> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/fundamental-sets.md).

# Fundamental sets

Numbers are classified into sets according to their properties:

* the set of natural numbers
* the set of integers
* the set of rational numbers
* the set of real numbers
* the set of complex numbers

The most fundamental number sets have their own unique identifier:

* 𝔹, the set of the Booleans, $$\mathbb{B}$$
* ℕ, the set of the natural numbers, $$\mathbb{N}$$
* ℤ, the set of the integers, $$\mathbb{Z}$$ (from ger. Zahl "number")
* ℚ, the set of the rational numbers, $$\mathbb{Q}$$ (from quotient)
* ℝ, the set of the real numbers, $$\mathbb{R}$$
* ℂ, the set of the complex numbers, $$\mathbb{C}$$
* ℍ, the set of the quaternions, $$\mathbb{H}$$
* 𝕆, the set of the octonions, $$\mathbb{O}$$
* Their relation: $$\varnothing\subset \mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R} \subset \mathbb{C}$$

and these number sets, being so fundamental in mathematics, that each has a name and a special symbol to denote it.

Each number set is infinite, but the set of natural numbers, the set of integers and the set of rational numbers is countably infinite, while the set of real numbers and the set of complex numbers numbers are bigger, uncountable, infinities.

* $$\mathbb{B}$$ is the finite set of boolean values (or Boolean domain)    

  $$\quad {\bot,\top}$$ or $${F,T}$$ or $${0,1}$$
* $$\mathbb{N}$$ is countable infinite set of natural numbers     

  $$\quad \mathbb{N}={0,1,2,3,\dots}$$
* $$\mathbb{Z}$$ is countable infinite set of integers     

  $$\quad \mathbb{Z}={\dots,-2,-1,0,1,2,\dots}$$.     
* $$\mathbb{Q}$$ is countable infinite set of rational numbers     

  $$\quad {p/q : p \in\mathbb{Z}, q\in \mathbb{N}\land q\neq 0}$$
* $$\mathbb{R}$$ is uncountable infinite set of real numbers.    

  It has various kinds of infinities.
* $$\mathbb{C}$$ is uncountable infinite set of complex numbers    

  $$\quad {a,b,i:a,b\in\mathbb{R}}, \quad i=\sqrt{-1}$$

Inclusion relation of number sets: $$\mathbb{N}\subseteq \mathbb{Z}\subseteq \mathbb{Q}\subseteq \mathbb{R} \subseteq \mathbb{C} \subseteq \mathbb{H}$$

N⊆Z⊆Q⊆R⊆C⊆H

N ⊆ Z ⊆ Q ⊆ R ⊆ C ⊆ H


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