在集合論和數學的其他分支中，一群集合的並集（Union），是以這群集合的所有元素來構成的集合。 有限聯集 聯集是由公理化集合論的分類公理來確保其唯一存在的特定集合 A cup B ： forall Aforall Bforall xleftx in A cup B Leftrightarrow left x in A vee x in B right right 也就是直觀上： 「對所有 x ， x in A cup B 等價於 x in A 或 x in B」 舉例： 集合 1 2 3 和 2 3 4 的並集是 1 2 3 4。數 9 不屬於素數集合 2 3 5 7 11ldots 和偶數集合 2 4 6 8 10ldots 的並集，因為 9 既不是素數，也不是偶數。 更通常的，多個集合的並集可以這樣定義： 例如，AB 和 C 的並集含有所有 A 的元素，所有 B 的元素和所有 C 的元素，而沒有其他元素。形式上： x 是 A cup B cup C 的元素，當且僅當 x 屬於 A 或 x 屬於 B 或 x 屬於 C。 代數性質 二元並集（兩個集合的並集）是一種結合運算，即 A cup B cup C A cup Bcup C。事實上，A cup B cup C 也等於這兩個集合，因此圓括號在僅進行並集運算的時候可以省略。 相似的，並集運算滿足交換律，即集合的順序任意。 空集是並集運算的單位元。即 varnothing cup A A，對任意集合 A。可以將空集當作零個集合的並集。 結合交集和補集運算，並集運算使任意冪整合為布林代數。例如，並集和交集相互滿足分配律，而且這三種運算滿足德摩根律。若將並集運算換成對稱差運算，可以獲得相應的布林環。 無限並集 由公理化集合論的並集公理，有唯一的集合 bigcupmathcal M 滿足： forall mathcal Mforall xleftleft x in bigcupmathcal Mright Leftrightarrow exists Aleft left A in mathcal Mright wedge x in A right right 也就是直觀上「對所有 mathcal M 和所有 x ， x in bigcupmathcal M 等價於有某個 mathcal M 的下屬集合 A ，使得 x in A」。以上的 mathcal M 可以直觀的視為一個集合族，而把 bigcupmathcal M 看成對 mathcal M 內的集合取並集，但這個公理並沒有對 mathcal M 下屬集合的數量做出任何限制，所以這個 bigcupmathcal M 被俗稱為任意並集或無限並集。 若 X subseteq bigcupmathcal M ，會稱 X 被 mathcal M 覆蓋（cover），也就是直觀上可以用 mathcal M 裡的所有集合疊起來蓋住 X。 例如： 對 mathcal M ABC，bigcupmathcal M A cup B cup C ，若 M 是空集， bigcupmathcal M 也是空集。 無限並集有多種表示方法： 可模仿求和符號記為 bigcup_Ain mathcal M A。 但大多數人會假設指標集 I 的存在，換句話說 若 I overset Acong mathcal M 則 bigcup_iin I A i bigcup mathcal M 在指標集 I 是自然數系 N 的情況下，更可以仿無窮級數來表示，也就是說： 若 N overset Acong mathcal M 則 bigcupinfty_i 0 A i bigcup mathcal M 也可以更粗略直觀的將 bigcupinfty_i 0 A i 寫作 A_0 cup A_1 cup A_2 cup ldots。 無限並集的性質 證明 1 forall xneg x in varnothing 空集公理 2 neg S in varnothing MP with A4 1 3S in varnothing Rightarrow neg x in S M0 with 2 4negneg S in varnothing Rightarrow neg x in S Equv with DN 3 5negneg S in varnothing wedge neg x in S Equv with De Morgan 4 6forall Sbig negneg S in varnothing wedge neg x in S big GEN with S 5 7neg exists Sneg S in varnothing wedge neg x in S Equv with DN 6 8forall xleftleft x in bigcupvarnothing right Leftrightarrow exists SS in varnothing wedge x in S right MP with 並集公理，A4 9left x in bigcupvarnothing right Leftrightarrow exists SS in varnothing wedge x in S MP with A4 8 10left x in bigcupvarnothing right Rightarrow exists SS in varnothing wedge x in S MP with AND 9 11neg exists SS in varnothing wedge x in S Rightarrow negleft x in bigcupvarnothing right MP with T 10 12negleft x in bigcupvarnothing right MP with 7 11 13forall xleft x notin bigcupvarnothing right GEN with x 12 14y varnothing Leftrightarrow forall xneg x in y E 15forall yy varnothing Leftrightarrow forall xneg x in y GEN with y 14 16left bigcupvarnothing varnothingright Leftrightarrow forall xleft negleft x in bigcupvarnothing rightright MP with A4 15 17 bigcupvarnothing varnothing Equv with 13 16 比較性質 證明 注意到可以從 AND 得到 mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal Rmathcal P vdash mathcal Q wedge mathcal R 換句話說，從演繹元定理有 u mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal R vdash mathcal P Rightarrow mathcal Q wedge mathcal R 1 forall Aleft A in mathcal M Rightarrow A in mathcal N right Hyp 2 A in mathcal M Rightarrow