Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 3/latex
\setcounter{section}{3}
\subtitle {Sets}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Georg_Cantor.jpg} }
\end{center}
\imagetext {Georg Cantor (1845-1918) is the creator of set theory.} }
\imagelicense { Georg Cantor 1894.jpg } {} {Taxiarchos228} {Commons} {PD} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {David_Hilbert_1886.jpg } }
\end{center}
\imagetext {David Hilbert (1862-1943) has called set theory a
\emphasize{paradise}{,} from where mathematicians should never be expelled.} }
\imagelicense { David Hilbert 1886.jpg } {} {} {Commons} {PD} {}
Mathematical structures like numbers are described as sets. A \keyword {set} {} is a collection of distinct objects which are called the \keyword {elements} {} of the set. By distinct we mean that it is clear which objects are considered to be equal and which are considered to be different. The \keyword {containment} {} of an element $x$ to a set $M$ is expressed by
\mathrelationchaindisplay
{\relationchain
{x
}
{ \in} {M
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
the noncontainment by
\mathrelationchaindisplay
{\relationchain
{x
}
{ \notin} { M
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For every element, exactly one of these possibilities holds. For example, we have
\mathrelationchain
{\relationchain
{ { \frac{ 3 }{ 7 } }
}
{ \notin }{ \N
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchain
{\relationchain
{ { \frac{ 3 }{ 7 } }
}
{ \in }{ \Q
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
An important principle for sets is the \keyword {principle of extensionality} {,} i.e. a set is determined by the elements it contains, beyond that it bears no further information. In particular, two sets coincide if they contain the same elements.
The set which does not contain any element is called the \keyword {empty set} {} and is denoted by
\mathdisp {\emptyset} { . }
A set $N$ is called a \keyword {subset} {} of a set $M$ if every element from $N$ does also belong to $M$. For this relation we write
\mathrelationchain
{\relationchain
{N
}
{ \subseteq }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{} (some people write
\mathrelationchain
{\relationchain
{ N
}
{ \subset }{ M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for this). One also says that the \keyword {inclusion} {}
\mathrelationchain
{\relationchain
{N
}
{ \subseteq }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds. For the number sets, the inclusions
\mathrelationchaindisplay
{\relationchain
{\N
}
{ \subseteq} {\Z
}
{ \subseteq} {\Q
}
{ \subseteq} {\R
}
{ } {
}
}
{}{}{}
hold. The subset relation
\mathrelationchain
{\relationchain
{N
}
{ \subseteq }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is a statement using for all, as it makes a claim about all elements from $N$. If we want to show
\mathrelationchain
{\relationchain
{N
}
{ \subseteq }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then we have to show for an arbitrary element
\mathrelationchain
{\relationchain
{x
}
{ \in }{N
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
that also the containment
\mathrelationchain
{\relationchain
{x
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds. In order to show this, we are only allowed to use the property
\mathrelationchain
{\relationchain
{x
}
{ \in }{N
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For us, sets will be either number sets or sets constructed from such number sets. A set is called \keyword {finite} {} if its elements may be counted by the natural numbers \mathl{1,2,3 , \ldots , n}{} for a certain
\mathrelationchain
{\relationchain
{n
}
{ \in }{ \N
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
In this case, the number $n$ is called the \keyword {number} {} (or the \keyword {cardinality} {}) of the set.
\subtitle {Possible descriptions for sets}
There are several ways to describe a set. The easiest one is to just list the elements of the set, here the order of the listing is not important. For finite sets, this is possible; however, for infinite sets, one has to describe a \quotationshort{construction rule}{} for the elements.
The most important set given by an infinite listing is the set of natural numbers
\mathrelationchaindisplay
{\relationchain
{ \N
}
{ =} { \{ 0,1,2,3, \ldots \}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Here, a certain set of numbers is described by the first elements in the hope that this indicates how the listing goes on and which numbers belong to the set. An important point is that $\N$ is not a set of certain digits, but the set of values represented by these digits or sequences of digits. For a natural number, there are many possibilities to represent it, the decimal expansion is just one of them.
