查看“︁并集”︁的源代码

{{noteTA
|2=zh-cn:布尔逻辑; zh-hk:布林運算; 
|G1=Math
}} 

[[File:set_union.png|thumb|A和B的并集]]

在[[集合论]]和[[数学]]的其他分支中，一群[[集合 (数学)|集合]]的'''并集'''（Union）<ref>{{cite book |author=程极泰 |title=[[集合论]] |year=1985 |publisher=国防工业出版社 |edition=第一版 |series=应用数学丛书 |id=15034.2766 |pages=14}}</ref>，是以这群集合的所有元素來构成的集合。

== 有限聯集 ==

聯集是由[[公理化集合论]]的[[策梅洛-弗兰克尔集合论#分類公理|分類公理]]來確保其唯一存在的特定集合 <math>A \cup B</math> ：
:<math>(\forall A)(\forall B)(\forall x)\left\{
    (x \in A \cup B)
    \Leftrightarrow
    \left[
        (x \in A)
        \vee
        (x \in B)
    \right]
\right\}  </math>

也就是直觀上：
:「對所有 <math>x</math> ， <math>x \in A \cup B</math> 等價於 <math>x \in A</math> 或 <math>x \in B</math>」

举例：

集合<math>\{1, 2, 3\}</math>和<math>\{2, 3, 4\} </math>的并集是<math>\{1, 2, 3, 4\}</math>。数<math>9</math>'''不属于'''[[素数]]集合<math>\{2, 3, 5, 7, 11,\ldots\}</math>和[[偶数]]集合<math>\{2, 4, 6, 8, 10,\ldots\}</math>的并集，因为<math>9</math>既不是素数，也不是偶数。

更通常的，多个集合的并集可以这样定义：
例如，<math>A,B</math>和<math>C</math>的并集含有所有<math>A</math>的元素，所有<math>B</math>的元素和所有<math>C</math>的元素，而没有其他元素。形式上：
:<math>x</math>是<math>A \cup B \cup C</math>的元素，当且仅当<math>x</math>属于<math>A</math>或<math>x</math>属于<math>B</math>或<math>x</math>属于<math>C</math>。

=== 代数性质 ===
二元并集（两个集合的并集）是一种[[结合律|结合]]运算，即
:<math>A \cup (B \cup C) =(A \cup B)\cup C</math>。事实上，<math>A \cup B \cup C</math>也等于这两个集合，因此圆括号在仅进行并集运算的时候可以省略。

相似的，并集运算满足[[交换律]]，即集合的顺序任意。

[[空集]]是并集运算的[[单位元]]。即<math>\varnothing \cup A = A</math>，对任意集合<math>A</math>。可以将空集当作[[0|零]]个集合的并集。

结合[[交集]]和[[补集]]运算，并集运算使任意[[幂集]]成为[[布尔代数]]。例如，并集和交集相互满足[[分配律]]，而且这三种运算满足[[德·摩根律]]。若将并集运算换成[[对称差]]运算，可以获得相应的[[布尔环]]。

== 无限并集 ==

由[[公理化集合论]]的[[并集公理]]，有唯一的集合 <math>\bigcup\mathcal{M}</math> 滿足：

: <math>(\forall \mathcal{M})(\forall x)\left\{
    \left(x \in \bigcup\mathcal{M}\right)
    \Leftrightarrow
    (\exists A)\left[
        \left(A \in \mathcal{M}\right)
        \wedge
        (x \in A)
    \right]
\right\}</math>

也就是直觀上「對所有 <math>\mathcal{M}</math> 和所有 <math>x</math> ， <math>x \in \bigcup\mathcal{M}</math> 等價於有某個 <math>\mathcal{M}</math> 的下屬集合 <math>A</math> ，使得<math>x \in A</math>」。以上的 <math>\mathcal{M}</math> 可以直觀的視為一個[[集合族]]，而把 <math>\bigcup\mathcal{M}</math> 看成對 <math>\mathcal{M}</math> 內的[[集合 (数学)|集合]]取并集，但這個公理並沒有對 <math>\mathcal{M}</math> 下屬集合的數量做出任何限制，所以這個 <math>\bigcup\mathcal{M}</math> 被俗稱為'''任意并集'''或'''无限并集'''。

若 <math>X \subseteq \bigcup\mathcal{M}</math> ，會稱 <math>X</math> 被 <math>\mathcal{M}</math>  '''覆蓋'''（cover），也就是直觀上可以用 <math>\mathcal{M}</math> 裡的所有集合疊起來蓋住 <math>X</math>。

