数学上布尔运算是什么意思？ 什么叫布尔运算？
\u4ec0\u4e48\u662f\u5e03\u5c14\u8fd0\u7b97\uff1f\u82f1\u56fd\u6709\u4e2a\u53eb\u5e03\u5c14\u7684\u6570\u5b66\u5e03\u5c14\u903b\u8f91
\u5e03\u5c14\u903b\u8f91\u5f97\u540d\u4e8e George Boole\uff0c\u4ed6\u662f\u8003\u514b\u5927\u5b66\uff08\u73b0\u7231\u5c14\u5170\u56fd\u7acb\u8003\u514b\u5927\u5b66\uff09\u7684\u82f1\u56fd\u6570\u5b66\u5bb6\uff0c\u4ed6\u5728\u5341\u4e5d\u4e16\u7eaa\u4e2d\u53f6\u9996\u6b21\u5b9a\u4e49\u4e86\u903b\u8f91\u7684\u4ee3\u6570\u7cfb\u7edf\u3002\u73b0\u5728\uff0c\u5e03\u5c14\u903b\u8f91\u5728\u7535\u5b50\u5b66\u3001\u8ba1\u7b97\u673a\u786c\u4ef6\u548c\u8f6f\u4ef6\u4e2d\u6709\u5f88\u591a\u5e94\u7528\u3002\u5728 1937 \u5e74\uff0cClaude Shannon \u5c55\u793a\u4e86\u5e03\u5c14\u903b\u8f91\u5982\u4f55\u5728\u7535\u5b50\u5b66\u4e2d\u4f7f\u7528\u3002
\u903b\u8f91-\u96c6\u5408\u4ee3\u6570\u548c\u6587\u6c0f\u56fe
\u4f7f\u7528\u96c6\u5408\u4ee3\u6570\u4f5c\u4e3a\u4ecb\u7ecd\u5e03\u5c14\u903b\u8f91\u7684\u4e00\u79cd\u65b9\u5f0f\u3002\u8fd8\u4f7f\u7528\u6587\u6c0f\u56fe\u6765\u5c55\u793a\u5404\u79cd\u5e03\u5c14\u903b\u8f91\u9648\u8ff0\u6240\u63cf\u8ff0\u7684\u96c6\u5408\u8054\u7cfb\u3002
\u8bbe X \u662f\u4e00\u4e2a\u96c6\u5408\uff1a
\u5143\u7d20\u662f\u4e00\u4e2a\u96c6\u5408\u7684\u6210\u5458\u3002\u8868\u793a\u4e3a \in\u3002\u5982\u679c\u5b83\u4e0d\u662f\u8fd9\u4e2a\u96c6\u5408\u7684\u5143\u7d20\uff0c\u8868\u793a\u4e3a \notin\u3002
\u5168\u96c6\u662f\u96c6\u5408 X\uff0c\u6709\u65f6\u8868\u793a\u4e3a 1\u3002\u6ce8\u610f\u4f7f\u7528\u5168\u96c6\u8fd9\u4e2a\u8bcd\u610f\u5473\u7740"\u8651\u53ca\u7684\u6240\u6709\u5143\u7d20"\uff0c\u540c"\u73b0\u6709\u7684\u6240\u6709\u5143\u7d20"\u4e00\u6837\u4e0d\u662f\u5fc5\u7136\u7684\u3002
\u7a7a\u96c6\u6216 null \u96c6\u5408\u662f\u6ca1\u6709\u5143\u7d20\u7684\u96c6\u5408\uff0c\u8868\u793a\u4e3a \varnothing\uff0c\u6709\u65f6\u8868\u793a\u4e3a 0\u3002
\u4e00\u5143\u7b97\u7b26\u5e94\u7528\u4e8e\u4e00\u4e2a\u5355\u4e00\u7684\u96c6\u5408\u3002\u6709\u4e00\u4e2a\u4e00\u5143\u7b97\u7b26\u53eb\u505a\u903b\u8f91\u975e(NOT)\u3002\u5b83\u7684\u4f5c\u7528\u662f\u91c7\u7528\u8865\u96c6\u3002
\u4e8c\u5143\u7b97\u7b26\u5e94\u7528\u4e8e\u4e24\u4e2a\u96c6\u5408\u3002\u57fa\u672c\u7684\u4e8c\u5143\u7b97\u7b26\u662f\u903b\u8f91\u6216(OR)\u548c\u903b\u8f91\u4e0e(AND)\u3002\u5b83\u4eec\u8fdb\u884c\u96c6\u5408\u7684\u4ea4\u96c6\u548c\u5e76\u96c6\u3002\u8fd8\u6709\u5176\u4ed6\u884d\u751f\u7684\u4e8c\u5143\u7b97\u7b26\uff0c\u6bd4\u5982\u903b\u8f91\u5f02\u6216(XOR)(\u6392\u4ed6\u7684\u6216)\u3002
