{"id":2115,"date":"2023-04-02T06:56:29","date_gmt":"2023-04-01T22:56:29","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2115"},"modified":"2023-04-05T20:43:21","modified_gmt":"2023-04-05T12:43:21","slug":"fundamentals-of-high-school-mathematics-sets-and-functions","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-high-school-mathematics-sets-and-functions\/","title":{"rendered":"\u9ad8\u4e2d\u6570\u5b66\u57fa\u7840\uff1a\u96c6\u5408\u4e0e\u51fd\u6570"},"content":{"rendered":"

\u96c6\u5408<\/h2>\n

\u7531\u4e00\u4e2a\u6216\u591a\u4e2a\u786e\u5b9a\u7684\u5143\u7d20\u6240\u6784\u6210\u7684\u6574\u4f53\u53eb\u505a\u96c6\u5408\u3002\u82e5$x$\u662f\u96c6\u5408$A$\u7684\u5143\u7d20\uff0c\u5219\u8bb0\u4f5cx \\in A<\/code>\u3002<\/p>\n

\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u6709\u4e09\u4e2a\u7279\u5f81\uff1a
\n1.\u786e\u5b9a\u6027\uff08\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u5fc5\u987b\u662f\u786e\u5b9a\u7684\uff09\u3002
\n2.\u4e92\u5f02\u6027\uff08\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u4e92\u4e0d\u76f8\u540c\uff09\u3002\u4f8b\u5982\uff1a\u96c6\u5408A=\\\\{1\uff0ca\\\\}<\/code>\uff0c\u5219$a$\u4e0d\u80fd\u7b49\u4e8e1\uff09\u3002
\n3.\u65e0\u5e8f\u6027\uff08\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u6ca1\u6709\u5148\u540e\u4e4b\u5206\uff09\uff0c\u5982\u96c6\u5408$\\{3,4,5\\}$\u548c$\\{3,5,4\\}$\u7b97\u4f5c\u540c\u4e00\u4e2a\u96c6\u5408\u3002<\/p>\n

<\/p>\n

\u96c6\u5408\u662f\u6307\u5177\u6709\u67d0\u79cd\u7279\u5b9a\u6027\u8d28\u7684\u5177\u4f53\u7684\u6216\u62bd\u8c61\u7684\u5bf9\u8c61\u6c47\u603b\u6210\u7684\u96c6\u4f53\uff0c\u8fd9\u4e9b\u5bf9\u8c61\u79f0\u4e3a\u8be5\u96c6\u5408\u7684\u5143\u7d20\u3002
\n\u4f8b\u5982\u5168\u4e2d\u56fd\u4eba\u7684\u96c6\u5408\uff0c\u5b83\u7684\u5143\u7d20\u5c31\u662f\u6bcf\u4e00\u4e2a\u4e2d\u56fd\u4eba\u3002\u6211\u4eec\u901a\u5e38\u7528\u5927\u5199\u5b57\u6bcd\u5982A<\/code>,B<\/code>,S<\/code>,T<\/code>,…\u8868\u793a\u96c6\u5408\uff0c\u800c\u7528\u5c0f\u5199\u5b57\u6bcd\u5982a<\/code>,b<\/code>,x<\/code>,y<\/code>,…\u8868\u793a\u96c6\u5408\u7684\u5143\u7d20\u3002
\n\u82e5x\u662f\u96c6\u5408$S$\u7684\u5143\u7d20\uff0c\u5219\u79f0$x$\u5c5e\u4e8e$S$\uff0c\u8bb0\u4e3a$x \\in S$\u3002\u82e5$y$\u4e0d\u662f\u96c6\u5408$S$\u7684\u5143\u7d20\uff0c\u5219\u79f0$y$\u4e0d\u5c5e\u4e8e$S$\uff0c\u8bb0\u4e3a$y \\notin S$\u3002\u4e00\u822c\u7684\u6211\u4eec\u628a\u542b\u6709\u6709\u9650\u4e2a\u5143\u7d20\u7684\u96c6\u5408\u53eb\u505a\u6709\u9650\u96c6<\/strong>\uff0c\u542b\u65e0\u9650\u4e2a\u5143\u7d20\u7684\u96c6\u5408\u53eb\u505a\u65e0\u9650\u96c6<\/strong>\u3002<\/p>\n

