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

Definition:Prime number

A natural number $n\ge 2$ is called a prime number if it is only divisible

by $1$ and by $n$.

Definition:Empty set

The set which does not contain any element is called the empty set, denoted by

$\mathrm{\varnothing }.$

Definition:Subset

Let $T$ and $M$ denote sets. $T$ is called a subset

of $M$ if every element of $T$ is also an element of $M$.

Definition:Intersection

For sets $L$ und $M$, we call

$L\cap M=\{x\mid x\in L\text{ and }x\in M\}$

the intersection

of the two sets.

Definition:Union

For sets $L$ und $M$, we call

$L\cup M=\{x\mid x\in L\text{ or }x\in M\}$

the union

of the sets.

Definition:Cartesian product

Suppose that two sets $L$ and $M$ are given. Then the set

$L\times M=\{(x,y)\mid x\in L,y\in M\}$

is called the product set

(or Cartesian product) of the sets.

Definition:Mapping

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

$F:L⟶M,x⟼F(x).$

Definition:Injective

Let $L$ and $M$ denote sets, and let

$F:L⟶M,x⟼F(x),$

be a mapping. Then $F$ is called injective if for two different elements $x,x^{\prime }\in L$, also $F(x)$ and $F(x^{\prime })$

are different.

Definition:Surjective

Let $L$ and $M$ denote sets, and let

$F:L⟶M,x⟼F(x),$

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:Bijective

Let $M$ and $L$ denote sets, and suppose that

$F:M⟶L,x⟼F(x),$

is a mapping. Then $F$ is called bijective if $F$ is injective as well as

surjective.

Definition:Inverse mapping

Let $F:L\to M$ denote a bijective mapping. Then the mapping

$G:M⟶L$

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 $F$.

Definition:Composition

Let $L,M$ and $N$ denote sets, let

$F:L⟶M,x⟼F(x),$

and

$G:M⟶N,y⟼G(y),$

be mappings. Then the mapping

$G\circ F:L⟶N,x⟼G(F(x)),$

is called the composition of the mappings

$F$ and $G$.

Definition:Operation

An operation (or binary operation) $\circ$ on a set $M$ is a mapping

$\circ :M\times M⟶M,(x,y)⟼\circ (x,y)=x\circ y.$

Definition:Field

A set $K$ is called a field if there are two binary operations (called addition and multiplication)

$+:K\times K⟶K\text{ and }\cdot :K\times K⟶K$

and two different elements $0,1\in K$ that fulfill the following properties.

1. Axioms for the addition:
   1. Associative law: $(a+b)+c=a+(b+c)$ holds for all $a,b,c\in K$.
   2. Commutative law: $a+b=b+a$ holds for all $a,b\in K$.
   3. $0$ is the neutral element of the addition, i.e., $a+0=a$ holds for all $a\in K$.
   4. Existence of the negative: For every $a\in K$, there exists an element $b\in K$ with $a+b=0$.
2. Axioms of the multiplication:
   1. Associative law: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds for all $a,b,c\in K$.
   2. Commutative law: $a\cdot b=b\cdot a$ holds for all $a,b\in K$.
   3. $1$ is the neutral element for the multiplication, i.e., $a\cdot 1=a$ holds for all $a\in K$.
   4. Existence of the inverse: For every $a\in K$ with $a\ne 0$, there exists an element $c\in K$ such that $a\cdot c=1$.
3. Distributive law: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ holds for all $a,b,c\in K$.

Definition:Factorial

For a natural number $n$, one puts

$n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1,$

and calls this $n$ factorial.

Definition:Binomial coefficient

Let $k$ and $n$ denote natural numbers with $k\le n$. Then

$(\genfrac{}{}{0ex}{}{n}{k}):=\frac{n!}{k!(n-k)!}$

is called the binomial coefficient $n$ choose $k$

Definition:Ordered field

A field $K$ is called an ordered field, if there is a relation $>$ (larger than) between the elements of $K$, fulfilling the following properties ($a\ge b$ means $a>b$ or $a=b$).