例如：

對 <math>\mathcal{M} = \{A,B,C\}</math>，<math>\bigcup\mathcal{M} = A \cup  B \cup  C</math> ，若 <math>M</math>是[[空集]]， <math>\bigcup\mathcal{M}</math> 也是空集。

无限并集有多种表示方法：

可模仿[[求和符号]]記為
: <math>\bigcup_{A\in \mathcal{M}} A</math>。
但大多數人會假設[[指标集]] <math>I</math> 的存在，換句話說
: 若 <math>I \,\overset{A}{\cong}\, \mathcal{M} </math> 則 <math>\bigcup_{i\in I} A(i) := \bigcup \mathcal{M}</math>
在[[指标集]] <math>I</math> 是[[自然数|自然数系]] <math>\N</math> 的情况下，更可以仿[[无穷级数]]來表示，也就是說：
: 若 <math>\N \,\overset{A}{\cong}\, \mathcal{M}</math> 則 <math>\bigcup^{\infty}_{i = 0} A(i) := \bigcup \mathcal{M}</math>
也可以更'''粗略直觀'''的將 <math>\bigcup^{\infty}_{i = 0} A(i)</math> 写作<math>A_{0} \cup  A_{1} \cup  A_{2} \cup \ldots</math>。

=== 无限并集的性質 ===
{{Math theorem
| name = 定理(0)
| math_statement = <br/>
<math>
\vdash
\bigcup\varnothing = \varnothing
</math>
}}
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|(1) <math>(\forall x)[\neg(x \in \varnothing)]</math> ([[空集公理]])

(2) <math>\neg(S \in \varnothing) </math>(MP with [[一阶逻辑#量词公理|A4]], 1)

(3)<math>(S \in \varnothing)
\Rightarrow
[\neg(x \in S)] </math>([[一阶逻辑#實質條件|M0]] with 2)

(4)<math>\neg\neg(S \in \varnothing)
\Rightarrow
[\neg(x \in S)]  </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#否定|DN]], 3)

(5)<math>\neg\{
[\neg(S \in \varnothing)]
\wedge
[\neg(x \in S)]
\}  </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#德摩根定律|De Morgan]], 4)

(6)<math>(\forall S)\big\{
    \neg\{
        [\neg(S \in \varnothing)]
        \wedge
        [\neg(x \in S)]
    \}
\big\}
  </math>([[一阶逻辑#普遍化元定理|GEN]] with <math>S</math> , 5)

(7)<math>\neg(\exists S)\{
    [\neg(S \in \varnothing)]
    \wedge
    [\neg(x \in S)]
\}

  </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#否定|DN]], 6)

(8)<math>(\forall x)\left\{
    \left(x \in \bigcup\varnothing \right)
    \Leftrightarrow
    (\exists S)\{
        (S \in \varnothing)
        \wedge
        (x \in S)
    \}
\right\}


  </math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]])

(9)<math>\left(x \in \bigcup\varnothing \right)
\Leftrightarrow
(\exists S)\{
    (S \in \varnothing)
    \wedge
    (x \in S)
\}


  </math>(MP with  [[一阶逻辑#量词公理|A4]], 8)

(10)<math>\left(x \in \bigcup\varnothing \right)
\Rightarrow
(\exists S)\{
    (S \in \varnothing)
    \wedge
    (x \in S)
\} 



  </math>(MP with [[一阶逻辑#且與或的直觀意義|AND]] ,9)

(11)<math>\neg(\exists S)\{
    (S \in \varnothing)
    \wedge
    (x \in S)
\}
\Rightarrow
\neg\left(x \in \bigcup\varnothing \right) 



  </math>(MP with [[一阶逻辑#否定|T]], 10)

(12)<math>\neg\left(x \in \bigcup\varnothing \right) 



  </math>(MP with 7, 11)

(13)<math>(\forall x)\left(x \not\in \bigcup\varnothing \right)  



  </math>([[一阶逻辑#普遍化元定理|GEN]] with <math>x</math> , 12)

(14)<math>(y = \varnothing)
\Leftrightarrow
(\forall x)[\neg(x \in y)]</math> ([[一阶逻辑#新理論的假設|E]])

(15)<math>(\forall y)\{
    (y = \varnothing)
    \Leftrightarrow
    (\forall x)[\neg(x \in y)]
\}</math> ([[一阶逻辑#普遍化元定理|GEN]] with <math>y</math> , 14)

(16)<math>\left(\bigcup\varnothing = \varnothing\right)
\Leftrightarrow
(\forall x)\left[\neg\left(x \in \bigcup\varnothing \right)\right] </math>(MP with  [[一阶逻辑#量词公理|A4]], 15)

