Standard Number Sets

These include the following in order:

• \mathbb{N}: Natural numbers from 1 to \infty, also described as counting numbers.
• \mathbb{Z}: Integers (ganze Zahlen) are the numbers from -\infty to \infty.
• \mathbb{Q}: Rational numbers also include fraction.
• \mathbb{I}: Irrational numbers have non-repeating, non-terminating decimal expansions (for example: \pi).
• \mathbb{R}: Real numbers on the number line (rational + irrational).

You could also express these as: \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} which helps to contextualize these number sets.

Quantifying Sets

A set M is countable if and only if there exists an injection from M into \mathbb{N}.

• Countable Infinite: If there exists a bijective function f: \mathbb{N} \to M, i.e., |M| = |\mathbb{N}|
• Uncountable Infinite: |M| > |\mathbb{N}|

Note: The Cartesian Product of two countable infinite is also countable infinite, due to the Cantor-Pairfunction: f(x,y)=\frac{1}{2}(x+y)(x+y+1)+y