1. For two elements $a,b\in K$, we have either $a>b$ or $a=b$ or $b>a$.
2. From $a\ge b$ and $b\ge c$, one may deduce $a\ge c$ (for any $a,b,c\in K$).
3. $a\ge b$ implies $a+c\ge b+c$ (for any $a,b,c\in K$).
4. From $a\ge 0$ and $b\ge 0$, one may deduce $ab\ge 0$ (for any $a,b\in K$).

Definition:Archimedean ordered field

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

$n\ge x.$

Definition:Real intervals

For real numbers $a,b$, $a\le b$, we call

1. $[a,b]=\{x\in ℝ\mid x\ge a\text{ and }x\le b\}$ the closed interval.
2. $]a,b[=\{x\in ℝ\mid x>a\text{ and }x<b\}$ the open interval.
3. $]a,b]=\{x\in ℝ\mid x>a\text{ and }x\le b\}$ the half-open interval (closed on the right).
4. $[a,b[=\{x\in ℝ\mid x\ge a\text{ and }x<b\}$ the half-open interval (closed on the left).

Definition:Floor

For a real number $x$, the floor $\lfloor x\rfloor$ is defined as

$\lfloor x\rfloor =n,\text{ if }x\in [n,n+1[\text{ and }n\in \mathbb{Z}.$

Definition:Modulus of a real number

For a real number $x\in ℝ$, the modulus is defined in the following way.

$|x|=\{\begin{array}{c}x,\text{ if }x\ge 0,\\ -x,\text{ if }x<0.\end{array}$

Definition:Increasing function

Let $I\subseteq ℝ$ denote an interval and let

$f:I⟶ℝ$

denote a function. Then $f$ is called increasing, if

$f(x^{\prime })\ge f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }\ge x.$

Definition:Decreasing function

Let $I\subseteq ℝ$ denote an interval and let

$f:I⟶ℝ$

denote a function. Then $f$ is called decreasing if

$f(x^{\prime })\le f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }\ge x.$

Definition:Strictly increasing function

Let $I\subseteq ℝ$ denote an interval and let

$f:I⟶ℝ$

denote a function. Then $f$ is called strictly increasing if

$f(x^{\prime })>f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }>x.$

Definition:Strictly decreasing function

Let $I\subseteq ℝ$ denote an interval and let

$f:I⟶ℝ$

denote a function. Then $f$ is called strictly decreasing if

$f(x^{\prime })<f(x)\text{ holds for all }x,x^{\prime }\in I\text{ with }{x}^{\prime }>x.$

Definition:Complex numbers

The set ${ℝ}^2$ 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

$ℂ.$

Definition:Real part, imaginary part

For a complex number

$z=a+b\mathrm{i},$

we call

$\mathrm{Re}\left(z\right)=a$

the real part of $z$ and

$\mathrm{Im}\left(z\right)=b$

the imaginary part of $z$.

Definition:Complex conjugation

The mapping

$ℂ⟶ℂ,z=a+b\mathrm{i}⟼\bar{z}:=a-b\mathrm{i},$

is called complex conjugation.

Definition:Modulus of a complex number

For a complex number

$z=a+b\mathrm{i},$

the modulus is defined by

$|z|=\sqrt{a^2+b^2}.$

Definition:Polynomial in one variable

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 $n\in \mathbb{N}$,

is called a polynomial in one variable over $K$.

Definition:Degree of a polynomial

The degree of a nonzero polynomial

$P=a_0+a_{1}X+a_2{X}^2+\cdots +a_n{X}^n$

with $a_n\ne 0$

is $n$.

Definition:Rational function

For polynomials $P,Q\in ℝ[X]$, $Q\ne 0$, the function

$D⟶ℝ,z⟼\frac{P(z)}{Q(z)},$

where $D$ is the complement of the zeroes

of $Q$, is called a rational function.