(17) <math>\bigcup\varnothing = \varnothing </math> ([[一阶逻辑#等價代換|Equv]] with 13, 16)
|}
==== 比較性質 ====
{{Math theorem
| name = 定理(1)
| math_statement = <br/>
<math>
(\mathcal{M} \subseteq \mathcal{N})
\vdash
\left(\bigcup\mathcal{M} \subseteq \bigcup\mathcal{N}\right)</math>
}}
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|注意到可以從([[一阶逻辑#且與或的直觀意義|AND]])得到
:<math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R}),\,\mathcal{P}
\vdash
\mathcal{Q} \wedge \mathcal{R} 
</math>
換句話說，從[[一阶逻辑#演繹元定理|演繹元定理]]有
:(u) <math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R})
\vdash
\mathcal{P} \Rightarrow (\mathcal{Q} \wedge \mathcal{R}) 
</math>

(1) <math>(\forall A)\left[
    (A \in \mathcal{M})
    \Rightarrow
    (A \in \mathcal{N})
\right]</math> (Hyp)

(2) <math>(A \in \mathcal{M})
\Rightarrow
(A \in \mathcal{N})</math>(MP with 1, [[一阶逻辑#量词公理|A4]])

(3) <math>[(a \in A) \wedge (A \in \mathcal{M})]
\Rightarrow
(A \in \mathcal{M})</math>([[一阶逻辑#且與或的直觀意義|AND]])

(4)<math>[(a \in A) \wedge (A \in \mathcal{M})]
\Rightarrow
(a \in A)</math>([[一阶逻辑#且與或的直觀意義|AND]])

(5)<math>[(a \in A) \wedge (A \in \mathcal{M})]
\Rightarrow
(A \in \mathcal{N})</math>([[一阶逻辑#演繹元定理|D1]] with 2, 3)

(6)<math>[(a \in A) \wedge (A \in \mathcal{M})]
\Rightarrow
[(a \in A) \wedge (A \in \mathcal{N})]   </math>(u with 4, 5)

(7)<math>(\exists A \in \mathcal{M})(a \in A)
\Rightarrow
(\exists A \in \mathcal{N})(a \in A)   </math>([[一阶逻辑#普遍化元定理|GENe]] with <math>A</math>,  6)

(8) <math>(\forall x)\left\{
    \left(x \in \bigcup\mathcal{M}\right)
    \Leftrightarrow
    (\exists A \in \mathcal{M})(x \in A)
\right\}</math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]])

(9) <math>(\forall x)\left\{
    \left(x \in \bigcup\mathcal{N}\right)
    \Leftrightarrow
    (\exists A \in \mathcal{N})(x \in A)
\right\}</math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]])

(10) <math>\left(x \in \bigcup\mathcal{M}\right)
\Leftrightarrow
(\exists A \in \mathcal{M})(x \in A)</math> (MP with 8, [[一阶逻辑#量词公理|A4]])

(11) <math>\left(x \in \bigcup\mathcal{N}\right)
\Leftrightarrow
(\exists A \in \mathcal{N})(x \in A)</math> (MP with 9, [[一阶逻辑#量词公理|A4]])

(12) <math>\left(x \in \bigcup\mathcal{M}\right)
\Rightarrow
(\exists A \in \mathcal{N})(x \in A)</math>([[一阶逻辑#演繹元定理|D1]] with 7, 10)

(13) <math>\left(x \in \bigcup\mathcal{M}\right)
\Rightarrow
\left(x \in \bigcup\mathcal{N}\right)</math>([[一阶逻辑#演繹元定理|D1]] with 11, 12)

(14) <math>(\forall x)\left[
    \left(x \in \bigcup\mathcal{M}\right)
    \Rightarrow
    \left(x \in \bigcup\mathcal{N}\right)
\right] </math>([[一阶逻辑#普遍化元定理|GEN]] with <math>a</math> , 13)
|}
==== 覆蓋性質 ====
{{Math theorem
| name = 定理(2)
| math_statement = <br/>
<math>
\vdash
A = \bigcup\mathcal{P}(A)
</math>
}}

「<math>A</math> 正好就是其[[冪集]]的聯集」，這個定理直觀上可理解成，因為[[冪集]] <math>\mathcal{P}(A)</math> 是以 <math>A</math> 和 <math>A</math> 的[[子集]]為元素，所以 <math>\mathcal{P}(A)</math> 的聯集理當是 <math>A</math> 。
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|注意到可以從([[一阶逻辑#且與或的直觀意義|AND]])得到
:<math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R}),\,\mathcal{P}
\vdash
\mathcal{Q} \wedge \mathcal{R} 
</math>
換句話說，從[[一阶逻辑#演繹元定理|演繹元定理]]有
:(u) <math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R})
\vdash
\mathcal{P} \Rightarrow (\mathcal{Q} \wedge \mathcal{R}) 
</math>