We discuss now the description of sets by properties. Let a set $M$ and a certain property $E$
\extrabracket {a predicate} {} {}
be given, such that the property can be applied to the elements of $M$. Hence, for the property $E$, we have in $M$ the subset consisting of all the elements from $M$ which fulfil this property. We write for this subset given by $E$
\mathrelationchaindisplay
{\relationchain
{ { \left\{ x \in M \mid E(x) \right\} }
}
{ =} { { \left\{ x \in M \mid x \text{ fulfils property } E \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
This only works for such properties for which the statement \mathl{E(x)}{} is well-defined for every
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
If one introduces such a subset, then one gives a name to it, which often reflects the name of the property, like
\mathrelationchaindisplay
{\relationchain
{ E
}
{ =} { { \left\{ x \in \N \mid x \text{ is even} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
\mathrelationchaindisplay
{\relationchain
{O
}
{ =} { { \left\{ x \in \N \mid x \text{ is odd} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
\mathrelationchaindisplay
{\relationchain
{ S
}
{ =} { { \left\{ x \in \N \mid x \text{ is a square number} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
\mathrelationchaindisplay
{\relationchain
{ { \mathbb P }
}
{ =} { { \left\{ x \in \N \mid x \text{ is a prime number} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For the sets occurring in mathematics, a multitude of mathematical properties is relevant, and, therefore, there is a multitude of relevant subsets. But also in the sets of everyday life like the set $C$ of the students in a course, there are many important properties which determine certain subsets, like
\mathrelationchaindisplay
{\relationchain
{ O
}
{ =} { { \left\{ x \in C \mid x \text{ lives in Osnabr}\ddot {\rm u}\text{ck} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
\mathrelationchaindisplay
{\relationchain
{ P
}
{ =} { { \left\{ x \in C \mid x \text{ studies physics} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
\mathrelationchaindisplay
{\relationchain
{ D
}
{ =} { { \left\{ x \in C \mid x \text{ has birthday in December} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The set $C$ itself is also given by a property, since
\mathrelationchaindisplay
{\relationchain
{ C
}
{ =} { { \left\{ x \mid x \text{ is a student in the course} \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
\subtitle {Set operations}
Similar to the construction of new statements from given statements by connecting them with logical connectives, there are operations to make new sets from old ones. The most important operations are the following:\extrafootnote {It is easy to memorize the symbols: the $\cup$ for union looks like u. The intersection is written as $\cap$. The corresponding logical operations or, and have the analogous forms $\vee$ and $\wedge$, respectively} {.} {}
\enumerationthree {\keyword {Union} {}
\mathrelationchaindisplay
{\relationchain
{A \cup B
}
{ \defeq} { { \left\{ x \mid x \in A \text{ or } x \in B \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {\keyword {Intersection} {}
\mathrelationchaindisplay
{\relationchain
{ A \cap B
}
{ \defeq} { { \left\{ x \mid x \in A \text{ and } x \in B \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {\keyword {Difference set} {}
\mathrelationchaindisplay
{\relationchain
{ A \setminus B
}
{ \defeq} { { \left\{ x \mid x \in A \text{ and } x \notin B \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
}
For these operations to make sense, the sets need to be subsets of a common basic set. This ensures that we are talking about the same elements. Quite often, this basic set is not mentioned explicitly and has to be understood from the context. A special case of the difference set is the \keyword {complement} {} of a subset
\mathrelationchain
{\relationchain
{A
}
{ \subseteq }{G
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in a given set $G$, also denoted as
\mathrelationchaindisplay
{\relationchain
{ \complement A
}
{ \defeq} { G \setminus A
}
{ =} { { \left\{ x \in G \mid x \not\in A \right\} }
}
{ } {
}
{ } {
}
}
{}{}{.}
If two sets have an empty intersection, meaning
\mathrelationchain
{\relationchain
{ A \cap B
}
{ = }{ \emptyset
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we also say that they are \definitionword {disjoint}{.}
\subtitle {Product set}
We want to describe within set theory the arithmetic operations on the number sets mentioned above like addition and multiplication. For the addition (say on $\N$), two natural numbers \mathcor {} {a} {and} {b} {} are added to yield another natural number, namely \mathl{a+b}{.} The set of pairs constitute the product set and the adding is interpreted as a mapping on the product set.