\u96c6\u5408\u7684\u8868\u793a\u65b9\u6cd5\uff1a\u5217\u4e3e\u6cd5\u3001\u63cf\u8ff0\u6cd5\u3001\u7b26\u5408\u6cd5<\/p>\n

\u6709\u4e00\u7c7b\u7279\u6b8a\u7684\u96c6\u5408\uff0c\u5b83\u4e0d\u5305\u542b\u4efb\u4f55\u5143\u7d20\uff0c\u5982$ \\{x|x \\in R x^2+1=0 \\} $\uff0c\u6211\u4eec\u79f0\u4e4b\u4e3a\u7a7a\u96c6<\/strong>\uff0c\u8bb0\u4e3a$\\emptyset$
\n\u7a7a\u96c6\u662f\u4e2a\u7279\u6b8a\u7684\u96c6\u5408\uff0c\u5b83\u67092\u4e2a\u7279\u70b9\uff1a
\n\u7a7a\u96c6$\\emptyset$\u662f\u4efb\u610f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\u7684\u771f\u5b50\u96c6<\/strong>\u3002
\n\u7a7a\u96c6\u662f\u4efb\u4f55\u4e00\u4e2a\u96c6\u5408\u7684\u5b50\u96c6<\/strong><\/p>\n

\u5982\u679c\u96c6\u5408$A$\u4e2d\u542b\u6709n\u4e2a\u5143\u7d20\uff0c\u5219\u96c6\u5408$A$\u67092\u7684n\u6b21\u65b9\u4e2a\u5b50\u96c6\uff0c2\u7684n\u6b21\u65b9-1\u4e2a\u771f\u5b50\u96c6<\/p>\n

\u6709\u4e9b\u96c6\u5408\u53ef\u4ee5\u7528\u4e00\u4e9b\u7279\u6b8a\u7b26\u53f7\u8868\u793a\uff0c\u6bd4\u5982\uff1a
\n$N$\uff1a\u975e\u8d1f\u6574\u6570\u96c6\u5408\u6216\u81ea\u7136\u6570\u96c6\u5408$\\{0,1,2,3,\u2026\\}$
\n$N*$\u6216$N+$\uff1a\u6b63\u6574\u6570\u96c6\u5408$\\{1,2,3,\u2026\\}$
\n$Z$\uff1a\u6574\u6570\u96c6\u5408$\\{\u2026,-1,0,1,\u2026\\}$
\n$Q$\uff1a\u6709\u7406\u6570\u96c6\u5408
\n$Q+$\uff1a\u6b63\u6709\u7406\u6570\u96c6\u5408
\n$Q-$\uff1a\u8d1f\u6709\u7406\u6570\u96c6\u5408
\n$R$\uff1a\u5b9e\u6570\u96c6\u5408(\u5305\u62ec\u6709\u7406\u6570\u548c\u65e0\u7406\u6570\uff09
\n$R+$\uff1a\u6b63\u5b9e\u6570\u96c6\u5408
\n$R-$\uff1a\u8d1f\u5b9e\u6570\u96c6\u5408
\n$C$\uff1a\u590d\u6570\u96c6\u5408
\n$\u2205$\uff1a\u7a7a\u96c6\uff08\u4e0d\u542b\u6709\u4efb\u4f55\u5143\u7d20\u7684\u96c6\u5408\uff09<\/p>\n

\u65e0\u7406\u6570<\/strong>\uff0c\u4e5f\u79f0\u4e3a\u65e0\u9650\u4e0d\u5faa\u73af\u5c0f\u6570\uff0c\u4e0d\u80fd\u5199\u4f5c\u4e24\u6574\u6570\u4e4b\u6bd4\u3002\u82e5\u5c06\u5b83\u5199\u6210\u5c0f\u6570\u5f62\u5f0f\uff0c\u5c0f\u6570\u70b9\u4e4b\u540e\u7684\u6570\u5b57\u6709\u65e0\u9650\u591a\u4e2a\uff0c\u5e76\u4e14\u4e0d\u4f1a\u5faa\u73af\u3002 \u5e38\u89c1\u7684\u65e0\u7406\u6570\u6709\u975e\u5b8c\u5168\u5e73\u65b9\u6570\u7684\u5e73\u65b9\u6839\u3001\u03c0\u548ce\uff08\u5176\u4e2d\u540e\u4e24\u8005\u5747\u4e3a\u8d85\u8d8a\u6570\uff09\u7b49<\/p>\n

