# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/List of definitions/Random

A natural number $n\geq 2$ is called a prime number if it is only divisible
$1$
$n$
The set which does not contain any element is called the empty set, denoted by
    $\emptyset .$
Let $T$ and $M$ denote sets. $T$ is called a subset
$M$
$T$
$M$
For sets $L$ und $M$, we call
    $L\cap M={\left\{x\mid x\in L{\text{ and }}x\in M\right\}}\,$
the intersection
For sets $L$ und $M$, we call
    $L\cup M={\left\{x\mid x\in L{\text{ or }}x\in M\right\}}\,$
the union
Suppose that two sets $L$ and $M$ are given. Then the set
    $L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,$
is called the product set
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 which is assigned to $x\in L$, is denoted by $F(x)$. For the mapping as a whole, we write
    $F\colon L\longrightarrow M,x\longmapsto F(x).$
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 $F(x')$
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
    $F(x)=y\,.$
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
Let $F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping
    $G\colon M\longrightarrow L$
which sends every element $y\in M$ to the uniquely determined element $x\in L$ with $F(x)=y$,
$F$
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
$F$
$G$
An operation (or binary operation) $\circ$ on a set $M$ is a mapping
    $\circ \colon M\times M\longrightarrow M,(x,y)\longmapsto \circ (x,y)=x\circ y.$
A set $K$ is called a field if there are two binary operations (called addition and multiplication)
    $+:K\times K\longrightarrow K{\text{ and }}\cdot :K\times K\longrightarrow K$
and two different elements $0,1\in K$, which fulfill the following properties.
* Axioms for the addition:
** Law of associativity: $(a+b)+c=a+(b+c)$ holds for all $a,b,c\in K$.
** Law of commutativity: $a+b=b+a$ holds for all $a,b\in K$.
** $0$ is the neutral element of the addition, i.e. $a+0=a$ holds for all $a\in K$.
** Existence of the negative: For every $a\in K$, there exists an element $b\in K$ with $a+b=0$.
* Axioms of the multiplication:
** Law of associativity: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds for all $a,b,c\in K$.
** Law of commutativity: $a\cdot b=b\cdot a$ holds for all $a,b\in K$.
** $1$ is the neutral element for the multiplication, i.e. $a\cdot 1=a$ holds for all $a\in K$.
** Existence of the inverse: For every $a\in K$ with $a\neq 0$, there exists an element $c\in K$ such that $a\cdot c=1$.
* Law of distributivity: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ holds for all $a,b,c\in K$.
For a natural number $n$, one puts
    $n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,$
$n$
Let $k$ and $n$ denote natural numbers with $k\leq n$. Then
    ${\binom {n}{k}}:={\frac {n!}{k!(n-k)!}}\,$
$n$
$k$
A field $K$ is called an ordered field, if there is a relation $>$ (larger than) between the elements of $K$, fulfilling the following properties ($a\geq b$ means $a>b$ or $a=b$).
* For two elements $a,b\in K$, we have either $a>b$ or $a=b$ or $b>a$.
* From $a\geq b$ and $b\geq c$, one may deduce $a\geq c$ (for any $a,b,c\in K$).
* $a\geq b$ implies $a+c\geq b+c$ (for any $a,b,c\in K$).
* From $a\geq 0$ and $b\geq 0$, one may deduce $ab\geq 0$ (for any $a,b\in K$).
Let $K$ be an ordered field. $K$ is called Archimedean, if the following Archimedean axiom holds, i.e. if for every $x\in K$ there exists a natural number $n$ such that
    $n\geq x\,.$
For real numbers $a,b$, $a\leq b$, we call
* $[a,b]={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x\leq b\right\}}$ the closed interval.
* $]a,b[={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x<b\right\}}$ the open interval.
* $]a,b]={\left\{x\in \mathbb {R} \mid x>a{\text{ and }}x\leq b\right\}}$ the half-open interval (closed on the right).
* $[a,b[={\left\{x\in \mathbb {R} \mid x\geq a{\text{ and }}x<b\right\}}$ the half-open interval (closed on the left).
For a real number $x$, the floor $\left\lfloor x\right\rfloor$ is defined as
    $\left\lfloor x\right\rfloor =n,{\text{ if }}x\in [n,n+1[{\text{ and }}n\in \mathbb {Z} .$
For a real number $x\in \mathbb {R}$, the modulus is defined in the following way.
