Set

set-theory

Definition

Set

A set is a collection of distinct objects called “elements”.

Cantor’s Definition

Cantor's Set

According to Cantor, a set is a collection of distinct objects of our intuition into a whole.

This is not, to be sure, a formal definition, but it already shows a characteristic: A set is not a “multiplicity”, but a single object, namely a “whole”.

Equivalence

Two sets $A$ and $B$ are called equivalent if they are subset of each other:

$$A=B:⟺A\subseteq B\wedge B\subseteq A$$

Power

Let $A,B$ be two sets, then $A^B$ is defined as the set of all functions from $A$ to $B$.

$$A^B=\{f\mid f:A\to B\}$$

Example: The following is used in real analysis for sequences.

$${\mathbb{R}}^{\mathbb{N}}=\{f\mid f:A\to B\}=\{(f_0,f_1,\dots )\mid f_i\in A\}$$