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子集个数

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求有限集的子集个数问题,有以下结论:

结论1 :含有 `n` 个元素的集合\(A = \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)的子集个数为\({2^n}\),真子集个数为 `2^n-1`,非空子集个数为 `2^n-1`,非空真子集个数为 `2^n-2`  \(({\rm{n}} \in {{\rm{N}}^ \star }\))

结论2 : 设 $m,n\in {{N}^{*}}~~,~m<n~~,B=\left\{ {{a}_{1}},{{a}_{2}},\ldots ,{{a}_{n}} \right\}$,则有,

①满足\(\left\{ {{a_1},{a_2}, \ldots ,{a_m}} \right\} \subseteq A \subseteq \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)的集合 `A` 的个数是\({2^{n - m}}\);

②满足\(\left\{ {{a_1},{a_2}, \ldots ,{a_m}} \right\} \subseteq A\left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)的集合 `A` 的个数是\({2^{n - m}}-1\);

③满足\(\left\{ {{a_1},{a_2}, \ldots ,{a_m}} \right\}A \subseteq \left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)的集合 `A` 的个数是\({2^{n - m}}-1\);

④满足\(\left\{ {{a_1},{a_2}, \ldots ,{a_m}} \right\}A\left\{ {{a_1},{a_2}, \ldots ,{a_n}} \right\}\)的集合 `A` 的个数是\({2^{n - m}}-2\).