Numbers

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]