{"id":82859,"date":"2026-07-25T13:48:50","date_gmt":"2026-07-25T05:48:50","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/82859.html"},"modified":"2026-07-25T13:48:50","modified_gmt":"2026-07-25T05:48:50","slug":"04-%e9%9b%86%e5%90%88%e8%ae%ba%ef%bc%9a%e4%b8%80%e5%88%87%e6%95%b0%e5%ad%a6%e7%bb%93%e6%9e%84%e7%9a%84%e5%ae%b9%e5%99%a8","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/82859.html","title":{"rendered":"04 \u96c6\u5408\u8bba\uff1a\u4e00\u5207\u6570\u5b66\u7ed3\u6784\u7684\u5bb9\u5668"},"content":{"rendered":"<h2>04 \u96c6\u5408\u8bba&#xff1a;\u4e00\u5207\u6570\u5b66\u7ed3\u6784\u7684\u5bb9\u5668<\/h2>\n<p>\u96c6\u5408\u662f\u6570\u5b66\u4e2d\u6700\u6734\u7d20\u3001\u6700\u57fa\u672c\u7684\u6982\u5ff5\u4e4b\u4e00\u3002\u5b83\u5982\u6b64\u57fa\u7840&#xff0c;\u4ee5\u81f3\u4e8e\u51e0\u4e4e\u4e0d\u53ef\u80fd\u7528\u66f4\u7b80\u5355\u7684\u6982\u5ff5\u6765\u5b9a\u4e49\u5b83\u3002\u5eb7\u6258\u5c14\u8bf4&#xff1a;\u201c\u96c6\u5408\u5c31\u662f\u6211\u4eec\u53ef\u4ee5\u660e\u786e\u611f\u77e5\u5176\u4e3a\u4e00\u6574\u4f53\u7684\u591a\u4e2a\u5bf9\u8c61\u7684\u603b\u548c\u3002\u201d \u5728\u8fd9\u7bc7\u535a\u5ba2\u4e2d&#xff0c;\u6211\u4eec\u5c06\u4ece\u7a7a\u96c6\u51fa\u53d1&#xff0c;\u4e00\u8def\u63a2\u7d22\u5230\u65e0\u7a77\u96c6\u5408\u7684\u5947\u5999\u4e16\u754c\u3002<\/p>\n<hr \/>\n<h3>\u4e00\u3001\u4ec0\u4e48\u662f\u96c6\u5408&#xff1f;<\/h3>\n<p>\u96c6\u5408&#xff08;Set&#xff09;\u662f\u7531\u786e\u5b9a\u7684\u4e0d\u540c\u5bf9\u8c61\u6784\u6210\u7684\u6574\u4f53\u3002\u8fd9\u4e9b\u5bf9\u8c61\u79f0\u4e3a\u96c6\u5408\u7684\u5143\u7d20&#xff08;Element&#xff09;\u3002<\/p>\n<h4>\u6734\u7d20\u5b9a\u4e49<\/h4>\n<p>\u96c6\u5408\u662f\u7531\u6ee1\u8db3\u67d0\u79cd\u7279\u5b9a\u6027\u8d28\u7684\u5bf9\u8c61\u6240\u7ec4\u6210\u7684\u6574\u4f53\u3002<\/p>\n<ul>\n<li>a \u2208 A&#xff1a;a \u662f\u96c6\u5408 A \u7684\u5143\u7d20&#xff08;\u5c5e\u4e8e&#xff09;<\/li>\n<li>a \u2209 A&#xff1a;a \u4e0d\u662f\u96c6\u5408 A \u7684\u5143\u7d20&#xff08;\u4e0d\u5c5e\u4e8e&#xff09;<\/li>\n<\/ul>\n<h4>\u96c6\u5408\u7684\u8868\u793a\u65b9\u6cd5<\/h4>\n<li>\n<p>\u5217\u4e3e\u6cd5&#xff1a;\u76f4\u63a5\u5217\u51fa\u6240\u6709\u5143\u7d20<\/p>\n<ul>\n<li>A &#061; {1, 2, 3, 4, 5}<\/li>\n<li>B &#061; {a, e, i, o, u}&#xff08;\u5143\u97f3\u5b57\u6bcd\u96c6\u5408&#xff09;<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>\u63cf\u8ff0\u6cd5&#xff1a;\u7528\u6027\u8d28\u63cf\u8ff0\u5143\u7d20<\/p>\n<ul>\n<li>A &#061; {x | x \u662f\u6b63\u6574\u6570\u4e14 x \u2264 5}<\/li>\n<li>B &#061; {x | x \u662f\u5076\u6570}<\/li>\n<li>C &#061; {x \u2208 \u2115 | x\u00b2 &lt; 20}<\/li>\n<\/ul>\n<\/li>\n<li>\n<p>Venn \u56fe&#xff1a;\u7528\u56fe\u5f62\u76f4\u89c2\u8868\u793a<\/p>\n<ul>\n<li>\u753b\u4e00\u4e2a\u77e9\u5f62\u8868\u793a\u5168\u96c6<\/li>\n<li>\u753b\u5706\u8868\u793a\u96c6\u5408<\/li>\n<li>\u5706\u7684\u4f4d\u7f6e\u548c\u91cd\u53e0\u8868\u793a\u96c6\u5408\u5173\u7cfb<\/li>\n<\/ul>\n<\/li>\n<hr \/>\n<h3>\u4e8c\u3001\u96c6\u5408\u7684\u57fa\u672c\u5173\u7cfb<\/h3>\n<h4>1. \u5b50\u96c6&#xff08;Subset&#xff09;<\/h4>\n<p>A \u2286 B \u5f53\u4e14\u4ec5\u5f53 \u2200x (x \u2208 A \u2192 x \u2208 B)<\/p>\n<p>\u5373&#xff1a;A \u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u662f B \u7684\u5143\u7d20\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;<\/p>\n<ul>\n<li>{1, 2} \u2286 {1, 2, 3}<\/li>\n<li>\u2115 \u2286 \u2124 \u2286 \u211a \u2286 \u211d<\/li>\n<\/ul>\n<h4>2. \u771f\u5b50\u96c6&#xff08;Proper Subset&#xff09;<\/h4>\n<p>A \u2282 B \u5f53\u4e14\u4ec5\u5f53 A \u2286 B \u4e14 A \u2260 B<\/p>\n<p>\u5373&#xff1a;A \u662f B \u7684\u5b50\u96c6&#xff0c;\u4f46 B \u81f3\u5c11\u6709\u4e00\u4e2a\u5143\u7d20\u4e0d\u5728 A \u4e2d\u3002<\/p>\n<h4>3. \u96c6\u5408\u76f8\u7b49<\/h4>\n<p>A &#061; B \u5f53\u4e14\u4ec5\u5f53 A \u2286 B \u4e14 B \u2286 A<\/p>\n<p>\u5373&#xff1a;\u4e24\u4e2a\u96c6\u5408\u76f8\u7b49&#xff0c;\u5f53\u4e14\u4ec5\u5f53\u5b83\u4eec\u6709\u5b8c\u5168\u76f8\u540c\u7684\u5143\u7d20\u3002\u8bc1\u660e\u96c6\u5408\u76f8\u7b49\u7684\u57fa\u672c\u65b9\u6cd5&#xff1a;\u8bc1\u660e\u4e92\u76f8\u5305\u542b\u3002<\/p>\n<hr \/>\n<h3>\u4e09\u3001\u96c6\u5408\u7684\u57fa\u672c\u8fd0\u7b97<\/h3>\n<h4>1. \u5e76\u96c6&#xff08;Union&#xff09;&#xff1a;\u222a<\/h4>\n<p>A \u222a B &#061; {x | x \u2208 A \u2228 x \u2208 B}<\/p>\n<p>\u201cA \u6216 B \u4e2d\u7684\u5143\u7d20\u201d\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;{1, 2, 3} \u222a {2, 3, 4} &#061; {1, 2, 3, 4}<\/p>\n<p>Venn \u56fe&#xff1a;\u4e24\u4e2a\u5706\u8986\u76d6\u7684\u5168\u90e8\u533a\u57df\u3002<\/p>\n<h4>2. \u4ea4\u96c6&#xff08;Intersection&#xff09;&#xff1a;\u2229<\/h4>\n<p>A \u2229 B &#061; {x | x \u2208 A \u2227 x \u2208 B}<\/p>\n<p>\u201c\u540c\u65f6\u5c5e\u4e8e A \u548c B \u7684\u5143\u7d20\u201d\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;{1, 2, 3} \u2229 {2, 3, 4} &#061; {2, 3}<\/p>\n<p>Venn \u56fe&#xff1a;\u4e24\u4e2a\u5706\u91cd\u53e0\u7684\u533a\u57df\u3002<\/p>\n<h4>3. \u5dee\u96c6&#xff08;Difference&#xff09;&#xff1a;\u2212 \u6216 \\\\<\/h4>\n<p>A \u2212 B &#061; {x | x \u2208 A \u2227 x \u2209 B}<\/p>\n<p>\u201c\u5728 A \u4e2d\u4f46\u4e0d\u5728 B \u4e2d\u7684\u5143\u7d20\u201d\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;{1, 2, 3} \u2212 {2, 3, 4} &#061; {1}<\/p>\n<p>Venn \u56fe&#xff1a;A \u5706\u4e2d\u4e0d\u4e0e B \u91cd\u53e0\u7684\u90e8\u5206\u3002<\/p>\n<h4>4. \u8865\u96c6&#xff08;Complement&#xff09;&#xff1a;A\u1d9c \u6216 A\u0304<\/h4>\n<p>A\u1d9c &#061; {x | x \u2208 U \u2227 x \u2209 A} &#061; U \u2212 A<\/p>\n<p>\u5176\u4e2d U \u662f\u5168\u96c6&#xff08;Universal Set&#xff09;&#xff0c;\u5373\u5f53\u524d\u8ba8\u8bba\u4e2d\u6240\u6709\u53ef\u80fd\u5143\u7d20\u7684\u96c6\u5408\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;\u82e5 U &#061; {1, 2, 3, 4, 5}&#xff0c;A &#061; {1, 2}&#xff0c;\u5219 A\u1d9c &#061; {3, 4, 5}<\/p>\n<p>Venn \u56fe&#xff1a;\u77e9\u5f62\u4e2d A \u5706\u4ee5\u5916\u7684\u533a\u57df\u3002<\/p>\n<h4>5. \u5bf9\u79f0\u5dee&#xff08;Symmetric Difference&#xff09;&#xff1a;\u2295 \u6216 \u0394<\/h4>\n<p>A \u2295 B &#061; (A \u2212 B) \u222a (B \u2212 A) &#061; (A \u222a B) \u2212 (A \u2229 B)<\/p>\n<p>\u201c\u53ea\u5c5e\u4e8e A \u6216\u53ea\u5c5e\u4e8e B \u7684\u5143\u7d20\u201d&#xff08;\u5373\u5f02\u6216&#xff09;\u3002<\/p>\n<p>\u4f8b\u5b50&#xff1a;{1, 2, 3} \u2295 {2, 3, 4} &#061; {1, 4}<\/p>\n<hr \/>\n<h3>\u56db\u3001\u96c6\u5408\u8fd0\u7b97\u7684\u4ee3\u6570\u6027\u8d28<\/h3>\n<p>\u96c6\u5408\u8fd0\u7b97\u6ee1\u8db3\u8bb8\u591a\u6f02\u4eae\u7684\u4ee3\u6570\u5f8b&#xff0c;\u548c\u5b9e\u6570\u7684\u8fd0\u7b97\u5f8b\u975e\u5e38\u76f8\u4f3c&#xff1a;<\/p>\n<h4>\u57fa\u672c\u8fd0\u7b97\u5f8b<\/h4>\n<table>\n<tr>\u540d\u79f0\u516c\u5f0f<\/tr>\n<tbody>\n<tr>\n<td>\u5e42\u7b49\u5f8b<\/td>\n<td>A \u222a A &#061; A&#xff1b;A \u2229 A &#061; A<\/td>\n<\/tr>\n<tr>\n<td>\u4ea4\u6362\u5f8b<\/td>\n<td>A \u222a B &#061; B \u222a A&#xff1b;A \u2229 B &#061; B \u2229 A<\/td>\n<\/tr>\n<tr>\n<td>\u7ed3\u5408\u5f8b<\/td>\n<td>(A \u222a B) \u222a C &#061; A \u222a (B \u222a C)&#xff1b;\u540c\u7406 \u2229<\/td>\n<\/tr>\n<tr>\n<td>\u5206\u914d\u5f8b<\/td>\n<td>A \u222a (B \u2229 C) &#061; (A \u222a B) \u2229 (A \u222a C)&#xff1b;A \u2229 (B \u222a C) &#061; (A \u2229 B) \u222a (A \u2229 C)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u5fb7\u6469\u6839\u5f8b&#xff08;De Morgan\u2019s Laws&#xff09;\u2b50<\/h4>\n<table>\n<tr>\u516c\u5f0f\u5bf9\u5e94\u903b\u8f91<\/tr>\n<tbody>\n<tr>\n<td>(A \u222a B)\u1d9c &#061; A\u1d9c \u2229 B\u1d9c<\/td>\n<td>\u00ac(p \u2228 q) \u2261 \u00acp \u2227 \u00acq<\/td>\n<\/tr>\n<tr>\n<td>(A \u2229 B)\u1d9c &#061; A\u1d9c \u222a B\u1d9c<\/td>\n<td>\u00ac(p \u2227 q) \u2261 \u00acp \u2228 \u00acq<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u8fd9\u5c31\u662f\u96c6\u5408\u4e0e\u903b\u8f91\u4e4b\u95f4\u7684\u6df1\u523b\u5bf9\u5e94&#xff1a;\u96c6\u5408\u8fd0\u7b97\u548c\u903b\u8f91\u8fd0\u7b97\u672c\u8d28\u4e0a\u662f\u540c\u6784\u7684&#xff01;<\/p>\n<hr \/>\n<h3>\u4e94\u3001\u5e42\u96c6&#xff1a;\u96c6\u5408\u7684&#034;\u96c6\u5408&#034;<\/h3>\n<h4>\u5b9a\u4e49<\/h4>\n<p>A \u7684\u5e42\u96c6&#xff08;Power Set&#xff09;&#xff0c;\u8bb0\u4f5c P(A) \u6216 2\u1d2c&#xff0c;\u662f A \u7684\u6240\u6709\u5b50\u96c6\u6240\u7ec4\u6210\u7684\u96c6\u5408\u3002<\/p>\n<p>P(A) &#061; {B | B \u2286 A}<\/p>\n<p>\u4f8b\u5b50&#xff1a;A &#061; {1, 2}<\/p>\n<p>P(A) &#061; {\u2205, {1}, {2}, {1, 2}}<\/p>\n<h4>\u5e42\u96c6\u7684\u5927\u5c0f<\/h4>\n<p>\u82e5 |A| &#061; n&#xff08;A \u6709 n \u4e2a\u5143\u7d20&#xff09;&#xff0c;\u5219 |P(A)| &#061; 2\u207f<\/p>\n<p>\u4e3a\u4ec0\u4e48&#xff1f;<\/p>\n<p>A \u7684\u6bcf\u4e2a\u5143\u7d20\u5728\u5b50\u96c6\u4e2d\u90fd\u6709&#034;\u51fa\u73b0&#034;\u6216&#034;\u4e0d\u51fa\u73b0&#034;\u4e24\u79cd\u9009\u62e9\u3002n \u4e2a\u5143\u7d20\u5c31\u6709 2 \u00d7 2 \u00d7 \u2026 \u00d7 2 &#061; 2\u207f \u79cd\u7ec4\u5408&#xff0c;\u6bcf\u79cd\u7ec4\u5408\u5bf9\u5e94\u4e00\u4e2a\u5b50\u96c6\u3002<\/p>\n<p>\u8fd9\u5c31\u662f\u5e42\u96c6\u8bb0\u4f5c 2\u1d2c \u7684\u539f\u56e0&#xff01;<\/p>\n<hr \/>\n<h3>\u516d\u3001\u7279\u6b8a\u7684\u96c6\u5408<\/h3>\n<h4>1. \u7a7a\u96c6&#xff08;Empty Set&#xff09;&#xff1a;\u2205<\/h4>\n<p>\u4e0d\u542b\u4efb\u4f55\u5143\u7d20\u7684\u96c6\u5408\u3002\u5b83\u6709\u4e00\u4e9b\u6709\u8da3\u7684\u6027\u8d28&#xff1a;<\/p>\n<ul>\n<li>\u2205 \u2286 A \u5bf9\u4efb\u4f55\u96c6\u5408 A \u90fd\u6210\u7acb&#xff08; vacuously true&#xff0c;\u56e0\u4e3a\u4e0d\u5b58\u5728 x \u2208 \u2205 \u4f7f\u5f97 x \u2209 A&#xff09;<\/li>\n<li>|\u2205| &#061; 0<\/li>\n<li>P(\u2205) &#061; {\u2205}&#xff08;\u6ce8\u610f&#xff1a;\u4e0d\u662f \u2205 \u672c\u8eab&#xff01;\u7a7a\u96c6\u7684\u5e42\u96c6\u5305\u542b\u4e00\u4e2a\u5143\u7d20\u2014\u2014\u7a7a\u96c6\u672c\u8eab&#xff09;<\/li>\n<li>|P(\u2205)| &#061; 2\u2070 &#061; 1<\/li>\n<\/ul>\n<h4>2. \u5355\u5143\u7d20\u96c6\u5408&#xff08;Singleton&#xff09;<\/h4>\n<p>\u53ea\u542b\u4e00\u4e2a\u5143\u7d20\u7684\u96c6\u5408&#xff0c;\u5982 {a}\u3001{\u2205}\u3002<\/p>\n<p>\u26a0\ufe0f \u6ce8\u610f\u533a\u5206&#xff1a;a \u548c {a} \u4e0d\u4e00\u6837&#xff01;<\/p>\n<ul>\n<li>a \u662f\u5143\u7d20<\/li>\n<li>{a} \u662f\u96c6\u5408&#xff0c;\u5b83\u7684\u552f\u4e00\u5143\u7d20\u662f a<\/li>\n<li>\u6240\u4ee5 a \u2208 {a} \u4e3a\u771f&#xff0c;\u4f46 a \u2286 {a} \u4e0d\u4e00\u5b9a\u4e3a\u771f&#xff08;\u9664\u975e a \u672c\u8eab\u4e5f\u662f\u96c6\u5408&#xff09;<\/li>\n<\/ul>\n<h4>3. \u5168\u96c6&#xff08;Universal Set&#xff09;<\/h4>\n<p>\u5f53\u524d\u8ba8\u8bba\u8303\u56f4\u5185\u6240\u6709\u5143\u7d20\u7684\u96c6\u5408\u3002\u5168\u96c6\u4e0d\u662f\u7edd\u5bf9\u7684&#xff0c;\u53d6\u51b3\u4e8e\u4e0a\u4e0b\u6587\u3002<\/p>\n<hr \/>\n<h3>\u4e03\u3001\u96c6\u5408\u7684\u57fa\u6570\u4e0e\u65e0\u7a77<\/h3>\n<h4>1. \u6709\u9650\u96c6\u4e0e\u65e0\u9650\u96c6<\/h4>\n<ul>\n<li>\u6709\u9650\u96c6&#xff1a;\u5143\u7d20\u4e2a\u6570\u662f\u4e00\u4e2a\u81ea\u7136\u6570<\/li>\n<li>\u65e0\u9650\u96c6&#xff1a;\u5143\u7d20\u4e2a\u6570\u4e0d\u662f\u4efb\u4f55\u81ea\u7136\u6570<\/li>\n<\/ul>\n<h4>2. \u57fa\u6570\u7684\u521d\u6b65\u6982\u5ff5<\/h4>\n<p>\u5bf9\u4e8e\u6709\u9650\u96c6&#xff0c;\u57fa\u6570\u5c31\u662f\u5143\u7d20\u7684\u4e2a\u6570\u3002\u4f46\u65e0\u9650\u96c6\u5462&#xff1f;<\/p>\n<p>\u5eb7\u6258\u5c14&#xff08;Georg Cantor&#xff09;\u7684\u4f1f\u5927\u53d1\u73b0&#xff1a;\u4e0d\u540c\u7684\u65e0\u9650\u96c6\u5408\u53ef\u4ee5\u6709&#034;\u5927\u5c0f&#034;\u4e4b\u5206&#xff01;<\/p>\n<h4>3. \u53ef\u6570\u65e0\u7a77&#xff08;\u2135\u2080&#xff09;<\/h4>\n<p>\u5982\u679c\u4e00\u4e2a\u65e0\u9650\u96c6\u5408\u7684\u5143\u7d20\u53ef\u4ee5\u6309\u67d0\u79cd\u987a\u5e8f&#034;\u4e00\u4e2a\u4e00\u4e2a\u6570\u51fa\u6765&#034;&#xff08;\u4e0e\u81ea\u7136\u6570\u96c6\u5efa\u7acb\u4e00\u4e00\u5bf9\u5e94&#xff09;&#xff0c;\u5c31\u79f0\u5b83\u662f\u53ef\u6570\u65e0\u7a77&#xff08;Countably Infinite&#xff09;\u3002<\/p>\n<p>\u7ecf\u5178\u4f8b\u5b50&#xff1a;<\/p>\n<ul>\n<li>\u81ea\u7136\u6570\u96c6 \u2115 &#061; {0, 1, 2, 3, \u2026}&#xff1a;\u663e\u7136\u53ef\u6570<\/li>\n<li>\u6574\u6570\u96c6 \u2124 &#061; {\u2026, -2, -1, 0, 1, 2, \u2026}&#xff1a;\u53ef\u6570&#xff01;\u53ef\u4ee5\u6309 0, 1, -1, 2, -2, \u2026 \u7684\u987a\u5e8f\u6392\u5217<\/li>\n<li>\u6709\u7406\u6570\u96c6 \u211a&#xff1a;\u53ef\u6570&#xff01;\u867d\u7136\u6709\u7406\u6570&#034;\u5bc6\u5bc6\u9ebb\u9ebb&#034;&#xff0c;\u4f46\u4ecd\u7136\u53ef\u4ee5\u4e00\u4e2a\u4e00\u4e2a\u6392\u51fa\u6765&#xff08;\u901a\u8fc7\u5bf9\u89d2\u7ebf\u679a\u4e3e\u6cd5&#xff09;<\/li>\n<\/ul>\n<p>\u8fd9\u4e9b\u96c6\u5408\u7684\u57fa\u6570\u90fd\u662f \u2135\u2080&#xff08;\u963f\u5217\u592b\u96f6&#xff09;\u3002<\/p>\n<h4>4. \u4e0d\u53ef\u6570\u65e0\u7a77&#xff08;&#x1d520;&#xff09;<\/h4>\n<p>\u4f46\u5eb7\u6258\u5c14\u7528\u5bf9\u89d2\u7ebf\u8bba\u8bc1\u6cd5\u8bc1\u660e\u4e86&#xff1a;<\/p>\n<p>\u5b9e\u6570\u96c6 \u211d \u662f\u4e0d\u53ef\u6570\u65e0\u7a77&#xff01;<\/p>\n<p>\u65e0\u8bba\u4f60\u600e\u4e48\u5c1d\u8bd5\u628a\u5b9e\u6570\u6392\u6210\u4e00\u5217&#xff0c;\u603b\u6709\u4e00\u4e2a\u5b9e\u6570\u88ab\u6f0f\u6389\u3002\u8fd9\u610f\u5473\u7740\u5b9e\u6570\u96c6&#034;\u6bd4&#034;\u81ea\u7136\u6570\u96c6&#034;\u5927&#034;\u2014\u2014\u5373\u4f7f\u5b83\u4eec\u90fd\u662f\u65e0\u9650\u96c6\u5408\u3002<\/p>\n<p>\u5b9e\u6570\u96c6\u7684\u57fa\u6570\u8bb0\u4e3a &#x1d520;&#xff08;\u8fde\u7eed\u7edf\u7684\u57fa\u6570&#xff09;&#xff0c;\u4e14 &#x1d520; &gt; \u2135\u2080\u3002<\/p>\n<h4>5. \u5e42\u96c6\u4e0e\u66f4\u5927\u7684\u65e0\u7a77<\/h4>\n<p>\u5eb7\u6258\u5c14\u8fd8\u8bc1\u660e\u4e86&#xff1a;<\/p>\n<p>\u5bf9\u4efb\u4f55\u96c6\u5408 A&#xff0c;|P(A)| &gt; |A|<\/p>\n<p>\u6240\u4ee5&#xff1a;<\/p>\n<ul>\n<li>|P(\u2115)| &gt; \u2135\u2080<\/li>\n<li>|P(P(\u2115))| &gt; |P(\u2115)|<\/li>\n<li>\u4f9d\u6b64\u7c7b\u63a8&#xff0c;\u65e0\u7a77\u96c6\u5408\u6709\u65e0\u7a77\u591a\u4e2a\u5c42\u6b21<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u96c6\u5408\u8bba\u4e2d\u6700\u4ee4\u4eba\u60ca\u53f9\u7684\u53d1\u73b0\u4e4b\u4e00&#xff1a;\u65e0\u7a77\u4e5f\u6709\u5927\u5c0f\u4e4b\u5206&#xff01;<\/p>\n<hr \/>\n<h3>\u516b\u3001\u96c6\u5408\u8bba\u4e0e\u8ba1\u7b97\u673a\u79d1\u5b66\u7684\u5173\u8054<\/h3>\n<table>\n<tr>\u96c6\u5408\u6982\u5ff5\u8ba1\u7b97\u673a\u5bf9\u5e94<\/tr>\n<tbody>\n<tr>\n<td>\u96c6\u5408<\/td>\n<td>\u6570\u636e\u7ed3\u6784\u4e2d\u7684\u96c6\u5408&#xff08;Set\/HashSet&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u5e76\u96c6 \u222a<\/td>\n<td>set1.union(set2)<\/td>\n<\/tr>\n<tr>\n<td>\u4ea4\u96c6 \u2229<\/td>\n<td>set1.intersection(set2)<\/td>\n<\/tr>\n<tr>\n<td>\u5dee\u96c6 \u2212<\/td>\n<td>set1.difference(set2)<\/td>\n<\/tr>\n<tr>\n<td>\u5bf9\u79f0\u5dee \u2295<\/td>\n<td>set1.symmetric_difference(set2)<\/td>\n<\/tr>\n<tr>\n<td>\u5b50\u96c6 \u2286<\/td>\n<td>set1.issubset(set2)<\/td>\n<\/tr>\n<tr>\n<td>\u5e42\u96c6<\/td>\n<td>\u5b50\u96c6\u679a\u4e3e\/\u4f4d\u8fd0\u7b97\u751f\u6210<\/td>\n<\/tr>\n<tr>\n<td>\u7a7a\u96c6 \u2205<\/td>\n<td>\u7a7a\u96c6\u5408 set()<\/td>\n<\/tr>\n<tr>\n<td>\u5168\u96c6<\/td>\n<td>\u5f53\u524d\u8bba\u57df\/\u7c7b\u578b\u7cfb\u7edf<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>\u7528\u4f4d\u8fd0\u7b97\u8868\u793a\u5b50\u96c6<\/h4>\n<p>\u5bf9\u4e8e\u6709\u9650\u96c6\u5408&#xff0c;\u5b50\u96c6\u53ef\u4ee5\u7528\u4e8c\u8fdb\u5236\u6570\u7f16\u7801&#xff1a;<\/p>\n<p>A &#061; {a, b, c}&#xff0c;\u7528 3 \u4f4d\u4e8c\u8fdb\u5236\u8868\u793a&#xff1a;<\/p>\n<ul>\n<li>\u7b2c 1 \u4f4d\u8868\u793a a \u662f\u5426\u5728\u5b50\u96c6\u4e2d<\/li>\n<li>\u7b2c 2 \u4f4d\u8868\u793a b \u662f\u5426\u5728\u5b50\u96c6\u4e2d<\/li>\n<li>\u7b2c 3 \u4f4d\u8868\u793a c \u662f\u5426\u5728\u5b50\u96c6\u4e2d<\/li>\n<\/ul>\n<table>\n<tr>\u5b50\u96c6\u4e8c\u8fdb\u5236\u5341\u8fdb\u5236<\/tr>\n<tbody>\n<tr>\n<td>\u2205<\/td>\n<td>000<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>{a}<\/td>\n<td>001<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>{b}<\/td>\n<td>010<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td>{a, b}<\/td>\n<td>011<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>{c}<\/td>\n<td>100<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>{a, c}<\/td>\n<td>101<\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td>{b, c}<\/td>\n<td>110<\/td>\n<td>6<\/td>\n<\/tr>\n<tr>\n<td>{a, b, c}<\/td>\n<td>111<\/td>\n<td>7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u5e42\u96c6\u6709 2\u207f \u4e2a\u5143\u7d20&#xff01;\u6bcf\u4e2a\u5b50\u96c6\u5bf9\u5e94\u4e00\u4e2a n \u4f4d\u4e8c\u8fdb\u5236\u6570&#xff0c;\u8303\u56f4\u662f 0 \u5230 2\u207f\u22121\u3002<\/p>\n<p>Python \u4ee3\u7801\u751f\u6210\u5e42\u96c6&#xff1a;<\/p>\n<p><span class=\"token keyword\">def<\/span> <span class=\"token function\">power_set<\/span><span class=\"token punctuation\">(<\/span>elements<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    n <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>elements<span class=\"token punctuation\">)<\/span><br \/>\n    result <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">]<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> mask <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">1<\/span> <span class=\"token operator\">&lt;&lt;<\/span> n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span>  <span class=\"token comment\"># 0 \u5230 2^n &#8211; 1<\/span><br \/>\n        subset <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span>elements<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">if<\/span> mask <span class=\"token operator\">&amp;<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">1<\/span> <span class=\"token operator\">&lt;&lt;<\/span> i<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><br \/>\n        result<span class=\"token punctuation\">.<\/span>append<span class=\"token punctuation\">(<\/span>subset<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> result<\/p>\n<p><span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span>power_set<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token string\">&#039;a&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#039;b&#039;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token string\">&#039;c&#039;<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token comment\"># \u8f93\u51fa: [[], [&#039;a&#039;], [&#039;b&#039;], [&#039;a&#039;, &#039;b&#039;], [&#039;c&#039;], [&#039;a&#039;, &#039;c&#039;], [&#039;b&#039;, &#039;c&#039;], [&#039;a&#039;, &#039;b&#039;, &#039;c&#039;]]<\/span><\/p>\n<hr \/>\n<h3>\u4e5d\u3001\u96c6\u5408\u8bba\u6096\u8bba\u4e0e\u516c\u7406\u5316<\/h3>\n<h4>\u7f57\u7d20\u6096\u8bba&#xff08;Russell\u2019s