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==Natural numbers==
==Natural numbers==
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We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.
We say that <math>\mathbb{N}_{0}</math> is the set of all '''natural numbers'''.
'''Induction:'''
[http://en.wikipedia.org/wiki/Mathematical_induction Mathematical induction] is a well-known method to prove a statement <math>S(n)</math> for all natural numbers <math>n\geq n_S</math>, where <math>n_S</math> is some integral number specific to <math>S</math>. Two types of induction are often distinguished:
# Ordinary induction: for <math>n>n_S</math>, <math>S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
# Strong induction: for <math>n>n_S</math>, <math>S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1)</math> is the induction hypothesis for the proof of <math>S(n)</math>.
Both types of induction are logically equivalent. Therefore, in this wiki, they are not distinguished from each other: induction always means strong induction.
==Real numbers==
<math>\mathbb{R}</math> denotes the set of all '''real numbers'''. We also define the following sets:
: <math>\mathbb{R}^+ := \{x\mid x\in \mathbb{R},x>0\}</math> (set of all positive real numbers)
: <math>\mathbb{R}^+_0 := \mathbb{R}^+ \cup \{ 0 \}</math>
Additionally <math>+\infty</math> or, for short, <math>\infty</math>, denotes the unique number that is larger than all real numbers. Analogously, <math>-\infty</math> is the unique number that is smaller than all real numbers. By convention the following properties hold for all <math>x\in\mathbb{R}</math>:
: <math>x+\infty = \infty + x = \infty + \infty = \infty</math>
We say that <math>+\infty</math> and <math>-\infty</math> are the '''neutral elements''' of the minimum and maximum operation, respectively:
: <math>\min\emptyset = +\infty</math>
: <math>\max\emptyset = -\infty</math>
==Empty sets and intervals==
# For <math>i>j</math>, we define <math>\{ x_i,...,x_j\}:=\emptyset</math>.
# For <math>a>b</math>, we define <math>[a,b]:=\emptyset</math>
==Boolean==
<math>\mathbb{B}</math> denotes the set of binary truth values ('''Boolean''' values).
: <math>\mathbb{B}:=\{ true,false\}</math>

Latest revision as of 18:09, 9 June 2015

Natural numbers

[math]\displaystyle{ \mathbb{N} }[/math] denotes the set of positive integral numbers:

[math]\displaystyle{ \mathbb{N} := \{1,2,3,...\} }[/math]
[math]\displaystyle{ \mathbb{N}_{0} := \mathbb{N} \cup \{0\} }[/math]

We say that [math]\displaystyle{ \mathbb{N}_{0} }[/math] is the set of all natural numbers.

Induction: Mathematical induction is a well-known method to prove a statement [math]\displaystyle{ S(n) }[/math] for all natural numbers [math]\displaystyle{ n\geq n_S }[/math], where [math]\displaystyle{ n_S }[/math] is some integral number specific to [math]\displaystyle{ S }[/math]. Two types of induction are often distinguished:

  1. Ordinary induction: for [math]\displaystyle{ n\gt n_S }[/math], [math]\displaystyle{ S(n-1) }[/math] is the induction hypothesis for the proof of [math]\displaystyle{ S(n) }[/math].
  2. Strong induction: for [math]\displaystyle{ n\gt n_S }[/math], [math]\displaystyle{ S(n_S)\wedge S(n_S+1)\wedge\ldots\wedge S(n-1) }[/math] is the induction hypothesis for the proof of [math]\displaystyle{ S(n) }[/math].

Both types of induction are logically equivalent. Therefore, in this wiki, they are not distinguished from each other: induction always means strong induction.

Real numbers

[math]\displaystyle{ \mathbb{R} }[/math] denotes the set of all real numbers. We also define the following sets:

[math]\displaystyle{ \mathbb{R}^+ := \{x\mid x\in \mathbb{R},x\gt 0\} }[/math] (set of all positive real numbers)
[math]\displaystyle{ \mathbb{R}^+_0 := \mathbb{R}^+ \cup \{ 0 \} }[/math]

Additionally [math]\displaystyle{ +\infty }[/math] or, for short, [math]\displaystyle{ \infty }[/math], denotes the unique number that is larger than all real numbers. Analogously, [math]\displaystyle{ -\infty }[/math] is the unique number that is smaller than all real numbers. By convention the following properties hold for all [math]\displaystyle{ x\in\mathbb{R} }[/math]:

[math]\displaystyle{ x+\infty = \infty + x = \infty + \infty = \infty }[/math]

We say that [math]\displaystyle{ +\infty }[/math] and [math]\displaystyle{ -\infty }[/math] are the neutral elements of the minimum and maximum operation, respectively:

[math]\displaystyle{ \min\emptyset = +\infty }[/math]
[math]\displaystyle{ \max\emptyset = -\infty }[/math]

Empty sets and intervals

  1. For [math]\displaystyle{ i\gt j }[/math], we define [math]\displaystyle{ \{ x_i,...,x_j\}:=\emptyset }[/math].
  2. For [math]\displaystyle{ a\gt b }[/math], we define [math]\displaystyle{ [a,b]:=\emptyset }[/math]

Boolean

[math]\displaystyle{ \mathbb{B} }[/math] denotes the set of binary truth values (Boolean values).

[math]\displaystyle{ \mathbb{B}:=\{ true,false\} }[/math]