A in mathcal NMP with 1 A4 3 a in A wedge A in mathcal M Rightarrow A in mathcal MAND 4a in A wedge A in mathcal M Rightarrow a in AAND 5a in A wedge A in mathcal M Rightarrow A in mathcal ND1 with 2 3 6a in A wedge A in mathcal M Rightarrow a in A wedge A in mathcal N u with 4 5 7exists A in mathcal Ma in A Rightarrow exists A in mathcal Na in A GENe with A 6 8 forall xleftleft x in bigcupmathcal Mright Leftrightarrow exists A in mathcal Mx in A rightMP with 並集公理，A4 9 forall xleftleft x in bigcupmathcal Nright Leftrightarrow exists A in mathcal Nx in A rightMP with 並集公理，A4 10 left x in bigcupmathcal Mright Leftrightarrow exists A in mathcal Mx in A MP with 8 A4 11 left x in bigcupmathcal Nright Leftrightarrow exists A in mathcal Nx in A MP with 9 A4 12 left x in bigcupmathcal Mright Rightarrow exists A in mathcal Nx in AD1 with 7 10 13 left x in bigcupmathcal Mright Rightarrow left x in bigcupmathcal NrightD1 with 11 12 14 forall xleft left x in bigcupmathcal Mright Rightarrow left x in bigcupmathcal Nright right GEN with a 13 覆蓋性質 「A 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 mathcal PA 是以 A 和 A 的子集為元素，所以 mathcal PA 的聯集理當是 A 。 證明 注意到可以從 AND 得到 mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal Rmathcal P vdash mathcal Q wedge mathcal R 換句話說，從演繹元定理有 u mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal R vdash mathcal P Rightarrow mathcal Q wedge mathcal R 1forall xleftleft x in bigcupmathcal PAright Leftrightarrow exists SS in mathcal PA wedge x in S right MP with 並集公理，A4 2 forall SS in mathcal PA Leftrightarrow S subseteq A 冪集公理 3 S in mathcal PA Leftrightarrow S subseteq A MP with A4 2 4 forall xleftleft x in bigcupmathcal PAright Leftrightarrow exists SS subseteq A wedge x in S right Equv with 1 3 5 S subseteq A wedge x in S Rightarrow S subseteq A AND 6 forall xx in S Rightarrow x in A Rightarrow x in S Rightarrow x in A A4 7 S subseteq A wedge x in S Rightarrow x in S Rightarrow x in A D1 with 5 6 8 S subseteq A wedge x in S Rightarrow x in S AND 9 S subseteq A wedge x in S Rightarrow x in S wedge x in S Rightarrow x in A u with 7 8 注意到 x in S x in S Rightarrow x in A vdash x in A 再對上式套用 AND 就有 x in S wedge x in S Rightarrow x in A vdash x in A a 10 S subseteq A wedge x in S Rightarrow x in A D1 with a 9 11 exists SS subseteq A wedge x in S Rightarrow x in A GENe with S 10 12 forall Sneg S subseteq A wedge x in S Rightarrow neg A subseteq A wedge x in A A4 13 A subseteq A wedge x in A Rightarrow exists SS subseteq A wedge x in S MP with T 12 14 x in A Rightarrow x in A I 15 A subseteq A GEN with x 14 注意到 AND 依據演繹定理可改寫為 A subseteq A vdash x in A Rightarrow A subseteq A wedge x in A b 16 x in A Rightarrow A subseteq A wedge x in A b with 15 17 x in A Rightarrow exists SS subseteq A wedge x in S D1 with 13 16 18 x in A Leftrightarrow exists SS subseteq A wedge x in S AND with 11 17 19 forall xleftleft x in bigcupmathcal PAright Leftrightarrow x in A right Equv with 4 18 直觀上，這個定理說「一群集合的聯集包含於 A ，則它們個個都包含於 A 」 證明 1 forall aleft exists M in mathcal Ma in M Rightarrow a in A right Hyp 2 M in mathcal M wedge a in M Rightarrow exists M in mathcal Ma in M A4 and T 3 exists M in mathcal Ma