% Naive Set Theory
% Mort Yao
% 2017-04-02
Basic set theory, with ZF axioms:
- Paul Halmos. Naive Set Theory.
Finite sets and infinite sets. A set is called finite if it contains finitely many elements; otherwise, it is called infinite.
Countable set and uncountable sets. A set is called countable if its elements can be enumerated; otherwise, it is called uncountable.
Clearly, all finite sets are countable. The set of natural numbers $\mathbb{N}$, the set of integers $\mathbb{Z}$ and the set of rational numbers $\mathbb{Q}$ are also countable. However, the set of real numbers $\mathbb{R}$ is uncountable.
Subset and superset. $A$ is a subset of $B$ (or: $B$ is a superset of $A$), denoted as $A \subseteq B$ (or: $B \supseteq A$), if and only if for every $x \in A$, there is $x \in B$.
$A$ and $B$ are said to be equal, denoted as $A = B$, if and only if $A \subseteq B$ and $B \subseteq A$; otherwise, $A$ and $B$ are said to be unequal, denoted as $A \neq B$.
$A$ is a proper subset of $B$ (or: $B$ is a proper superset of $A$), denoted as $A \subset B$ (or: $B \supset A$), if and only if $A \subseteq B$ and $A \neq B$.
Union. $A \cup B = { x : x \in A \lor x \in B }$.
Intersection. $A \cap B = { x : x \in A \land x \in B }$.
Difference. $A \setminus B = { x : x \not\in A \land x \in B }$.
Symmetric difference. $A \triangle B = (A \setminus B) \cup (B \setminus A) = { x : x \in A \oplus x \in B }$.
Cartesian product (cross product). $A \times B = { (x,y) : x \in A \land y \in B }$.
Power set. $\mathcal{P}(A) = { X : X \subseteq A }$.
Empty set. The empty set ${}$ is denoted as $\varnothing$. $| \varnothing | = 0$.
Disjoint sets. Two sets $A$ and $B$ are said to be disjoint, if and only if $A \cap B = \varnothing$.