We define\extrafootnote {In mathematics, definitions are usually presented as such and get a number so that it is easy to refer to them. The definition contains the description of a situation where only concepts are used that have been defined before. In this situation, a new concept together with a name for it is introduced. This name is printed in a certain font, typically in
\emphasize{italic}{.} The new concept can be formulated without the new name; the new name is an abbreviation for the new concept. Quite often, the concepts depend on parameters, like the product set depends on its component sets. The names are often chosen arbitrarily; the meaning of the word within the mathematical context can be understood only via the explicit definition and not via its meaning in everyday life.} {} {.}
\inputdefinition
{ }
{
Suppose that two sets
\mathcor {} {L} {and} {M} {}
are given. Then the set
\mathrelationchaindisplay
{\relationchain
{ L \times M
}
{ =} { { \left\{ (x,y) \mid x \in L , \, y \in M \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is called the \definitionword {product set}{}
}
The elements of a product set are called \keyword {pairs} {} and denoted by \mathl{(x,y)}{.} Here the ordering is essential. The product set consists of all pair combinations, where in the first \keyword {component} {} there is an element of the first set and in the second component there is an element of the second set. Two pairs are equal if and only if they are equal in both components.
If one of the sets is empty, then so is the product set. If both sets are finite, say the first with $n$ elements and the second with $k$ elements, then the product set has \mathl{n \cdot k}{} elements. It is also possible to form the product set of more than two sets.
\inputexample{}
{
Let $F$ be the set of all first names, and $L$ be the set of all last names. Then
\mathdisp {F \times L} { }
is the set of all names. The elements of this set are, in pair notation, \mathl{(\text{Heinz},\text{Miller})}{,} \mathl{(\text{Petra}, \text{Miller})}{} and \mathl{(\text{Lucy},\text{Sonnenschein})}{.} From a name, one can easily deduce the first name and the last name by looking at the first or the second component. Even if all first names and all last names do really occur, not every combination of a first name and a last name does occur. For the product set, all possible combinations are allowed.
}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {SquareLattice.svg} }
\end{center}
\imagetext {} }
\imagelicense { SquareLattice.svg } {} {Jim.belk} {Commons} {PD} {}
For a product set it is also possible that both sets are equal. Then one has to be careful not to confuse the components.
\inputexample{}
{
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Chess board blank.svg} }
\end{center}
\imagetext {} }
\imagelicense { Chess board blank.svg } {} {Beao} {Commons} {CC-by-sa 3.0} {}
A chess board
\extrabracket {meaning the set of squares of a chess board where a chess piece may stand} {} {}
is the
product set
\mathdisp {\{a,b,c,d,e,f,g,h\} \times\{1,2,3,4,5,6,7,8\}} { . }
Every square is a pair, e.g., \mathl{(a,1), (d,4), (c,7)}{.} Because the two component sets are different, one may write instead of pair notation simply \mathl{a1,d4,c7}{.} This notation is the starting point to describe chess positions and complete chess games.
}
The product set \mathl{\R \times \R}{} is thought of as a plane, one denotes it also by $\R^2$. The product set \mathl{\Z \times \Z}{} is a set of lattice points.