Definition:Real sequence

A real sequence is a mapping

$\mathbb{N}⟶ℝ,n⟼x_n.$

Definition:Heron sequence

Let $c\in {ℝ}_{+}$ denote a positive real number. The Heron-sequence, with the positive initial value $x_0$, is defined recursively by

$x_{n+1}:=\frac{x_n+\frac{c}{x_n}}{2}$

Definition:Convergent sequence

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ denote a real sequence, and let $x\in ℝ$. We say that the sequence converges to $x$, if the following property holds.

For every positive $ϵ>0$, $ϵ\in ℝ$, there exists some $n_0\in \mathbb{N}$, such that for all $n\ge n_0$, the estimate

$|x_n-x|\le ϵ$

holds.

If this condition is fulfilled, then $x$ is called the limit of the sequence. For this we write

$\underset{n\to \mathrm{\infty }}{lim}{x}_n:=x.$

If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence diverges.

Definition:Bounded subset

A subset $M\subseteq ℝ$ of the real numbers is called bounded, if there exist real numbers $s\le S$ such that

$M\subseteq [s,S]$.

Definition:Increasing sequence

A real sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is called increasing, if $x_{n+1}\ge x_n$ holds for all

$n\in \mathbb{N}$.

Definition:Decreasing sequence

A real sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is called decreasing, if $x_{n+1}\le x_n$ holds for all

$n\in \mathbb{N}$.

Definition:Cauchy sequence

A real sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is called a Cauchy sequence, if the following condition holds.

For every $ϵ>0$, there exists an $n_0\in \mathbb{N}$, such that for all $n,m\ge n_0$, the estimate

$|x_n-x_m|\le ϵ$

holds.

Definition:Subsequence

Let ${\left(x_n\right)}_{n\in \mathbb{N}}$ be a real sequence. For any strictly increasing mapping $\mathbb{N}\to \mathbb{N},i\mapsto n_i$, the sequence

$i\mapsto x_{n_i}$

is called a subsequence of the sequence.

Definition:Completely ordered field

An ordered field $K$ is called complete or completely ordered, if every Cauchy sequence in $K$

converges.

Definition:Nested intervals

A sequence of closed intervals

$I_n=[a_n,b_n],n\in \mathbb{N},$

in $ℝ$ 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.

${(b_n-a_n)}_{n\in \mathbb{N}},$

converges

to $0$.

Definition:Tending to $+\mathrm{\infty }$

A real sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is said to tend to $+\mathrm{\infty }$, if for every $s\in ℝ$, there exists some $N\in \mathbb{N}$, such that

$x_n\ge s\text{ holds for all }n\ge N.$

Definition:Tending to $-\mathrm{\infty }$

A real sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ is said to tend to $-\mathrm{\infty }$, if for every $s\in ℝ$, there exists some $N\in \mathbb{N}$. such that

$x_n\le s\text{ holds for all }n\ge N.$

Definition:Series

Let ${\left(a_k\right)}_{k\in \mathbb{N}}$ be a sequence of real numbers. The series $\sum _{k=0}^{\mathrm{\infty }}{a}_k$ 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 also write

$\sum _{k=0}^{\mathrm{\infty }}{a}_k$

for its limit,

and this limit is called the sum of the series.

Definition:Absolute convergence of a series

A series

$\sum _{k=0}^{\mathrm{\infty }}{a}_k$

of real numbers is called absolutely convergent, if the series

$\sum _{k=0}^{\mathrm{\infty }}|a_k|$

converges.

Definition:Geometric series

For every $x\in ℝ$, the series

$\sum _{k=0}^{\mathrm{\infty }}{x}^k$

is called the geometric series in $x$.