(1)<math>(\forall x)\left\{
    \left[x \in \bigcup\mathcal{P}(A)\right]
    \Leftrightarrow
    (\exists S)\{
        [S \in \mathcal{P}(A)]
        \wedge
        (x \in S)
    \}
\right\}
</math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]])

(2) <math>(\forall S)\{
    [S \in \mathcal{P}(A)]
    \Leftrightarrow
    (S \subseteq A)
\}
</math>([[幂集公理]])

(3) <math>[S \in \mathcal{P}(A)]
\Leftrightarrow
(S \subseteq A)
</math>(MP with [[一阶逻辑#量词公理|A4]] ,2)

(4) <math>(\forall x)\left\{
    \left[x \in \bigcup\mathcal{P}(A)\right]
    \Leftrightarrow
    (\exists S)[
        (S \subseteq A)
        \wedge
        (x \in S)
    ]
\right\}
</math> ([[一阶逻辑#等價代換|Equv]] with 1, 3)

(5) <math>[(S \subseteq A)
\wedge
(x \in S)]
\Rightarrow
(S \subseteq A)




</math>([[一阶逻辑#且與或的直觀意義|AND]])

(6) <math>(\forall x)[
    (x \in S)
    \Rightarrow
    (x \in A)
]
\Rightarrow
[(x \in S) \Rightarrow (x \in A)]






</math>([[一阶逻辑#量词公理|A4]])

(7) <math>[(S \subseteq A)
\wedge
(x \in S)]
\Rightarrow
[(x \in S) \Rightarrow (x \in A)]




</math>([[一阶逻辑#演繹元定理|D1]] with 5, 6)

(8) <math>[(S \subseteq A)
\wedge
(x \in S)]
\Rightarrow
(x \in S)




</math>([[一阶逻辑#且與或的直觀意義|AND]])

(9) <math>[(S \subseteq A)
\wedge
(x \in S)]
\Rightarrow
\{
(x \in S)
\wedge
[(x \in S) \Rightarrow (x \in A)]
\}




</math>(u with 7, 8)

注意到
:<math>
(x \in S),\, [(x \in S) \Rightarrow (x \in A)]
\vdash
(x \in A)
 
</math>
再對上式套用([[一阶逻辑#且與或的直觀意義|AND]])就有
:<math>
\{
(x \in S)
\wedge
[(x \in S) \Rightarrow (x \in A)]
\}
\vdash
(x \in A)
 
</math>(a)
(10') <math>[(S \subseteq A)
\wedge
(x \in S)]
\Rightarrow
(x \in A)




</math>([[一阶逻辑#演繹元定理|D1]] with a, 9)

(11') <math>(\exists S)[
    (S \subseteq A)
    \wedge
    (x \in S)
]
\Rightarrow
(x \in A)





</math>([[一阶逻辑#普遍化元定理|GENe]] with <math>S</math>, 10')

(12') <math>(\forall S)\{\neg[
    (S \subseteq A)
    \wedge
    (x \in S)
]\}
\Rightarrow
\{\neg[
    (A \subseteq A)
    \wedge
    (x \in A)
]\}





</math> ([[一阶逻辑#量词公理|A4]])

(13') <math>[
(A \subseteq A)
\wedge
(x \in A)
]
\Rightarrow
(\exists S)[
    (S \subseteq A)
    \wedge
    (x \in S)
]





</math> (MP with [[一阶逻辑#否定|T]], 12')

(14') <math>(x \in A) \Rightarrow (x \in A)</math> ([[一阶逻辑#定理與證明|I]])

(15') <math>A \subseteq A</math> ([[一阶逻辑#普遍化元定理|GEN]] with <math>x</math> , 14')

注意到([[一阶逻辑#且與或的直觀意義|AND]])依據[[一阶逻辑#演繹元定理|演繹定理]]可改寫為
:<math>
(A \subseteq A)
\vdash
(x \in A)
\Rightarrow
[(A \subseteq A) \wedge (x \in A)]
 
</math>(b)
(16<nowiki>''</nowiki>) <math>
(x \in A)
\Rightarrow
[(A \subseteq A) \wedge (x \in A)]

 
</math> (b with 15')

(17<nowiki>''</nowiki>) <math>(x \in A)
\Rightarrow
(\exists S)[
    (S \subseteq A)
    \wedge
    (x \in S)
]





</math> ([[一阶逻辑#演繹元定理|D1]] with 13<nowiki>', 16''</nowiki>)

