Set

A set is a well-defined collection of distinct objects, called elements or members of the set.

Notation

• $x \in A$ means $x$ is an element of $A$
• $x \notin A$ means $x$ is not an element of $A$
• $\{a, b, c\}$ denotes the set containing elements $a$, $b$, and $c$
• $\{x : P(x)\}$ or $\{x \mid P(x)\}$ denotes the set of all $x$ satisfying property $P$

Basic Sets

• $\emptyset$ or $\{\}$ — the empty set
• $\mathbb{N}$ — natural numbers
• $\mathbb{Z}$ — integers
• $\mathbb{Q}$ — rational numbers
• $\mathbb{R}$ — real numbers
• $\mathbb{C}$ — complex numbers

Operations

• Union: $A \cup B = \{x : x \in A \text{ or } x \in B\}$
• Intersection: $A \cap B = \{x : x \in A \text{ and } x \in B\}$
• Difference: $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$
• Complement: $A^c = \{x : x \notin A\}$

Related

• cardinality
• function
• subset