Definition:Continuous function

Let $D\subseteq ℝ$ be a subset,

$f:D⟶ℝ$

a function, and $x\in D$ a point. We say that $f$ is continuous in the point $x$, if for every $ϵ>0$, there exists a $\delta >0$, such that for all $x^{\prime }\in D$ fulfilling $|x-x^{\prime }|\le \delta$, the estimate $|f(x)-f(x^{\prime })|\le ϵ$ holds. We say that $f$ continuous, if it is continuous in every point

$x\in D$

Definition:Limit of a function

Let $T\subseteq ℝ$ denote a subset and $a\in ℝ$ a point. Let

$f:T⟶ℝ$

be a function. Then $b\in ℝ$ is called limit of $f$ in $a$, if for every $ϵ>0$ there exists some $\delta >0$ such that for all $x\in T$ fulfilling

$|x-a|\le \delta ,$

the estimate

$|f(x)-b|\le ϵ$

holds. In this case, we write

${\lim }_{x\to a}f(x)=b.$

Definition:Maximum

Let $M$ denote a set, and

$f:M⟶ℝ$

a function. We say that $f$ attains in a point $x\in M$ its maximum, if

$f(x)\ge f(x^{\prime })\text{ holds for all }{x}^{\prime }\in M.$

Definition:Minimum

Let $M$ denote a set, and

$f:M⟶ℝ$

a function. We say that $f$ attains in a point $x\in M$ its minimum, if

$f(x)\le f(x^{\prime })\text{ holds for all }{x}^{\prime }\in M.$

Definition:Power series

Let ${\left(c_n\right)}_{n\in \mathbb{N}}$ be a sequence of real numbers and $x$ another real number. Then the series

$\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$

is called the power series in $x$ for the coefficients ${\left(c_n\right)}_{n\in \mathbb{N}}$.

Definition:Cauchy product

For two series $\sum _{i=0}^{\mathrm{\infty }}{a}_i$ and $\sum _{j=0}^{\mathrm{\infty }}{b}_j$ of real numbers, the series

$\sum _{k=0}^{\mathrm{\infty }}{c}_k\text{ with }{c}_k:=\sum _{i=0}^k{a}_i{b}_{k-i}$

is called the Cauchy-product of the series.

Definition:Exponential series

For every $x\in ℝ$, the series

$\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$

is called the exponential series in $x$.

Definition:Exponential function

The function

$ℝ⟶ℝ,x⟼\exp x:=\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!},$

is called the (real)

exponential function.

Definition:Euler's number

The real number

$e:=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}$

is called Euler's number.

Definition:Natural logarithm

The natural logarithm

$\ln :{ℝ}_{+}⟶ℝ,x⟼\ln x,$

is defined as the inverse function of the

real exponential function.

Definition:Exponential function to base

For a positive real number $b>0$, the exponential function for the base $b$ is defined as

$b^x:=\exp (x\ln b).$

Definition:Logarithm to base

For a positive real number $b>0$, $b\ne 1$, the logarithm to base $b$ of $x\in {ℝ}_{+}$ is defined by

${\log }_{b}x:=\frac{\ln x}{\ln b}.$

Definition:Hyperbolic sine

The function defined for $x\in ℝ$ by

$\sinh x:=\frac{1}{2}(e^x-e^{-x}),$

is called hyperbolic sine.

Definition:Hyperbolic cosine

The function defined for $x\in ℝ$ by

$\cosh x:=\frac{1}{2}(e^x+e^{-x}),$

is called hyperbolic cosine.

Definition:Hyperbolic tangent

The function

$ℝ⟶ℝ,x⟼\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}},$

is called hyperbolic tangent.

Definition:Even function

A function $f:ℝ\to ℝ$ is called even if, for all $x\in ℝ$, the identity

$f(x)=f(-x)$

holds.

Definition:Odd function

A function $f:ℝ\to ℝ$ is called odd if, for all $x\in ℝ$, the identity

$f(x)=-f(-x)$

holds.