(18<nowiki>''</nowiki>) <math>(x \in A)
\Leftrightarrow
(\exists S)[
    (S \subseteq A)
    \wedge
    (x \in S)
]





</math> ([[一阶逻辑#且與或的直觀意義|AND]] with 11<nowiki>', 17''</nowiki>)

(19<nowiki>''</nowiki>) <math>(\forall x)\left\{
    \left[x \in \bigcup\mathcal{P}(A)\right]
    \Leftrightarrow
    (x \in A)
\right\}

</math>([[一阶逻辑#等價代換|Equv]] with 4, 18<nowiki>'''</nowiki>)

|}
{{Math theorem
| name = 定理(3)
| math_statement = <br/>
<math>\left(\bigcup\mathcal{M} \subseteq A \right)
\vdash
(\forall M \in \mathcal{M})(M \subseteq A)</math>
}}

直觀上，這個定理說「一群集合的聯集包含於  <math>A</math> ，則它們個個都包含於 <math>A</math> 」
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|(1) <math>(\forall a)\left[
    (\exists M \in \mathcal{M})(a \in M)
    \Rightarrow
    (a \in A)
\right]</math> (Hyp)

(2) <math>[(M \in \mathcal{M}) \wedge (a \in M)]
\Rightarrow
(\exists M \in \mathcal{M})(a \in M)</math> ([[一阶逻辑#量词公理|A4]] and [[一阶逻辑#否定|T]])

(3) <math>(\exists M \in \mathcal{M})(a \in M)
\Rightarrow
(a \in A)</math> (MP with 1, [[一阶逻辑#量词公理|A4]])

(4) <math>[(M \in \mathcal{M}) \wedge (a \in M)]
\Rightarrow
(a \in A) </math> ([[一阶逻辑#演繹元定理|D1]] with 2, 3)

(5) <math>(M \in \mathcal{M})
\Rightarrow
[(a \in M) \Rightarrow (a \in A)]  </math> (MP with [[一阶逻辑#量詞的簡寫|abb]], 4)

(6) <math>(\forall a)\{
    (M \in \mathcal{M})
    \Rightarrow
    [(a \in M) \Rightarrow (a \in A)]
\}   </math> ([[一阶逻辑#普遍化元定理|GEN]] with <math>a</math> , 5)

(7) <math>(M \in \mathcal{M})
\Rightarrow
(\forall a)[(a \in M) \Rightarrow (a \in A)]   </math>  (MP with [[一阶逻辑#量词公理|A5]] , 6)

(8) <math>(\forall M)\{
    (M \in \mathcal{M})
    \Rightarrow
    (\forall a)[(a \in M) \Rightarrow (a \in A)]
\}   </math> ([[一阶逻辑#普遍化元定理|GEN]] with <math>M</math> , 7)
|}
{{Math theorem
| name = 定理(4)
| math_statement = <br/>
<math>
\vdash
\left(A = \bigcup\mathcal{M}\right)
\Rightarrow
\left\{A \subseteq  \bigcup\mathcal[\mathcal{M} \cup \mathcal{P}(A)]\right\}
</math>
}}直觀上，這個定理說「集族 <math>\mathcal{M}</math> 的聯集為 <math>A</math> ，則對 <math>A</math> 的每點 <math>a</math> ，都可從 <math>\mathcal{M}</math> 裡找到一個 <math>a</math> 的鄰域 <math>M</math> ，且這個鄰域不會比 <math>A</math> 大 」
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|注意到可以從([[一阶逻辑#且與或的直觀意義|AND]])得到
:<math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R}),\,\mathcal{P}
\vdash
\mathcal{Q} \wedge \mathcal{R} 
</math>
換句話說，從[[一阶逻辑#演繹元定理|演繹元定理]]有
:(u) <math>
(\mathcal{P} \Rightarrow \mathcal{Q}),\,(\mathcal{P} \Rightarrow \mathcal{R})
\vdash
\mathcal{P} \Rightarrow (\mathcal{Q} \wedge \mathcal{R}) 
</math>
(1) <math>(\forall a)\left[
    (a \in A)
    \Leftrightarrow
    (\exists M \in \mathcal{M})(a \in M)
\right]</math> (Hyp)

(2) <math>(\forall M \in \mathcal{M})(M \subseteq A)</math>(MP with 1, 定理3)

(3) <math>(M \in \mathcal{M})
\Rightarrow
(M \subseteq A)</math>(MP with  [[一阶逻辑#量词公理|A4]], 2)

(4) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
(M \in \mathcal{M})</math>([[一阶逻辑#且與或的直觀意義|AND]])