\u5b50\u96c6\u8868\u793a\u4e3a A \subseteq B\uff0c\u610f\u5473\u8fd9\u5728\u96c6\u5408 A \u4e2d\u6240\u6709\u5143\u7d20\u90fd\u5728\u96c6\u5408 B \u4e2d\u3002
\u771f\u5b50\u96c6\u8868\u793a\u4e3a A \subset B\uff0c\u610f\u5473\u7740\u5728\u96c6\u5408 A \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5728\u96c6\u5408 B \u4e2d\uff0c\u5e76\u4e14\u4e24\u4e2a\u96c6\u5408\u4e0d\u7b49\u540c\u3002
\u8d85\u96c6\u8868\u793a\u4e3a A \supseteq B\uff0c\u610f\u5473\u7740\u5728\u96c6\u5408 B \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5728\u96c6\u5408 A \u4e2d\u3002
\u771f\u8d85\u96c6 \u8868\u793a\u4e3a A \supset B\uff0c\u610f\u5473\u7740\u5728\u96c6\u5408 B \u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u5728\u96c6\u5408 A \u4e2d\uff0c\u5e76\u4e14\u4e24\u4e2a\u96c6\u5408\u4e0d\u7b49\u540c\u3002
\u4f8b\u5b50
\u8bbe\u56fe\u50cf\u4e3a\u96c6\u5408 A \u5305\u542b"\u5168\u96c6"\u4e2d\u6240\u6709\u5076\u6570(\u4e8c\u7684\u500d\u6570)\uff0c\u96c6\u5408 B \u5305\u542b"\u5168\u96c6"\u4e2d\u6240\u6709\u4e09\u7684\u500d\u6570\u3002\u5219\u4e24\u4e2a\u96c6\u5408\u7684\u4ea4\u96c6(\u5728\u96c6\u5408 A AND B \u4e2d\u6240\u6709\u7684\u5143\u7d20)\u5c06\u662f"\u5168\u96c6"\u4e2d\u6240\u6709\u516d\u7684\u500d\u6570\u3002
\u96c6\u5408 A \u7684\u8865\u96c6(\u6240\u6709\u4e0d\u5728\u96c6\u5408 A \u4e2d\u7684\u5143\u7d20)\u662f"\u5168\u96c6"\u4e2d\u6240\u6709\u7684\u5947\u6570\u3002
\u628a\u8fd0\u7b97\u8fde\u63a5\u8d77\u6765
\u5c3d\u7ba1\u5728\u4efb\u4f55\u5e03\u5c14\u8fd0\u7b97\u4e2d\u90fd\u6700\u591a\u6709\u4e24\u4e2a\u96c6\u5408\u53c2\u4e0e\uff0c\u4ece\u8fd9\u4e2a\u8fd0\u7b97\u6240\u5f62\u6210\u7684\u65b0\u96c6\u5408\u53ef\u4ee5\u63a5\u7740\u4e0e\u5176\u4ed6\u96c6\u5408\u8054\u5408\u8d77\u6765\u5b9e\u73b0\u53e6\u5916\u7684\u5e03\u5c14\u8fd0\u7b97\u3002\u4f7f\u7528\u524d\u9762\u7684\u4f8b\u5b50\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u65b0\u96c6\u5408 C \u4f5c\u4e3a"\u5168\u96c6"\u4e2d\u6240\u6709\u4e94\u7684\u500d\u6570\u7684\u96c6\u5408\u3002\u6240\u4ee5 "\u96c6\u5408 A AND B AND C" \u5c06\u662f"\u5168\u96c6"\u4e2d\u6240\u6709 30 \u7684\u500d\u6570\u3002\u5982\u679c\u4e3a\u4e86\u66f4\u65b9\u4fbf\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u96c6\u5408 AB \u5f53\u4f5c\u96c6\u5408 A \u548c B \u7684\u4ea4\u96c6\uff0c\u6216\u8005\u8bf4"\u5168\u96c6"\u4e2d\u6240\u6709\u516d\u7684\u500d\u6570\u7684\u96c6\u5408\u3002\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u79f0 "\u96c6\u5408 AB AND C" \u662f"\u5168\u96c6"\u4e2d\u6240\u6709 30 \u7684\u500d\u6570\u7684\u96c6\u5408\u3002\u6211\u4eec\u63a5\u7740\u8fdb\u4e00\u6b65\u7684\u628a\u8fd9\u4e2a\u7ed3\u679c\u53eb\u505a\u96c6\u5408 ABC\u3002
