Set Theory

The study of collections of objects, where elements either belong to a set or they don’t—simple, yet powerful! (Hammack 2013) 🌍

Learning Objectives

Learning objectives of the Set Theory section.

• Sets
• Subsets
• Set Operations
• Cartesian Product

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}

Hammack, Richard H. 2013. Book of Proof. Richard Hammack.