(5) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
(a \in M)</math>([[一阶逻辑#且與或的直觀意義|AND]])

(6) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
(M \in \mathcal{M})</math>([[一阶逻辑#且與或的直觀意義|AND]])

(7) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
(M \subseteq A)</math> ([[一阶逻辑#演繹元定理|D1]] with 3, 4)

(8) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
[(a \in M) \wedge (M \in \mathcal{M})]</math>(a with 5, 6)

(9) <math>[(a \in M) \wedge (M \in \mathcal{M})]
\Rightarrow
[(a \in M) \wedge (M \in \mathcal{M}) \wedge (M \subseteq A)]</math>(a with 7, 8)

(10) <math>(\exists M \in \mathcal{M})(a \in M)
\Rightarrow
(\exists M \in \mathcal{M})[(a \in M) \wedge (M \subseteq A)]</math>([[一阶逻辑#普遍化元定理|GENe]] with <math>M</math>,  9)

(11) <math>(a \in A)
\Leftrightarrow
(\exists M \in \mathcal{M})(a \in M)
</math>(MP with  [[一阶逻辑#量词公理|A4]], 1)

(12) <math>(a \in A)
\Rightarrow
(\exists M \in \mathcal{M})(a \in M)
</math>([[一阶逻辑#且與或的直觀意義|AND]] with  11)

(13) <math>(a \in A)
\Rightarrow
(\exists M \in \mathcal{M})[(a \in M) \wedge (M \subseteq A)]
</math>([[一阶逻辑#演繹元定理|D1]] with 10, 12)

(14) <math>(\forall a \in A)(\exists M \in \mathcal{M})[(a \in M) \wedge (M \subseteq A)]
</math>([[一阶逻辑#普遍化元定理|GEN]] with <math>a</math>,  13) 

(15)<math>(\forall S)\{
    [S \in \mathcal{P}(A)]
    \Leftrightarrow
    (S \subseteq A)
\}
</math>([[幂集公理]]) 

(16)<math>[M \in \mathcal{P}(A)]
\Leftrightarrow
(M \subseteq A)
</math>(MP with  [[一阶逻辑#量词公理|A4]], 15) 

(17)<math>(\forall a \in A)(\exists M \in \mathcal{M})\{(a \in M) \wedge [M \in \mathcal{P}(A)]\}
</math>([[一阶逻辑#等價代換|Equv]] with 14, 16) 

(18) <math>(\forall A)(\forall B)(\forall x)\left\{
    (x \in A \cap B)
    \Leftrightarrow
    \left[
        (x \in A)
        \wedge
        (x \in B)
    \right]
\right\}  </math>([[交集#有限交集|有限交集]]) 

(19)<math>(\forall B)(\forall x)\left\{
    (x \in \mathcal{M} \cap B)
    \Leftrightarrow
    \left[
        (x \in \mathcal{M})
        \wedge
        (x \in B)
    \right]
\right\}
  </math>(MP with  [[一阶逻辑#量词公理|A4]], 18) 

(20)<math>(\forall x)\big\{
    [x \in \mathcal{M} \cap \mathcal{P}(A)]
    \Leftrightarrow
    \left\{
        (x \in \mathcal{M})
        \wedge
        [x \in \mathcal{P}(A)]
    \right\}
\big\}

  </math>(MP with  [[一阶逻辑#量词公理|A4]], 19) 

(21)<math>[M \in \mathcal{M} \cap \mathcal{P}(A)]
\Leftrightarrow
\left\{
    (M \in \mathcal{M})
    \wedge
    [M \in \mathcal{P}(A)]
\right\}

  </math>(MP with  [[一阶逻辑#量词公理|A4]], 20) 

(22)<math>(\forall a \in A)(\exists M)\{
    (a \in M)
    \wedge
    [M \in \mathcal{M} \cap \mathcal{P}(A)]
\}

</math>([[一阶逻辑#等價代換|Equv]] with 17, 21) 

(23)<math>\left\{a \in \bigcup[\mathcal{M} \cap \mathcal{P}(A)]\right\}
\Leftrightarrow
(\exists M)\{
    (a \in M)
    \wedge
    [M \in \mathcal{M} \cap \mathcal{P}(A)]
\}

</math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]]) 

(24)<math>(\forall a)\left\{
    (a \in A)
    \Rightarrow
    \left\{a \in \bigcup[\mathcal{M} \cap \mathcal{P}(A)]\right\}
\right\}