    $\vert {x}\vert ={\begin{cases}x\,,{\text{ if }}x\geq 0\,,\\-x,\,{\text{ if }}x<0\,.\end{cases}}\,$
Let $I\subseteq \mathbb {R}$ denote an interval and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Then $f$ is called increasing, if
    $f(x')\geq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x.$
Let $I\subseteq \mathbb {R}$ denote an interval and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Then $f$ is called decreasing if
    $f(x')\leq f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'\geq x.$
Let $I\subseteq \mathbb {R}$ denote an interval and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Then $f$ is called strictly increasing if
    $f(x')>f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'>x.$
Let $I\subseteq \mathbb {R}$ denote an interval and let
    $f\colon I\longrightarrow \mathbb {R}$
denote a function. Then $f$ is called strictly decreasing if
    $f(x')<f(x){\text{ holds for all }}x,x'\in I{\text{ with }}x'>x.$
The set $\mathbb {R} ^{2}$ with $0:=(0,0)$ and $1:=(1,0)$, with componentwise addition and the multiplication defined by
    $(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,,$
is called the field of complex numbers. We denote it by
    $\mathbb {C} .$
For a complex number
    $z=a+b{\mathrm {i} }\,,$
we call
    $\operatorname {Re} \,{\left(z\right)}=a\,$
the real part of $z$ and
    $\operatorname {Im} \,{\left(z\right)}=b\,$
$z$
The mapping
    $\mathbb {C} \longrightarrow \mathbb {C} ,z=a+b{\mathrm {i} }\longmapsto {\overline {z}}:=a-b{\mathrm {i} },$
For a complex number
    $z=a+b{\mathrm {i} }\,,$
the modulus is defined by
    $\vert {z}\vert ={\sqrt {a^{2}+b^{2}}}\,.$
Let $K$ be a field. An expression of the form
    $P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,$
with $a_{i}\in K$ and $n\in \mathbb {N}$,
$K$
The degree of a nonzero polynomial
    $P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$
with $a_{n}\neq 0$
$n$
For polynomials $P,Q\in \mathbb {R} [X]$, $Q\neq 0$, the function
    $D\longrightarrow \mathbb {R} ,z\longmapsto {\frac {P(z)}{Q(z)}},$
where $D$ is the complement of the zeroes
$Q$
A real sequence is a mapping
    $\mathbb {N} \longrightarrow \mathbb {R} ,n\longmapsto x_{n}.$
Let $c\in \mathbb {R} _{+}$ denote a positive real number. The Heron-sequence, with the positive initial value $x_{0}$, is defined recursively by
    $x_{n+1}:={\frac {x_{n}+{\frac {c}{x_{n}}}}{2}}\,$
Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ denote a real sequence, and let $x\in \mathbb {R}$. We say that the sequence converges to $x$, if the following property holds.
For every positive $\epsilon >0$, $\epsilon \in \mathbb {R}$, there exists some $n_{0}\in \mathbb {N}$, such that for all $n\geq n_{0}$, the estimate
    $\vert {x_{n}-x}\vert \leq \epsilon \,$
holds.
If this condition is fulfilled, then $x$ is called the limit of the sequence. For this we write
    $\lim _{n\rightarrow \infty }x_{n}:=x\,.$
A subset $M\subseteq \mathbb {R}$ of the real numbers is called bounded, if there exist real numbers $s\leq S$ such that
$M\subseteq [s,S]$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called increasing, if $x_{n+1}\geq x_{n}$ holds for all
$n\in \mathbb {N}$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called decreasing, if $x_{n+1}\leq x_{n}$ holds for all
$n\in \mathbb {N}$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is called a Cauchy sequence, if the following condition holds.
For every $\epsilon >0$, there exists an $n_{0}\in \mathbb {N}$, such that for all $n,m\geq n_{0}$, the estimate
    $\vert {x_{n}-x_{m}}\vert \leq \epsilon \,$
Let ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence. For any strictly increasing mapping $\mathbb {N} \rightarrow \mathbb {N} ,i\mapsto n_{i}$, the sequence
    $i\mapsto x_{n_{i}}$
An ordered field $K$ is called complete or completely ordered, if every Cauchy sequence in $K$
A sequence of closed intervals
    $I_{n}=[a_{n},b_{n}],\,n\in \mathbb {N} ,$
in $\mathbb {R}$ is called (a sequence of) nested intervals, if $I_{n+1}\subseteq I_{n}$ holds for all $n\in \mathbb {N}$, and if the sequence of the lengths of the intervals, i.e.
