Introduction

Table of Content

Set

\[(| A | \geq | B |) \land (| A | \leq | B |) \implies | A | = | B |\]

\[| \mathbb{C} | = | \mathbb{R} | ? \tag{1}\]

Proof

Define the mapping \(\phi: [0, 1) \times [0, 1) \to [0, 1)\), where

\[\phi((0.a_1a_2\cdots, 0.b_1b_2\cdots)) = 0.a_1b_1a_2b_2.\]

This is a bijection. ◼

\[| 2^{\mathbb{N}} | = | \mathbb{R} | \tag{2}\]

Define the mapping \(\rho: 2^{\mathbb{N}} \to [0, 1)\), where

\[\rho(A) = (0.a_1a_2a_3\cdots)_2,\ a_i = [i \in A].\]

This is a bijection. ◼

Note: \(\mathbb{R}\) 和 \([0, 1)\) 之間存在 bijection；

甚至 \(\vert \mathbb{R}\times\mathbb{R}\times\cdots\times\mathbb{R}\vert = \vert\mathbb{R}\vert\)，不論多少（finite?）\(\mathbb{R}\) 乘在一起。

\[\mathbb{Q}^{\times} = \mathbb{Q} \backslash \{0\}\] \[\mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} = \{\bar 0, \bar 1, \cdots, \overline{n-1} \}\]

\(\bar 0, \bar 1, \cdots, \overline{n-1}\) 是 \(\mathbb{Z}_n\) 的 elements。

見 Relation