</math>([[一阶逻辑#等價代換|Equv]] with 22, 23) 
|}
==== 運算性質 ====
{{Math theorem
| name = 定理(5)
| math_statement = <br/>
若
:<math>\mathcal{M}_A := \left\{
    B \,|\,
    (\exists M \in \mathcal{M})(B = M \cap A)
\right\}</math>
則
:<math>\vdash
\bigcup\mathcal{M}_A = A \cap \left(\bigcup\mathcal{M}\right)</math>
}}
{| class="mw-collapsible mw-collapsed wikitable"
!'''證明'''
|-
|(1)<math>(\forall B)[
    (B \in \mathcal{M}_A)
    \Leftrightarrow
    (\exists M \in \mathcal{M})(B = M \cap A)
]</math> (<math>\mathcal{M}_A </math>的定義)

(2) <math>(\forall x)\left\{
    \left(x \in \bigcup\mathcal{M}\right)
    \Leftrightarrow
    (\exists B \in \mathcal{M})(x \in B)
\right\}  </math>(MP with [[并集公理]],  [[一阶逻辑#量词公理|A4]])

(3) <math>(\forall A)(\forall B)(\forall x)\left\{
    (x \in A \cap B)
    \Leftrightarrow
    \left[
        (x \in A)
        \wedge
        (x \in B)
    \right]
\right\}  </math>([[交集#有限交集|有限交集]]) 

(4)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists B)[ (B \in \mathcal{M}_A) \wedge (x \in B)]</math>(MP with  [[一阶逻辑#量词公理|A4]], 2) 

(5) <math>(B \in \mathcal{M}_A)
\Leftrightarrow
(\exists M \in \mathcal{M})(B = M \cap A) </math>(MP with  [[一阶逻辑#量词公理|A4]], 1) 

(6) <math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists B)[
    (x \in B)
    \wedge
    (\exists M \in \mathcal{M})(B = M \cap A)
] </math>([[一阶逻辑#等價代換|Equv]] with 4, 5) 

(7)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists B)(\exists M)[
    (x \in B)
    \wedge
    ( M \in \mathcal{M})
    \wedge
    (B = M \cap A)
] </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#量詞的簡寫|Ce]], 6) 

(8)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists M)(\exists B)[
    (x \in B)
    \wedge
    ( M \in \mathcal{M})
    \wedge
    (B = M \cap A)
] </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#量詞的可交換性|量詞可交換性]] ,7) 

(9) <math>(B = M \cap A)
\Rightarrow\{
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]
\}
 </math>([[一阶逻辑#等式定理|E2]]) 

(10)<math>[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
(B = M \cap A)
 </math>([[一阶逻辑#且與或的直觀意義|AND]]) 

(11)<math>[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
 </math><math>\Rightarrow\{
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]
\}
 </math>([[一阶逻辑#演繹元定理|D1]] with 9,10) 

(12)<math>\{
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\}
 </math> 

<math>\Rightarrow\{
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]
\}
 </math>(MP with [[一阶逻辑#邏輯公理|A2]], 11) 

(13)<math>[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
 </math>([[一阶逻辑#定理與證明|I]]) 

(14)<math>[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)] 
 </math>(MP with 12, 13) 

(15)<math>[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]  
 </math>([[一阶逻辑#且與或的直觀意義|AND]]) 

(16)<math>[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]   
 </math>([[一阶逻辑#演繹元定理|D1]] with 14,15) 

(17)<math>(\exists M)(\exists B)[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Rightarrow
(\exists M)[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]    
 </math>([[一阶逻辑#普遍化元定理|GENe]] with <math>B</math> then <math>M</math>) 

(18)<math>M \cap A = M \cap A  
 </math> ([[一阶逻辑#等式定理|E1]]) 

注意到配合([[一阶逻辑#且與或的直觀意義|AND]])和[[一阶逻辑#演繹元定理|演繹定理]]有 

<math>\mathcal{P}
\vdash
\mathcal{R} \Rightarrow (\mathcal{R} \wedge \mathcal{P})   
 </math>(a) 

(19)<math>[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]
\Rightarrow
[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]    
 </math>(a with 18) 

(20)<math>(\forall B)\{\neg[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]\}
\Rightarrow
\{\neg[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]\}    
 </math>([[一阶逻辑#量词公理|A4]]) 

(21)<math>[(x \in M \cap A)
\wedge
( M \in \mathcal{M})
\wedge
(M \cap A = M \cap A)]
\Rightarrow
(\exists B)[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]     
 </math>(MP with [[一阶逻辑#否定|T]], 20) 

(22)<math>[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]
\Rightarrow
(\exists B)[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]     
 </math>([[一阶逻辑#演繹元定理|D1]] with 19, 21) 