\u4f7f\u7528\u5706\u62ec\u53f7
\u5c3d\u7ba1\u4efb\u4f55\u6570\u76ee\u7684\u903b\u8f91 AND(\u6216\u4efb\u4f55\u6570\u76ee\u7684\u903b\u8f91 OR)\u53ef\u4ee5\u88ab\u8fde\u63a5\u5728\u4e00\u8d77\u800c\u6ca1\u6709\u6b67\u4e49\uff0cAND \u548c OR \u548c NOT \u7684\u7ec4\u5408\u53ef\u4ee5\u5bfc\u81f4\u6b67\u4e49\u7684\u60c5\u51b5\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u4f7f\u7528\u5706\u62ec\u53f7\u6765\u5206\u6e05\u8fd0\u7b97\u7684\u6b21\u5e8f\u3002\u6c38\u8fdc\u662f\u6700\u5185\u7684\u62ec\u53f7\u5185\u7684\u8fd0\u7b97\u5148\u8fdb\u884c\uff0c\u968f\u540e\u662f\u5916\u5c42\u7684\u62ec\u53f7\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u5728\u6240\u6709\u7684\u62ec\u53f7\u5185\u8fd0\u7b97\u90fd\u5b8c\u6210\u3002\u63a5\u7740\u8fdb\u884c\u62ec\u53f7\u5916\u7684\u8fd0\u7b97\u3002
\u6027\u8d28
\u4e3a\u4e24\u4e2a\u4e3b\u8981\u7684\u4e8c\u5143\u8fd0\u7b97\u7684\u7b26\u53f7\u5b9a\u4e49\u4e3a \land / \cap (\u903b\u8f91\u4e0e/\u4ea4\u96c6)\u548c \lor / \cup (\u903b\u8f91\u6216/\u5e76\u96c6)\uff0c\u628a\u5355\u4e00\u7684\u4e00\u5143\u8fd0\u7b97\u7684\u7b26\u53f7\u5b9a\u4e49\u4e3a \lnot / ~ (\u903b\u8f91\u975e/\u8865\u96c6)\u3002\u6211\u4eec\u8fd8\u4f7f\u7528\u503c 0 (\u903b\u8f91\u5047/\u7a7a\u96c6)\u548c 1 (\u903b\u8f91\u771f/\u5168\u96c6)\u3002\u4e0b\u5217\u6027\u8d28\u9002\u7528\u4e8e\u5e03\u5c14\u4ee3\u6570\u548c\u5e03\u5c14\u903b\u8f91\u4e8c\u8005\uff1a
a \lor (b \lor c) = (a \lor b) \lor c a \land (b \land c) = (a \land b) \land c \u7ed3\u5408\u5f8b
a \lor b = b \lor a a \land b = b \land a \u4ea4\u6362\u5f8b
a \lor (a \land b) = a a \land (a \lor b) = a \u5438\u6536\u5f8b
a \lor (b \land c) = (a \lor b) \land (a \lor c) a \land (b \lor c) = (a \land b) \lor (a \land c) \u5206\u914d\u5f8b
a \lor \lnot a = 1 a \land \lnot a = 0 \u4e92\u8865\u5f8b
a \lor a = a a \land a = a \u7b49\u5e42\u5f8b
a \lor 0 = a a \land 1 = a \u6709\u754c\u5f8b
a \lor 1 = 1 a \land 0 = 0
\lnot 0 = 1 \lnot 1 = 0 0 \u548c 1 \u662f\u4e92\u8865\u7684
\lnot (a \lor b) = \lnot a \land \lnot b \lnot (a \land b) = \lnot a \lor \lnot b de Morgan \u5b9a\u5f8b
\lnot \lnot a = a \u5377\u7ed5\u5f8b(involution
\u771f\u503c\u8868