Paradox&#xff09;<\/h4>\n<p>\u96c6\u5408\u8bba\u7684\u6734\u7d20\u5b9a\u4e49\u5bfc\u81f4\u4e86\u8457\u540d\u7684\u6096\u8bba&#xff1a;<\/p>\n<p>\u8bbe R &#061; {x | x \u2209 x}&#xff08;\u6240\u6709\u4e0d\u5305\u542b\u81ea\u8eab\u7684\u96c6\u5408\u6240\u6784\u6210\u7684\u96c6\u5408&#xff09;<\/p>\n<p>\u95ee&#xff1a;R \u2208 R \u5417&#xff1f;<\/p>\n<ul>\n<li>\u5982\u679c R \u2208 R&#xff0c;\u90a3\u4e48\u6839\u636e\u5b9a\u4e49 R \u2209 R&#xff08;\u56e0\u4e3a R \u53ea\u5305\u542b\u90a3\u4e9b\u4e0d\u5305\u542b\u81ea\u8eab\u7684\u96c6\u5408&#xff09;<\/li>\n<li>\u5982\u679c R \u2209 R&#xff0c;\u90a3\u4e48\u6839\u636e\u5b9a\u4e49 R \u2208 R&#xff08;\u56e0\u4e3a R \u4e0d\u5305\u542b\u81ea\u8eab&#xff09;<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u6096\u8bba&#xff01;<\/p>\n<p>\u8fd9\u4e2a\u6096\u8bba\u8bf4\u660e\u4e0d\u80fd\u968f\u610f\u6784\u9020\u96c6\u5408\u3002\u4e3a\u4e86\u907f\u514d\u8fd9\u6837\u7684\u6096\u8bba&#xff0c;\u6570\u5b66\u5bb6\u5efa\u7acb\u4e86\u516c\u7406\u5316\u96c6\u5408\u8bba&#xff08;ZFC \u516c\u7406\u7cfb\u7edf&#xff09;&#xff0c;\u5bf9\u96c6\u5408\u7684\u6784\u9020\u52a0\u4ee5\u9650\u5236\u3002<\/p>\n<p>\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d&#xff0c;\u6211\u4eec\u4e3b\u8981\u5904\u7406\u6709\u9650\u96c6\u5408&#xff0c;\u6240\u4ee5\u901a\u5e38\u4e0d\u4f1a\u9047\u5230\u8fd9\u4e9b\u6096\u8bba\u3002\u4f46\u4e86\u89e3\u5b83\u4eec\u6709\u52a9\u4e8e\u7406\u89e3\u96c6\u5408\u8bba\u7684\u6df1\u5c42\u7ed3\u6784\u3002<\/p>\n<hr \/>\n<h3>\u5341\u3001\u672c\u7ae0\u5c0f\u7ed3<\/h3>\n<table>\n<tr>\u6982\u5ff5\u8981\u70b9<\/tr>\n<tbody>\n<tr>\n<td>\u96c6\u5408<\/td>\n<td>\u786e\u5b9a\u5bf9\u8c61\u7684\u603b\u4f53<\/td>\n<\/tr>\n<tr>\n<td>\u5b50\u96c6 \u2286<\/td>\n<td>\u4e00\u4e2a\u96c6\u5408\u7684\u6240\u6709\u5143\u7d20\u5c5e\u4e8e\u53e6\u4e00\u4e2a\u96c6\u5408<\/td>\n<\/tr>\n<tr>\n<td>\u5e76\u96c6 \u222a<\/td>\n<td>\u5c5e\u4e8e A \u6216 B<\/td>\n<\/tr>\n<tr>\n<td>\u4ea4\u96c6 \u2229<\/td>\n<td>\u540c\u65f6\u5c5e\u4e8e A \u548c B<\/td>\n<\/tr>\n<tr>\n<td>\u5dee\u96c6 \u2212<\/td>\n<td>\u5728 A \u4e2d\u4f46\u4e0d\u5728 B \u4e2d<\/td>\n<\/tr>\n<tr>\n<td>\u8865\u96c6 \u1d9c<\/td>\n<td>\u5168\u96c6\u4e2d\u4e0d\u5728 A \u4e2d\u7684\u5143\u7d20<\/td>\n<\/tr>\n<tr>\n<td>\u5bf9\u79f0\u5dee \u2295<\/td>\n<td>\u53ea\u5c5e\u4e8e A \u6216\u53ea\u5c5e\u4e8e B<\/td>\n<\/tr>\n<tr>\n<td>\u5e42\u96c6 P(A)<\/td>\n<td>\u6240\u6709\u5b50\u96c6\u6784\u6210\u7684\u96c6\u5408&#xff0c;\u5927\u5c0f\u4e3a 2^<\/td>\n<\/tr>\n<tr>\n<td>\u5fb7\u6469\u6839\u5f8b<\/td>\n<td>(A\u222aB)\u1d9c &#061; 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