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!