Set Theory
Learning Objectives
Learning objectives of the Set Theory section.
Summary Table
Summary of the Set Theory section.
| Notation | Description | Example |
|---|---|---|
| \(\in\) | Element of | \(2 \in A\) |
| \(\notin\) | Not an element of | \(5 \notin A\) |
| \(|A|\) | Cardinality | \(|Colors| = 3\) |
| \(\{\}\) or \(\emptyset\) | Empty set | \(\emptyset = \{\}\) |
| \(\mathbb{N}\) | Set of natural numbers | \(\{1, 2, 3, ...\}\) |
| \(\mathbb{Z}\) | Set of integers | \(\{..., -2, -1, 0, 1, 2, ...\}\) |
| \(\mathbb{R}\) | Set of real numbers | \(\{..., -0.22,...,0,...,1,..., \pi, ... \}\) |
| Set-builder notation | Describes set by a rule | \(\{2x : x \in \mathbb{Z}\}\) |
| \(\subseteq\) | Subset | \(\mathbb{N} \subseteq \mathbb{Z}\) |
| \(\not\subseteq\) | Not a subset | \(\{1,2\} \not\subseteq \{2,3\}\) |
| \(\cup\) | Union | \(A \cup B = \{x: x \in A \text{ or } B\}\) |
| \(\cap\) | Intersection | \(A \cap B = \{x: x \in A \text{ and } B\}\) |
| \(A - B\) | Set difference | \(A - B = \{x: x \in A, x \notin B\}\) |
| \(\bar{A}\) or \(A^c\) | Complement (relative to universal set \(U\)) | \(\bar{P} = \mathbb{N} - P\) |
| \((x, y)\) | Ordered pair | \((1, 2) \neq (2, 1)\) |
| \(A \times B\) | Cartesian product | \(\{(a,b) : a \in A, b \in B\}\) |
| \(A^n\) | Cartesian power | \(\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}\) |