(23)<math>(\exists M)[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]
\Rightarrow
(\exists M)(\exists B)[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]     
 </math>([[一阶逻辑#普遍化元定理|GENe]] with <math>M</math>) 

(24)<math>(\exists M)(\exists B)[(x \in B)
\wedge
( M \in \mathcal{M})
\wedge
(B = M \cap A)]
\Leftrightarrow
(\exists M)[(x \in M \cap A)
\wedge
( M \in \mathcal{M})]    
 </math>([[一阶逻辑#且與或的直觀意義|AND]] with 17, 23) 

(25)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists M)[(x \in M \cap A)
\wedge
( M \in \mathcal{M})] </math>([[一阶逻辑#等價代換|Equv]] with 8, 24) 

(26) <math>(x \in M \cap A)
\Leftrightarrow
[(x \in M) \wedge (x \in A)]</math>(MP with  [[一阶逻辑#量词公理|A4]], 3) 

(27)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
(\exists M)[
    (x \in M)
    \wedge
    (x \in A)
    \wedge
    (M \in \mathcal{M})
]  </math>([[一阶逻辑#等價代換|Equv]] with 25, 26) 

(28)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
\{
(\exists M)[
    (x \in M)
    \wedge
    (M \in \mathcal{M})
]
\wedge
(x \in A)
\}  </math>([[一阶逻辑#等價代換|Equv]] with [[一阶逻辑#量詞的簡寫|Ce]], 27) 

(30) <math>\left(x \in \bigcup\mathcal{M}\right)
\Leftrightarrow
(\exists M \in \mathcal{M})(x \in M)  </math>(MP with  [[一阶逻辑#量词公理|A4]], 2) 

(31)<math>\left(x \in \bigcup\mathcal{M}_A\right)
\Leftrightarrow
\left[
\left(x \in \bigcup\mathcal{M}\right)
\wedge
(x \in A)
\right]    </math>([[一阶逻辑#等價代換|Equv]] with 28, 30) 

(32)<math>\left[x \in A \cap \left(\bigcup\mathcal{M}\right) \right]
\Leftrightarrow
\left[
    (x \in A)
    \wedge
    \left(x \in \bigcup\mathcal{M}\right)
\right]    </math>(MP with  [[一阶逻辑#量词公理|A4]], 3) 

(33)<math>\left[x \in A \cap \left(\bigcup\mathcal{M}\right) \right]
\Leftrightarrow
\left(x \in \bigcup\mathcal{M}_A\right)     </math>([[一阶逻辑#等價代換|Equv]] with 31, 32) 

(34)<math>(\forall x)\left\{
    \left[x \in A \cap \left(\bigcup\mathcal{M}\right) \right]
    \Leftrightarrow
    \left(x \in \bigcup\mathcal{M}_A\right)
\right\}     </math>([[一阶逻辑#普遍化元定理|GEN]] with <math>x</math>, 33) 
|}
直觀上這個定理說，交集在「无限并集满足分配律」，一般會不正式的寫為
: <math>\bigcup_{i\in I}\left(A \cap B_{i}\right) = A \cap \bigcup_{i\in I} B_{i}</math>
{{Math theorem
| name = 定理(6)
| math_statement = <br/>
<math>\N \,\overset{A}{\cong}\, \mathcal{A} </math>，若對[[自然数]] <math>m \in \N </math> 做以下的符號定義：
: <math>\mathcal{A}_m
:=
\left\{
S \in \mathcal{A} \,|\,
    A^{-1}(S) \geq m
\right\} </math>
: <math>\mathcal{I}
:=
\left\{
S \,\bigg|\,
    (\exists m \in \N)\left(
        S = \bigcap \mathcal{A}_m
    \right)
\right\} </math>
: <math>\mathcal{S}
:=
\left\{
S \,\bigg|\,
    (\exists m \in \N)\left(
        S = \bigcup \mathcal{A}_m
    \right)
\right\}  </math>
那有
: <math>\vdash
\bigcup \mathcal{I} \subseteq \bigcap \mathcal{S}</math>。
}}

這個定理一般會被不正式的寫為
: <math>\bigcup^{\infty}_{i=0}\left(\bigcap^{\infty}_{j=i} A_j \right)
\subseteq
\bigcap^{\infty}_{i=0}\left(\bigcup^{\infty}_{j=i} A_j \right)</math>。
== 参考 ==
* [[朴素集合论]]
* [[交集]]
* [[补集]]
* [[对称差]]
* [[不交并]]
* [[布尔逻辑]]

== 参考文献 ==
<references />

[[Category:抽象代数|B]]
[[Category:集合論基本概念|B]]
[[Category:二元運算|B]]

{{集合论}}