Set Theory
A set is an unordered collection of distinct objects.
Notation
A set can be written in a few different ways.
Special Sets
There are a collection of well known sets that we can use to classify other sets.
- \(\mathbb{N}\) or the natural numbers is the set containing \(\lbrace 0, 1, 2, 3, ...\rbrace\)
- \(\mathbb{Z}\) or all the integers \(\lbrace 0, -1, 1, -2, 2, -3, 3, ...\rbrace\)
- Rationals, all fractions with integer numerators and denominators.
- Real Number, all numbers including \(\pi, \pi, e\)
- Complex Numbers, \(\lbrace a + bi : a \in \mathbb{R}, b \in \mathbb{R}} \rbrace\)
- The empty set
Using the special sets, we can set up more complicated sets.
Set Operations
We can preform operations on sets using the operators below to find a set
Union
A Union(\(\cup\)) is used to combine two sets, resulting in a set that has everything in both sets.
Intersection
An Intersection(\(\cap\)) contains only the elements that are in both sets.
Difference
A Difference (\(\setminus\)) of sets is everything in one set and not the other.
Complement
A Complement (\(\complement\)) is everything that isn’t in a set, everything that isn’t in it.
Subsets
Some sets are entirely contained in another set, or if everything in \(R\) was in \(S\), \(R\) would be a subset of \(S\). This is denoted as \(R \subset S\). And if \(R\) and \(S\) aren’t the same, then \(R\) is a proper subset of \(S\) or \(R \subseteq S\).
This comes with some weird relationships:
- \(\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}\)
- For any set \(S\), \(\emptyset \subseteq S\).
Cardinality
The Cardinality of a set is essentially its length or how many elements are inside it.
Power Sets
The Power Set of a set is a set of all its subsets. \(\mathcal{P}(\lbrace 1, 2, 3 \rbrace) = \lbrace \emptyset, \lbrace 1 \rbrace, \lbrace 1, 2 \rbrace, \lbrace 2, 3 \rbrace, \lbrace 1, 3 \rbrace, \lbrace 1, 2, 3 \rbrace \rbrace\)
- The \(\mathcal{P}\) is the same as POW.