\u5e03\u5c14\u903b\u8f91\u53ea\u4f7f\u7528\u4e24\u4e2a\u503c 0 \u548c 1\uff0c\u8fd9\u4e24\u4e2a\u503c\u7684\u4ea4\u96c6\u548c\u5e76\u96c6\u53ef\u4ee5\u4f7f\u7528\u771f\u503c\u8868\u5b9a\u4e49\u5982\u4e0b\uff1a
\cap 0 1
0 0 0
1\uff0e0 1
\cup 0 1
0 0 1
1\uff0e1 1
\u4e5f\u53ef\u4ee5\u5efa\u7acb\u6d89\u53ca\u591a\u4e2a\u8f93\u5165\u548c\u5176\u4ed6\u5e03\u5c14\u8fd0\u7b97\u7684\u66f4\u590d\u6742\u7684\u771f\u503c\u8868\u3002
\u771f\u503c\u8868\u5e94\u7528\u5728\u903b\u8f91\u4e2d\uff0c\u89e3\u91ca 0 \u4e3a\u5047\uff0c1 \u4e3a\u771f\uff0c\cap \u4e3a\u4e0e\uff0c\cup \u4e3a\u6216\uff0c\u800c ¬ \u4e3a\u975e\u3002
\u5176\u4ed6\u8bb0\u53f7
\u53ef\u4ee5\u4f7f\u7528\u5404\u79cd\u6837\u5f0f\u7684\u57fa\u672c\u7b97\u7b26\u6765\u8868\u8fbe\u5e03\u5c14\u903b\u8f91\u3002AND(\u4e0e)\u3001OR(\u6216)\u3001NOT(\u975e)\u662f\u6700\u76f4\u89c9\u7684\u3002\u6570\u5b66\u5bb6\u3001\u5de5\u7a0b\u5e08\u548c\u7a0b\u5e8f\u5458\u7ecf\u5e38\u4f7f\u7528 + \u8868\u793a\u6216\uff0c\cdot \u8868\u793a\u4e0e(\u56e0\u4e3a\u5728\u67d0\u4e9b\u65b9\u9762\u8fd9\u4e9b\u8fd0\u7b97\u7c7b\u4f3c\u4e8e\u5728\u5176\u4ed6\u4ee3\u6570\u7ed3\u6784\u4e2d\u7684\u52a0\u6cd5\u548c\u4e58\u6cd5\uff0c\u5e76\u4e14\u8fd9\u79cd\u8bb0\u53f7\u4f7f\u719f\u6089\u666e\u901a\u4ee3\u6570\u7684\u4eba\u6613\u4e8e\u5f97\u5230\u79ef\u4e4b\u548c\u8303\u5f0f)\u3002\u975e\u4e5f\u8868\u793a\u4e3a\u5728\u8981\u5426\u5b9a\u7684\u8868\u8fbe\u5f0f\u9876\u4e0a\u7684\u4e00\u4e2a\u6a2a\u7ebf\u3002
\u53e6\u4e00\u79cd\u8bb0\u53f7\u4f7f\u7528"\u4ea4"\u8868\u793a\u4e0e\u4f7f\u7528"\u5e76"\u8868\u793a\u6216\u3002\u4f46\u662f\u8fd9\u4f1a\u5bfc\u81f4\u6df7\u6dc6\uff0c\u56e0\u4e3a\u672f\u8bed"\u5e76"\u4e5f\u7ecf\u5e38\u7528\u4e8e\u5408\u5e76\u96c6\u5408\u7684\u53e6\u4e00\u4e2a\u5e03\u5c14\u8fd0\u7b97\uff0c\u5b83\u5305\u62ec\u4e86\u4e0e\u548c\u6216\u4e8c\u8005\u3002
\u5e03\u5c14\u672f\u8bed\u7684\u57fa\u672c\u6570\u5b66\u4f7f\u7528
\u5728\u8054\u7acb\u65b9\u7a0b\u7684\u60c5\u51b5\u4e0b\uff0c\u5b83\u4eec\u662f\u7528\u6697\u542b\u7684\u903b\u8f91\u4e0e\u8fde\u63a5\u7684\uff1a
x + y = 2
AND
x - y = 2
\u540c\u6837\u9002\u7528\u4e8e\u8054\u7acb\u4e0d\u7b49\u5f0f\uff1a
x + y < 2
AND
x - y < 2
\u5927\u4e8e\u7b49\u4e8e\u53f7(\ge)\u548c\u5c0f\u4e8e\u7b49\u4e8e\u53f7(\le)\u53ef\u4ee5\u5047\u5b9a\u5305\u542b\u4e86\u4e00\u4e2a\u903b\u8f91\u6216\uff1a
X < 2
OR
X = 2
\u52a0/\u51cf\u53f7(\pm)\uff0c\u5728\u5e73\u65b9\u6839\u7684\u89e3\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u903b\u8f91\u6216\uff1a
WIDTH = 3
OR
