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集合运算中常用的结论

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1集合中的逻辑关系

(1)交集的运算性质.

\(A \cap B = B \cap A\),\(A \cap B \subseteq A\),\(A \cap B \subseteq B\)  ;   \(A \cap I = A\),\(A \cap A = A\),\(A \cap \emptyset  = \emptyset \).

(2)并集的运算性质.

\(A \cup B = B \cup A\),\(A \subseteq A \cup B\),\(B \subseteq A \cup B\)  ;   \(A \cup I = I\),\(A \cup A = A\),\(A \cup \emptyset  = A\).

(3)补集的运算性质.

${{\complement }_{I}}({{\complement }_{I}}A)=A$,${{\complement }_{I}}\varnothing =I$,${{\complement }_{I}}I=\varnothing $      ;        $({{\complement }_{I}}A)\cap A=\varnothing $,$A\cup ({{\complement }_{I}}A)I$.

补充性质:$A\cap B=A\Leftrightarrow A\cup B=B\Leftrightarrow A\subseteq B\Leftrightarrow {{\complement }_{I}}B\subseteq {{\complement }_{I}}A\Leftrightarrow A\cap {{\complement }_{I}}B=\varnothing $

(4)结合律与分配律.

结合律:\(A \cup (B \cup C) = (A \cup B) \cup C\)    ;     \(A \cap (B \cap C) = (A \cap B) \cap C\).

分配律:\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)   ;  \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).

(5)反演律(德摩根定律).

${{\complement }_{I}}(A\cap B)=({{\complement }_{I}}A)\cup ({{\complement }_{I}}B)$       ;        ${{\complement }_{I}}(A\cup B)=({{\complement }_{I}}A)\cap ({{\complement }_{I}}B)$.

即“交的补\( = \)补的并”,“并的补\( = \)补的交”.

2.由\(n(n \in {{\rm{N}}^*})\)个元素组成的集合\(A\)的子集个数

\(A\)的子集有\({2^n}\)个,非空子集有\({2^n} - 1\)个,真子集有\({2^n} - 1\)个,非空真子集有\({2^n} - 2\)个.

3.容斥原理

\(Card(A \cup B) = Card(A) + Card(B) - Card(A \cap B)\).