\inputexample{}
{
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Geometri_cylinder.png} }
\end{center}
\imagetext {The cylinder (its surface) is the product set of a circle and a line segment.} }
\imagelicense { Geometri cylinder.png } {} {Anp} {sv Wikipedia} {PD} {}
Let $S$ denote a circle
\extrabracket {the circumference} {} {,}
and let $I$ be a line segment. The circle is a subset of a plane $E$, and the line segment is a subset of a line $G$, so that for the product sets, we have the relation
\mathrelationchaindisplay
{\relationchain
{ S \times I
}
{ \subseteq} { E \times G
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The product set \mathl{E \times G}{} is the three-dimensional space, and the product set \mathl{S \times I}{} is the surface of a cylinder.
}
\subtitle {Mappings}
Suppose that a particle is moving in space. This process is described by assigning to every point in time
\mathrelationchain
{\relationchain
{t
}
{ \in }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
the point in space
\mathrelationchain
{\relationchain
{z(t)
}
{ \in }{\R^3
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
where the particle is at that time. The running of a computer program that uses altogether $s$ memory units can be described by saying for every computing step (which means the execution of a program line) what the content of the memory units is. So to the number $n$ counting the computing steps, we assign the tuple
\mathrelationchaindisplay
{\relationchain
{(b_1 (n) , \ldots , b_s(n))
}
{ \in} { \N^s
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
In an election, every voter has to decide for exactly one party ( or for not voting). The temperature profile on the earth is described by assigning to every point in time and every point on the surface its temperature. Such and many other situations are captured by the concept of a mapping.
\inputdefinition
{ }
{
Let
\mathcor {} {L} {and} {M} {}
denote sets. A \definitionword {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
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is denoted by \mathl{F(x)}{.} For the mapping as a whole, we write
\mathdisp {F \colon L \longrightarrow M
}
If a mapping
$F \colon L \rightarrow M$
is given, then $L$ is called the \definitionword {domain}{} (or domain of definition) of the map, and $M$ is called the \definitionword {codomain}{} (or \keyword {target range} {}) of the map. For an element
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the element
\mathrelationchaindisplay
{\relationchain
{ F(x)
}
{ \in} { M
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is called the \keyword {value} {} of $F$ at the \keyword {place} {} (or \keyword {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
\mathrelationchain
{\relationchain
{x
}
{ \in }{L_1
}
{ = }{L_2
}
{ }{
}
{ }{
}
}
{}{}{}
the equality
\mathrelationchain
{\relationchain
{ F(x)
}
{ = }{ G(x)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in
\mathrelationchain
{\relationchain
{M_1
}
{ = }{ M_2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called \keyword {functions} {.} However, we will usually reserve the term \keyword {function} {} for mappings where the codomain is a number set like the real numbers $\R$.
For every set $L$, the mapping
\mathdisp {L \longrightarrow L
, x \longmapsto x} { , }
which sends every element to itself, is called the \keyword {identity} {} (on $L$). We denote it by
$\operatorname{Id}_{ L }$. For another set $M$ and a fixed element
\mathrelationchain
{\relationchain
{c
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the mapping
\mathdisp {L \longrightarrow M
, x \longmapsto c} { }
that sends every element
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
to the \keyword {constant value} {} $c$ is called the \definitionword {constant mapping}{}
\extrabracket {with value $c$} {} {.}
It is usually again denoted by $c$\extrafootnote {Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things} {.} {.}
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., \extrabracket {from $\R$ to $\R$} {} {} \mathl{x \mapsto x^2}{,} \mathl{x \mapsto x^3- e^x + \sin x}{,} etc. In the sciences and in sociology, also \keyword {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.