WIDTH = -3
\u5728\u8ba1\u7b97\u673a\u4e2d\u5e03\u5c14\u903b\u8f91\u5b9a\u4e49\u82e5\u5e72\u5e03\u5c14\u903b\u8f91\u51fd\u6570\uff0c\u6709\u65f6\u5019\u79f0\u4e3a\u64cd\u4f5c\u7b26\u3002\u6bcf\u4e2a\u51fd\u6570\u6839\u636e\u4e00\u4e2a\u6216\u8005\u66f4\u591a\u7684\u8f93\u5165\uff0c\u7528\u4e00\u4e2a\u903b\u8f91\u7b97\u6cd5\u6765\u8ba1\u7b97\u8f93\u51fa\u503c\u3002\u8be5\u7b97\u6cd5\u6839\u636e\u8f93\u5165\u6240\u53d6\u771f\u548c\u5047\u7684\u7ec4\u5408\u6765\u51b3\u5b9a\u4ec0\u4e48\u65f6\u5019\u8f93\u51fa\u771f\uff080\u771f1\u5047\uff1b1\u771f0\u5047\u3002\u76f8\u5bf9\u7684\uff09\u3002\u6bcf\u4e2a\u903b\u8f91\u51fd\u6570\u7c7b\u4f3c\u4e8e\u4e00\u4e2a\u73b0\u5b9e\u4e16\u754c\u7684\u903b\u8f91\u8fd0\u7b97\uff0c\u53ef\u4ee5\u7528\u6765\u5b9a\u4e49\u5404\u79cd\u903b\u8f91\u7684\u60c5\u51b5\u3002
1 \u975e\uff08NOT)
\u51fd\u6570\uff1aNOT \u4ec5\u662f\u4e00\u4e2a\u5426\u5b9a\uff1b\u8f93\u51fa\u4e0e\u8f93\u5165\u7684\u76f8\u53cd\u3002\uff08NOT\u51fd\u6570\u4ec5\u6709\u4e00\u4e2a\u8f93\u5165\uff0c\u6545\u79f0\u4e3a\u4e00\u5143\u51fd\u6570\u6216\u8005\u4e00\u5143\u64cd\u4f5c\u7b26\uff09\u3002\u5f53\u8f93\u5165\u4e3a\u5047\uff0c\u8f93\u51fa\u662f\u771f\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002NOT\u51fd\u6570\u903b\u8f91\u4e0a\u8868\u8fbe\u4e00\u4e2a\u6761\u4ef6\u7684\u53cd\u9762\u3002
2 \u4e0e ( AND)
\u51fd\u6570\uff1aAND \u53ef\u4ee5\u6709\u4efb\u610f\u591a\u4e2a\u8f93\u5165\uff0c\u4f46\u6700\u5c11\u662f\u4e24\u4e2a\u3002\u4ec5\u5f53AND\u51fd\u6570\u7684\u7b2c\u4e00\u4e2a\u3001\u7b2c\u4e8c\u4e2a\u548c\u7b2c\u4e09\u4e2a\u8f93\u5165\u7b49\u90fd\u662f\u771f\uff0c\u5b83\u7684\u8f93\u51fa\u624d\u662f\u771f\u3002
3 \u6216 \uff08OR)
\u51fd\u6570\uff1aOR\u53ef\u4ee5\u6709\u4efb\u610f\u591a\u4e2a\u8f93\u5165\uff0c\u4f46\u6700\u5c11\u662f\u4e24\u4e2a\u3002OR\u51fd\u6570\u65e0\u8bba\u4f55\u65f6\u53ea\u8981\u4e00\u4e2a\u8f93\u5165\u4e2d\u51fa\u73b0\u4e86\u771f\uff0c\u8f93\u51fa\u5c31\u662f\u771f\u3002
4 \u5f02\u6216 \uff08XOR)
\u51fd\u6570\uff1aXOR\u662fOR\u7684\u53d8\u4f53\u3002\u4ec5\u5f53\u4e00\u4e2a\u8f93\u5165\u6216\u8005\u53e6\u4e00\u4e2a\u8f93\u5165\u662f\u771f\uff0c\u4f46\u4e0d\u662f\u4e24\u8005\u90fd\u4e3a\u771f\uff08\u65e2\u5982\u679c\u8f93\u5165\u662f\u4e0d\u540c\u7684\uff09\uff0c\u5b83\u7684\u8f93\u51fa\u624d\u4e3a\u771f\u3002
表示方法
"∨" 表示"或"
"∧" 表示"且".
"┐"表示"非".
"=" 表示"等价".
1和0表示"真"和"假"
(还有一种表示,"+"表示"或", "·"表示"与"）
在百度百科，文库里查找一下，应该解释的很详尽。
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