Description

TODO! Write a small description.

Standard number systems

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 spaces.

The empty set \emptyset

Represents the absents of any elements. M \cap \emptyset \neq M (if M is not empty itself) M \cup \emptyset = M

Operations

Their are a couple operations that are specific to sets which are as follows:

A\cup B: Written as "union of A and B"; these are all elements present in A or B.

A \cap B: Written as "intersection of A and B"; these are all elements that are present in both A and B.

A \subset B: Written as "A is a subset of B"; all elements of A are contained in B.

A \supset B: Written as "A is the super set of B"; A also contains all elements in B.

A \setminus B: Written as "difference between A and B"; all elements that are only in A and not in B.

A^C: Written as "the compliment of A"; all elements that are not in A.

De Morgan's law

De Morgan's laws describe how complement, union, and intersection relate to each other.

1. (A \cup B)^C = A^C \cap B^C
2. (A \cap B)^C = A^C \cup B^C