Set

Overview

Set theory concerns itself with using sets as the axiomatic foundation for math. These are outlined in ZFC.

Index Sets

Let $I$ be a set, called the index set. Let $F$ be a function whose domain includes $I$ . Then we define

⋃_{i \in I} F (i) = ⋃ {F (i) ∣ i \in I}

and, if $I \neq \emptyset$ ,

⋂_{i \in I} F (i) = ⋂ {F (i) ∣ i \in I}

Cartesian Product

Provided that the sets are suitably indexed, we can form (something like) the Cartesian product of infinitely many sets. Let $I$ be an index set and $H$ a function whose domain includes $I$ . Then

\underset{i \in I}{\times} H (i) = {f ∣ f is a function with domain I and \forall i \in I, f (i) \in H (i)}

Function Sets

For sets $A$ and $B$ , the collection of functions $F$ from $A$ into $B$ is:

^{A} B = {F ∣ F : A \to B}

$^{A} B$ is read as " $B$ -pre- $A$ ". It is often written as $B^{A}$ instead.