Set Theory
Intro
Mathematicians make a big hay out of set theory, but it's an intuitive and simple field. The hardest part is the notation. Unfortunately, I don't have a better system. Even if I did, you'd still have to learn how to read other people's notations.
What is a set?
A "set" is a fancy word for a group. It is simply stuff that you put together in an arbitrarily defined group.
A set may have zero, one, or many items. Items can be anything you can think of. Mathematicians like to think of numbers as members of a set, but that's a very limited view.
Set Definitions
Sets are defined by their members. These may either be enumerated (such as, "This set contains "A", "B", and "C" and that's it.") or classification (such as, "These sets contain all vowels in the English alphabet.")
Care must be taken not to define sets in contradictory terms. The classic example is, "The set of all sets that do not contain themselves." At first, this makes sense, but careful thought will reveal that the set can neither contain itself nor not contain itself, and so the definition simply makes no sense. (This is similar to the logical statement, "This statement is false.")
A good rule of thumb is not to allow sets to belong in themselves, or rather, don't define sets in relation to themselves.
Set Operations and Notation
All the obvious operations are obvious. The notation is not.
- Whether an item is in a set.
- <math>x \in A</math>
- The union of two sets is the set of all items belonging in either set.
- <math>A \cup B := \left\{{x: x \in A \land x \in B}\right\}</math>
- The intersection of two sets is the items that belong to both sets.
- <math>A \cap B := \left\{{x: x \in A \lor x \in B}\right\}</math>