in M Rightarrow a in A MP with 1 A4 4 M in mathcal M wedge a in M Rightarrow a in A D1 with 2 3 5 M in mathcal M Rightarrow a in M Rightarrow a in A MP with abb 4 6 forall aM in mathcal M Rightarrow a in M Rightarrow a in A GEN with a 5 7 M in mathcal M Rightarrow forall aa in M Rightarrow a in A MP with A5 6 8 forall MM in mathcal M Rightarrow forall aa in M Rightarrow a in A GEN with M 7 直觀上，這個定理說「集族 mathcal M 的聯集為 A ，則對 A 的每點 a ，都可從 mathcal M 裡找到一個 a 的鄰域 M ，且這個鄰域不會比 A 大 」 證明 注意到可以從 AND 得到 mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal Rmathcal P vdash mathcal Q wedge mathcal R 換句話說，從演繹元定理有 u mathcal P Rightarrow mathcal Qmathcal P Rightarrow mathcal R vdash mathcal P Rightarrow mathcal Q wedge mathcal R 1 forall aleft a in A Leftrightarrow exists M in mathcal Ma in M right Hyp 2 forall M in mathcal MM subseteq AMP with 1 定理 3 3 M in mathcal M Rightarrow M subseteq AMP with A4 2 4 a in M wedge M in mathcal M Rightarrow M in mathcal MAND 5 a in M wedge M in mathcal M Rightarrow a in MAND 6 a in M wedge M in mathcal M Rightarrow M in mathcal MAND 7 a in M wedge M in mathcal M Rightarrow M subseteq A D1 with 3 4 8 a in M wedge M in mathcal M Rightarrow a in M wedge M in mathcal Ma with 5 6 9 a in M wedge M in mathcal M Rightarrow a in M wedge M in mathcal M wedge M subseteq Aa with 7 8 10 exists M in mathcal Ma in M Rightarrow exists M in mathcal Ma in M wedge M subseteq AGENe with M 9 11 a in A Leftrightarrow exists M in mathcal Ma in M MP with A4 1 12 a in A Rightarrow exists M in mathcal Ma in M AND with 11 13 a in A Rightarrow exists M in mathcal Ma in M wedge M subseteq A D1 with 10 12 14 forall a in Aexists M in mathcal Ma in M wedge M subseteq A GEN with a 13 15forall SS in mathcal PA Leftrightarrow S subseteq A 冪集公理 16M in mathcal PA Leftrightarrow M subseteq A MP with A4 15 17forall a in Aexists M in mathcal Ma in M wedge M in mathcal PA Equv with 14 16 18 forall Aforall Bforall xleftx in A cap B Leftrightarrow left x in A wedge x in B right right 有限交集 19forall Bforall xleftx in mathcal M cap B Leftrightarrow left x in mathcal M wedge x in B right right MP with A4 18 20forall xbigx in mathcal M cap mathcal PA Leftrightarrow leftx in mathcal M wedge x in mathcal PA right big MP with A4 19 21M in mathcal M cap mathcal PA Leftrightarrow leftM in mathcal M wedge M in mathcal PA right MP with A4 20 22forall a in Aexists Ma in M wedge M in mathcal M cap mathcal PA Equv with 17 21 23lefta in bigcup mathcal M cap mathcal PAright Leftrightarrow exists Ma in M wedge M in mathcal M cap mathcal PA MP with 並集公理，A4 24forall alefta in A Rightarrow lefta in bigcup mathcal M cap mathcal PAright right Equv with 22 23 運算性質 證明 1forall BB in mathcal M_A Leftrightarrow exists M in mathcal MB M cap A mathcal M_A 的定義 2 forall xleftleft x in bigcupmathcal Mright Leftrightarrow exists B in mathcal Mx in B right MP with 並集公理，A4 3 forall Aforall Bforall xleftx in A cap B Leftrightarrow left x in A wedge x in B right right 有限交集 4left x in bigcupmathcal M_Aright Leftrightarrow exists BB in mathcal M_A wedge x in BMP with A4 2 5 B in mathcal M_A Leftrightarrow exists M in mathcal MB M cap A MP with A4 1 6 left x in bigcupmathcal