# 集合論

(由元素集合論跳轉過嚟)

${\displaystyle \{1,3,5,7,9,\cdots \}}$

## 常見嘅集

• 自然數${\displaystyle \mathbb {N} :=\{1,2,3,4,\cdots \}}$，有啲數學家會將零包括埋入自然數呢個集入面。
• 整數${\displaystyle \mathbb {Z} :=\{\cdots ,-2,-1,0,1,2,3,4,\cdots \}=\{0,\pm 1,\pm 2,\pm 3,\cdots \}}$。由整數出嚟，可以有一個叫正整數嘅集。${\displaystyle \mathbb {Z} ^{+}:=\{0,1,2,3,4,\cdots \}}$
• 有理數${\displaystyle \mathbb {Q} :=\{{\frac {a}{b}}|a,b\in \mathbb {Z} \land b\neq 0\}}$
• 實數${\displaystyle \mathbb {R} :=\mathbb {Q} \cup \{{\text{all irrational number}}\}}$
• 複數${\displaystyle \mathbb {C} :=\{a+bi|a,b\in \mathbb {R} ,i={\sqrt {-1}}\}}$

### 例子

• ${\displaystyle \{5,6,7\}=\{7,5,6\}}$
• ${\displaystyle \{1,2,3\}\neq \{1,2\}}$
• ${\displaystyle \{\varnothing \}\neq \varnothing }$
• ${\displaystyle \{\{2\},4\}\neq \{2,4\}}$
• ${\displaystyle \mathbb {R} \neq \mathbb {N} }$
• ${\displaystyle 2\mathbb {N} =\{0,\pm 2,\pm 4,\cdots \}}$

### 例子

• ${\displaystyle \{\varnothing ,1,2\}}$嘅基數係${\displaystyle 3}$
• ${\displaystyle \{\varnothing ,1,\{4,2\}\}}$嘅基數都係${\displaystyle 3}$

## 定義集

### 例子

• ${\displaystyle \{x\in \mathbb {Z} :4\leq x\leq 6\}}$就係呢個集${\displaystyle \{4,5,6\}}$
• ${\displaystyle \{x\in \mathbb {R} |x^{2}+1=0\}}$就係呢個集${\displaystyle \varnothing }$
• ${\displaystyle \{x\in \mathbb {C} |x^{2}+1=0\}}$就係呢個集${\displaystyle \{i,-i\}}$

## 做數學基礎

### 用集合論定義自然數字

${\displaystyle 0=\{\}}$

${\displaystyle 1=\{0\}=\{\{\}\}}$

${\displaystyle 2=\{0,1\}=\{\{\},\{\{\}\}\}}$

${\displaystyle ...}$

## 攷

1. Jech, T. (2013). Set theory. Springer Science & Business Media.
2. Roitman, J. (1990). Introduction to modern set theory (Vol. 8). John Wiley & Sons.