}^{n}a_{1j}x_{j}\\\sum _{j=1}^{n}a_{2j}x_{j}\\\vdots \\\sum _{j=1}^{n}a_{mj}x_{j}\end{pmatrix}}\,.$
The $i$-th component of the image vector is
    $y_{i}=\left(a_{i1},\,a_{i2},\,\ldots ,\,a_{in}\right){\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}=\sum _{j=1}^{n}a_{ij}x_{j}\,.$
So one has to apply the $i$-th row of the matrix to the column vector $x$ in the described way.
It is a goal of linear algebra to determine, in dependence of the entries $a_{ij}$, whether the mapping defined by the matrix is injective, surjective, or bijective, and how, in the bijective case, the inverse mapping looks like.

## Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium and magnesium various fruits have (in milligram with respect to 100 gram of the fruit).
This table yields a mapping, which assigns to a $4$-tuple ${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}$, representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium and magnesium, in the form of a $3$-tuple ${\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}$. This mapping can be described with the matrix
    ${\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$
using matrix multiplication as
    ${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}12x_{1}+53x_{2}+4x_{3}+9x_{4}\\7x_{1}+40x_{2}+12x_{3}+5x_{4}\\6x_{1}+10x_{2}+8x_{3}+27x_{4}\end{pmatrix}}={\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}.$
# Fruit salad/Minerals and vitamins/Example

A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium, and magnesium various fruits have (in milligrams with respect to 100 grams of the fruit).
This table yields a mapping, which assigns to a $4$-tuple ${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}$, representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium, and magnesium, in the form of a $3$-tuple ${\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}$. This mapping can be described with the matrix
    ${\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}$
using matrix multiplication as
    ${\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}\longmapsto {\begin{pmatrix}12&53&4&9\\7&40&12&5\\6&10&8&27\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\end{pmatrix}}={\begin{pmatrix}12x_{1}+53x_{2}+4x_{3}+9x_{4}\\7x_{1}+40x_{2}+12x_{3}+5x_{4}\\6x_{1}+10x_{2}+8x_{3}+27x_{4}\end{pmatrix}}={\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\end{pmatrix}}.$
# Mapping/Composition/Section
## Definition

Let $L,\,M$ and $N$ denote sets, let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
and
    $G\colon M\longrightarrow N,y\longmapsto G(y),$
be mappings. Then the mapping
    $G\circ F\colon L\longrightarrow N,x\longmapsto G(F(x)),$
is called the composition of the mappings
$F$
$G$
So we have
    $(G\circ F)(x):=G(F(x))\,,$
where the left-hand side is defined by the right-hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression, if possible).
For a bijective mapping $\varphi \colon M\rightarrow N$, the inverse mapping $\varphi ^{-1}\colon N\rightarrow M$ is characterized by the conditions
    $\varphi \circ \varphi ^{-1}=\operatorname {Id} _{N}\,$
and
    $\varphi ^{-1}\circ \varphi =\operatorname {Id} _{M}\,.$

## Lemma

Let $L,M,N$ and $P$ be sets, and let
    $F\colon L\longrightarrow M,x\longmapsto F(x),$
    $G\colon M\longrightarrow N,y\longmapsto G(y),$
and
    $H\colon N\longrightarrow P,z\longmapsto H(z),$
be mappings. Then
    $H\circ (G\circ F)=(H\circ G)\circ F\,$

### Proof

Two mappings $\alpha ,\beta \colon L\rightarrow P$ are the same if and only if the equality $\alpha (x)=\beta (x)$ holds for every $x\in L$. So let $x\in L$. Then
    ${\begin{aligned}(H\circ (G\circ F))(x)&=H((G\circ F)(x))\\&=H(G(F(x)))\\&=(H\circ G)(F(x))\\&=((H\circ G)\circ F)(x).\end{aligned}}$
$\Box$