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Aplicacion_2.svg} }
\end{center}
\imagetext {} }
\imagelicense { Aplicación 2.svg } {} {HiTe} {Commons} {PD} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Beliebteste_Eissorten_in_Deutschland.svg} }
\end{center}
\imagetext {} }
\imagelicense { Beliebteste Eissorten in Deutschland.svg } {} {Doofi} {Commons} {PD} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Tiefkühlkonsum.svg} }
\end{center}
\imagetext {} }
\imagelicense { Tiefkühlkonsum.svg } {} {SInner1} {Commons} {CC-by-sa 3.0} {}
\valuetablesixfromsubrows { $x$ }
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{ {6
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{ $\pi(x)$ }
{\mathrowandfive {2} {4} {6} {5} {3} }
{ {1} }
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Exp.svg} }
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\imagetext {} }
\imagelicense { Exp.svg } {} {} {Commons} {CC-by-sa 3.0} {}
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\tableleadsevenxseven
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Monkey_Saddle_Surface_(Shaded).png} }
\end{center}
\imagetext {} }
\imagelicense { Monkey Saddle Surface (Shaded).png } {} {Inductiveload} {Commons} {PD} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Schoenberg-ebringen-isohypsen.png} }
\end{center}
\imagetext {} }
\imagelicense { Schoenberg-ebringen-isohypsen.png } {} {W-j-s} {Commons} {CC-by-sa 3.0} {}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Elliptic_orbit.gif} }
\end{center}
\imagetext {} }
\imagelicense { Elliptic orbit.gif } {} {Brandir} {Commons} {CC-BY-SA 2.5} {}
\subtitle {Injective and surjective mappings}
\inputdefinition
{ }
{
Let $L$ and $M$ denote sets, and let
\mathdisp {F \colon L \longrightarrow M
, x \longmapsto F(x)} { , }
be a
mapping.
Then $F$ is called \definitionword {injective}{} if for two different elements
\mathrelationchain
{\relationchain
{ x,x'
}
{ \in }{ L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
also
\mathcor {} {F(x)} {and} {F(x')} {}
}
If we want to show that a certain mapping is injective then we may show the following: For any two elements
\mathcor {} {x} {and} {x'} {}
fulfilling the condition
\mathrelationchain
{\relationchain
{F(x)
}
{ = }{F(x')
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we can deduce that
\mathrelationchain
{\relationchain
{x
}
{ = }{x'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
This is often easier to show than the statement that
\mathrelationchain
{\relationchain
{x
}
{ \neq }{x'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
implies
\mathrelationchain
{\relationchain
{F(x)
}
{ \neq }{F(x')
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
\inputdefinition
{ }
{
Let $L$ and $M$ denote sets, and let
\mathdisp {F \colon L \longrightarrow M
, x \longmapsto F(x)} { , }
be a
mapping.
Then $F$ is called \definitionword {surjective}{} if, for every
\mathrelationchain
{\relationchain
{y
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
there exists at least one element
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
such that
\mathrelationchaindisplay
{\relationchain
{F(x)
}
{ =} {y
}
{ } {
}
{ } {
}
{ } {
}
}
}
\inputexample{}
{
We consider a football game as the mapping which assigns, to every goal of team $A$, the corresponding goal scorer. Suppose that there are no own goals and no changes. The goals of $A$ are numbered by \mathl{1,2 , \ldots , n}{.} Then we have a mapping
\mathdisp {\psi \colon \{ 1 , \ldots , n \} \longrightarrow A = \{ \text{player } 1, \,\text{player } 2 , \ldots , \text{player } 11 \}} { , }
given by
\mathrelationchaindisplay
{\relationchain
{ \psi(i)
}
{ =} { \text{ the player who has scored the } i\text{-th goal}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The injectivity of $\psi$ means that every player has scored at most one goal, the surjectivity means that every player has scored at least one goal.
}
\inputexample{}
{
Let $H$ denote the set of all
\extrabracket {living or dead} {} {}
people. We study the mapping
\mathdisp {\varphi \colon H \longrightarrow H} { , }
which assigns to every person his or her
\extrabracket {biological} {} {}
mother. This is well-defined, as every person has a uniquely determined mother. This mapping is not
injective,
since there exists different people
\extrabracket {brothers and sisters} {} {}
with the same mother. It is also not
surjective,
since not every person is a mother of somebody.