    ${\left(b_{n}-a_{n}\right)}_{n\in \mathbb {N} },$
converges
$0$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to $+\infty$, if for every $s\in \mathbb {R}$, there exists some $N\in \mathbb {N}$, such that
    $x_{n}\geq s{\text{ holds for all }}n\geq N.$
A real sequence ${\left(x_{n}\right)}_{n\in \mathbb {N} }$ is said to tend to $-\infty$, if for every $s\in \mathbb {R}$, there exists some $N\in \mathbb {N}$. such that
    $x_{n}\leq s{\text{ holds for all }}n\geq N.$
Let ${\left(a_{k}\right)}_{k\in \mathbb {N} }$ be a sequence of real numbers. The series $\sum _{k=0}^{\infty }a_{k}$ is the sequence ${\left(s_{n}\right)}_{n\in \mathbb {N} }$ of the partial sums
    $s_{n}:=\sum _{k=0}^{n}a_{k}\,.$
If the sequence ${\left(s_{n}\right)}_{n\in \mathbb {N} }$ converges, then we say that the series converges. In this case, we write also
    $\sum _{k=0}^{\infty }a_{k}$
for its limit,
A series
    $\sum _{k=0}^{\infty }a_{k}$
of real numbers is called absolutely convergent, if the series
    $\sum _{k=0}^{\infty }\vert {a_{k}}\vert$
For every $x\in \mathbb {R}$, the series
    $\sum _{k=0}^{\infty }x^{k}$
$x$
Let $D\subseteq \mathbb {R}$ be a subset,
    $f\colon D\longrightarrow \mathbb {R}$
a function, and $x\in D$ a point. We say that $f$ is continuous in the point $x$, if for every $\epsilon >0$, there exists a $\delta >0$, such that for all $x'\in D$ fulfilling $\vert {x-x'}\vert \leq \delta$, the estimate $\vert {f(x)-f(x')}\vert \leq \epsilon$ holds. We say that $f$ continuous, if it is continuous in every point
$x\in D$
Let $T\subseteq \mathbb {R}$ denote a subset and $a\in \mathbb {R}$ a point. Let
    $f\colon T\longrightarrow \mathbb {R}$
be a function. Then $b\in \mathbb {R}$ is called limit of $f$ in $a$, if for every $\epsilon >0$ there exists some $\delta >0$ such that for all $x\in T$ fulfilling
    $\vert {x-a}\vert \leq \delta \,,$
the estimate
    $\vert {f(x)-b}\vert \leq \epsilon \,$
holds. In this case, we write
    $\operatorname {lim} _{x\rightarrow a}\,f(x)=b\,.$
Let $M$ denote a set, and
    $f\colon M\longrightarrow \mathbb {R}$
a function. We say that $f$ attains in a point $x\in M$ its maximum, if
    $f(x)\geq f(x'){\text{ holds for all }}x'\in M.$
Let $M$ denote a set, and
    $f\colon M\longrightarrow \mathbb {R}$
a function. We say that $f$ attains in a point $x\in M$ its minimum, if
    $f(x)\leq f(x'){\text{ holds for all }}x'\in M.$
Let ${\left(c_{n}\right)}_{n\in \mathbb {N} }$ be a sequence of real numbers and $x$ another real number. Then the series
    $\sum _{n=0}^{\infty }c_{n}x^{n}$
$x$
${\left(c_{n}\right)}_{n\in \mathbb {N} }$
For two series $\sum _{i=0}^{\infty }a_{i}$ and $\sum _{j=0}^{\infty }b_{j}$ of real numbers, the series
    $\sum _{k=0}^{\infty }c_{k}{\text{ with }}c_{k}:=\sum _{i=0}^{k}a_{i}b_{k-i}$
For every $x\in \mathbb {R}$, the series
    $\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$
$x$
The function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \exp x:=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},$
is called the (real)
The real number
    $e:=\sum _{k=0}^{\infty }{\frac {1}{k!}}\,$
The natural logarithm
    $\ln \colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto \ln x,$
is defined as the inverse function of the
For a positive real number $b>0$, the exponential function for the base $b$ is defined as
    $b^{x}:=\exp(x\ln b)\,.$
For a positive real number $b>0$, $b\neq 1$, the logarithm to base $b$ of $x\in \mathbb {R} _{+}$ is defined by
    $\log _{b}x:={\frac {\ln x}{\ln b}}\,.$
The function defined for $x\in \mathbb {R}$ by
    $\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,$
The function defined for $x\in \mathbb {R}$ by
    $\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,$
The function
    $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},$
A function $f\colon \mathbb {R} \rightarrow \mathbb {R}$ is called even, if for all $x\in \mathbb {R}$, the identity
    $f(x)=f(-x)\,$
A function $f\colon \mathbb {R} \rightarrow \mathbb {R}$ is called odd, if for all $x\in \mathbb {R}$, the identity
    $f(x)=-f(-x)\,$