M_Aright Leftrightarrow exists Bx in B wedge exists M in mathcal MB M cap A Equv with 4 5 7left x in bigcupmathcal M_Aright Leftrightarrow exists Bexists Mx in B wedge M in mathcal M wedge B M cap A Equv with Ce 6 8left x in bigcupmathcal M_Aright Leftrightarrow exists Mexists Bx in B wedge M in mathcal M wedge B M cap A Equv with 量詞可交換性，7 9 B M cap A Rightarrowx 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 E2 10x in B wedge M in mathcal M wedge B M cap A Rightarrow B M cap A AND 11x in B wedge M in mathcal M wedge B M cap A Rightarrowx 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 D1 with 910 12x 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 Rightarrowx 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 MP with A2 11 13x 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 I 14x 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 MP with 12 13 15x 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 AND 16x in B wedge M in mathcal M wedge B M cap A Rightarrow x in M cap A wedge M in mathcal M D1 with 1415 17exists Mexists Bx in B wedge M in mathcal M wedge B M cap A Rightarrow exists Mx in M cap A wedge M in mathcal M GENe with B then M 18 M cap A M cap A E1 注意到配合 AND 和演繹定理有 mathcal P vdash mathcal R Rightarrow mathcal R wedge mathcal P a 19x 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 a with 18 20forall Bneg 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 A4 21x in M cap A wedge M in mathcal M wedge M cap A M cap A Rightarrow exists Bx in B wedge M in mathcal M wedge B M cap A MP with T 20 22x in M cap A wedge M in mathcal M Rightarrow exists Bx in B wedge M in mathcal M wedge B M cap A D1 with 19 21 23exists Mx in M cap A wedge M in mathcal M Rightarrow exists Mexists Bx in B wedge M in mathcal M wedge B M cap A GENe with M 24exists Mexists Bx in B wedge M in mathcal M wedge B M cap A Leftrightarrow exists Mx in M cap A wedge M in mathcal M AND with 17 23 25left x in bigcupmathcal M_Aright Leftrightarrow exists Mx in M cap A wedge M in mathcal M Equv with 8 24 26 x in M cap A Leftrightarrow x in M wedge x in AMP with A4 3 27left x in bigcupmathcal M_Aright Leftrightarrow exists Mx in M wedge x in A wedge M in mathcal M Equv with 25 26 28left x in bigcupmathcal M_Aright Leftrightarrow exists Mx in M wedge M in mathcal M wedge x in A Equv with Ce 27 30 left x in bigcupmathcal Mright Leftrightarrow exists M in mathcal Mx in M MP with A4 2 31left x in bigcupmathcal M_Aright Leftrightarrow left left x in bigcupmathcal Mright wedge x in A right Equv with 28 30 32left x in A cap left bigcupmathcal Mright right Leftrightarrow left x in A wedge left x in bigcupmathcal Mright right MP with A4 3 33left x in A cap left bigcupmathcal Mright right Leftrightarrow left x in bigcupmathcal M_Aright Equv with 31 32 34forall xleftleft x in A cap left bigcupmathcal Mright right Leftrightarrow left x in bigcupmathcal M_Aright right GEN with x 33 直觀上這個定理說，交集在「無限並集滿足分配律」，一般會不正式的寫為 bigcup_iin Ileft A cap B_iright A cap bigcup_iin I B_i 這個定理一般會被不正式的寫為 bigcupinfty_i0left bigcapinfty_ji A_j right subseteq bigcapinfty_i0left bigcupinfty_ji A_j right。 參考 樸素集合論 交集 補集 對稱差 不交併 布林邏輯 參考文獻 B B B