}
\inputexample{}
{
The
mapping
\mathdisp {\R \longrightarrow \R
, x \longmapsto x^2} { , }
is neither injective nor surjective. It is not
injective
because the different numbers
\mathcor {} {2} {and} {-2} {}
are both sent to $4$. It is not
surjective
because only nonnegative elements are in the image
\extrabracket {a negative number does not have a real square root} {} {.}
The mapping
\mathdisp {\R_{\geq 0} \longrightarrow \R
, x \longmapsto x^2} { , }
is injective, but not surjective. The injectivity can be seen as follows: If
\mathrelationchain
{\relationchain
{x
}
{ \neq }{y
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then one number is larger, say
\mathrelationchaindisplay
{\relationchain
{x
}
{ >} {y
}
{ \geq} {0
}
{ } {
}
{ } {
}
}
{}{}{.}
But then also
\mathrelationchain
{\relationchain
{x^2
}
{ > }{ y^2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and in particular
\mathrelationchain
{\relationchain
{x^2
}
{ \neq }{y^2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
The mapping
\mathdisp {\R \longrightarrow \R_{\geq 0}
, x \longmapsto x^2} { , }
is not injective, but surjective, since every nonnegative real number has a square root. The mapping
\mathdisp {\R_{\geq 0} \longrightarrow \R_{\geq 0}
, x \longmapsto x^2} { , }
is injective and surjective.
}
\inputdefinition
{ }
{
Let $M$ and $L$ denote sets, and suppose that
\mathdisp {F \colon M \longrightarrow L
, x \longmapsto F(x)} { , }
is a
mapping.
Then $F$ is called \definitionword {bijective}{} if $F$ is
injective
as well as
}
\inputremark {}
{
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
\mathrelationchaindisplay
{\relationchain
{F(x)
}
{ =} {y
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
(in the two variables $x$ and $y$). The surjectivity means that for every
\mathrelationchain
{\relationchain
{y
}
{ \in }{ M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
there exists at least one solution
\mathrelationchaindisplay
{\relationchain
{x
}
{ \in} { L
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
for this equation; the injectivity means that for every
\mathrelationchain
{\relationchain
{y
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
there exists at most one solution
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for this equation. The bijectivity means that for every
\mathrelationchain
{\relationchain
{y
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
there exists exactly one solution
\mathrelationchain
{\relationchain
{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.
}
\inputdefinition
{ }
{
Let
$F \colon L \rightarrow M$
denote a
bijective mapping.
Then the mapping
\mathdisp {G \colon M \longrightarrow L} { }
that sends every element
\mathrelationchain
{\relationchain
{y
}
{ \in }{M
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
to the uniquely determined element
\mathrelationchain
{\relationchain
{x
}
{ \in }{L
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mathrelationchain
{\relationchain
{F(x)
}
{ = }{y
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
}
\inputdefinition
{ }
{
Let
$L,\, M$ and $N$
denote sets, let
\mathdisp {F \colon L \longrightarrow M
, x \longmapsto F(x)} { , }
and
\mathdisp {G \colon M \longrightarrow N
, y \longmapsto G(y)} { , }
be
mappings.
Then the mapping
\mathdisp {G \circ F \colon L \longrightarrow N
, x \longmapsto G(F(x))} { , }
is called the \definitionword {composition}{} of the mappings
}
So we have
\mathrelationchaindisplay
{\relationchain
{ (G \circ F)(x)
}
{ \defeq} { 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).
\inputfactproof
{Mapping/Composition/Associativity/Fact}
{Lemma}
{}
{
\factsituation {Let
$L, M, N$ and $P$
be sets, and let
\mathdisp {F \colon L \longrightarrow M
, x \longmapsto F(x)} { , }
\mathdisp {G \colon M \longrightarrow N
, y \longmapsto G(y)} { , }
and
\mathdisp {H \colon N \longrightarrow P
, z \longmapsto H(z)} { , }
be
mappings.}
\factconclusion {Then
\mathrelationchaindisplay
{\relationchain
{ H \circ (G \circ F)
}
{ =} { (H \circ G) \circ F
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\factextra {}
{See Exercise 3.27 .}