## Introduction to Set Theory

### Intro

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some." - Paul Halmos

### What is a set

A set is a collection of "things".

What "things" you may ask? Almost anything you'd like ...

A collection of three colours, red, blue, and yellow is a set. So is the collection of the names of capital cities of Australia (or the world for that matter).

We write a set by enclosing the collection of "things" inside \(\{\text{brackets}\}\) with commas separating the different objects.

e.g. \(\{\text{North}, \text{East}, \text{South}, \text{West}\}\) is a set

Often we consider sets that contain numbers. Other times we consider sets that contain sets!

e.g. \(\{17, 42, 108\}\) is also a set

### Properties of sets

Sets do not care about multiples of the same "thing", nor do they care about order

e.g. \(\{\text{cat}, \text{dog}\}\) is the same as \(\{\text{cat}, \text{dog}, \text{cat}\}\)

e.g. \(\{1, 2, 3\}\) is the same as \(\{3,2,1\}\)

We use the symbol \(\in\) to represent "membership" to a set.

We write \(1 \in \{1,2,3\}\) to say, 1 is a "member of" (or "belongs to") the set \(\{1, 2, 3\}\). \(4 \notin \{1,2,3\}\).

We usually denote a set with a capital letter (usually starting at the start of the alphabet).

We generally denote members of the set with a lower case letter (usually from the end of the alphabet).

For example you will often see something like. Let \(x\) be a member of the set \(A\)

If all the members in one set, say set \(A\) , also belong to another set, say set \(B\) , we say that \(A\) is a subset of \(B\) We write this as \(A \subseteq B\)

\(B\) may very well contain more "things" than \(A\)

For example the set \(\{1,2,3\} \subseteq \{\cdots, -2, -1,0,1,2,\cdots\}.\)

### Some famous sets

- The counting numbers \(\{0, 1, 2, 3, 4, 5, \cdots\}\), otherwise known as the Natural Numbers, denoted by \(\mathbb{N}\)
- The Integers \(\{\cdots, -2, -1, 0, 1, 2, \cdots\}\) denoted by \(\mathbb{Z}\)
- The Rational Numbers, i.e. those numbers that can be expressed as a ratio of two integers, denoted by \(\mathbb{Q}\)
- The Real Numbers, denoted by \(\mathbb{R}\)
- The set with no members \(\{\}\), otherwise known as the Empty Set, denoted by \(\emptyset\)

### What can we do with sets

Create a new set by combining the members of other sets. We call this the **Union** of
sets.
We write the union of two sets \(A\) and \(B\) as \(A \cup B\).

e.g. Let \(A = \{a, b, c\}\) and let \(B = \{x, y\}\), then \(A \cup B = \{a, b, c, x, y\}\)

Create a new set by including only the common members of other sets. We call this the
**Intersection** of sets.

We write the intersection of two sets \(C\) and \(D\) as \(C \cap D\).

e.g. Let \(C = \{-2, 0, 2\}\) and let \(D = \{-1, 0, 1\}\), then \(C \cap D = \{0\}\)

Count them ... well, kind of, ... We call this the **Cardinality** of a set.

Formally it is not really a "count", but we will get to that another day!

The number of elements (the Cardinality) of a set \(A\) is is denoted by \(|A|\)

e.g. Let \(A = \{a, b, c\}\), then \(|A| = 3\).

### Generating rules

We often write a set explicitly, by listing out all of the elements in the set.

This doesn't work when we want to write out infinite sets.

In these cases we use a "generating rule" (or rules) to define how a set is generated.

For example, we could define the even integers the following way: \(\{2x \text{ such that } x \in \mathbb{Z}\}\)

### Some of the big results of set theory

I'd like to end this post by briefly bringing up some of the most famous results of Set Theory. I want to give the beginning of an answer to the question "Why do we care about sets?"

- Formal ways of building the real numbers (via Dedekind cuts for example)
- There are more real numbers than natural numbers (Cantor's Diagonalisation Argument)
- Mathematics doesn't play as nice as we would have liked (Godel's Incompleteness Theorems)

### Further Reading

I'd recommend the following two resources!