\u96c6\u5408\u4e4b\u95f4\u7684\u57fa\u672c\u8fd0\u7b97\uff1a
\n\u5e76\u96c6\u5b9a\u4e49\uff1a\u7531\u6240\u6709\u5c5e\u4e8e\u96c6\u5408$A$\u6216\u5c5e\u4e8e\u96c6\u5408$B$\u7684\u5143\u7d20\u6240\u7ec4\u6210\u7684\u96c6\u5408\uff0c\u8bb0\u4f5c$A \\bigcup B$\uff08\u6216$B \\bigcup A$\uff09\uff0c\u8bfb\u4f5c$A$\u5e76$B$\uff08\u6216$B$\u5e76$A$\uff09\uff0c\u5373$A \\bigcup B=\\{x|x \\in A,\u6216x \\in B\\}$\u3002\u5e76\u96c6\u8d8a\u5e76\u8d8a\u591a\u3002
\n\u4ea4\u96c6\u5b9a\u4e49\uff1a\u7531\u5c5e\u4e8e$A$\u4e14\u5c5e\u4e8e$B$\u7684\u76f8\u540c\u5143\u7d20\u7ec4\u6210\u7684\u96c6\u5408\uff0c\u8bb0\u4f5c$A$ \\bigcap $B$\uff08\u6216$B$ \\bigcap $A$\uff09\uff0c\u8bfb\u4f5c$A$\u4ea4$B$\uff08\u6216$B$\u4ea4$A$\uff09\uff0c\u5373$A \\bigcap B=\\{x|x \\in A,\u4e14x \\in B\\}$\u3002\u4ea4\u96c6\u8d8a\u4ea4\u8d8a\u5c11\u3002
\n\u82e5$A$\u5305\u542b$B$($B$\u5305\u542b\u4e8e$A$)\uff0c\u5219$A \\bigcap B=B$\uff0c$A \\bigcup B=A$<\/p>\n

\u8865\u96c6
\n\u76f8\u5bf9\u8865\u96c6\u5b9a\u4e49\uff1a\u7531\u5c5e\u4e8e$A$\u800c\u4e0d\u5c5e\u4e8e$B$\u7684\u5143\u7d20\u7ec4\u6210\u7684\u96c6\u5408\uff0c\u79f0\u4e3a$B$\u5173\u4e8e$A$\u7684\u76f8\u5bf9\u8865\u96c6\uff0c\u8bb0\u4f5c$A-B$\u6216$A \\backslash B$\uff0c\u5373$A-B={x|x \\in A\uff0c\u4e14x \\notin B}$
\n\u7edd\u5bf9\u8865\u96c6\u5b9a\u4e49\uff1a$A$\u5173\u4e8e\u5168\u96c6\u5408$U$\u7684\u76f8\u5bf9\u8865\u96c6\u79f0\u4f5c$A$\u7684\u7edd\u5bf9\u8865\u96c6\uff0c\u8bb0\u4f5c$A’$\u6216$C_uA$\u6216$\\~A$\u3002\u6709$U’=\\emptyset$\uff1b$\\emptyset’=U$<\/p>\n

\u51fd\u6570<\/h2>\n

\u51fd\u6570\u7684\u5b9a\u4e49\uff1a\u8bbe$A$\u3001$B$\u662f\u975e\u7a7a\u7684\u6570\u96c6\uff0c\u5982\u679c\u6309\u7167\u67d0\u79cd\u786e\u5b9a\u7684\u5bf9\u5e94\u5173\u7cfb$f$\uff0c\u4f7f\u5f97\u5bf9\u4e8e\u96c6\u5408$A$\u4e2d\u7684\u4efb\u610f\u4e00\u4e2a\u5143\u7d20$x$\uff0c\u5728\u96c6\u5408$B$\u4e2d\u90fd\u6709\u552f\u4e00\u786e\u5b9a\u7684\u6570$y$\u548c\u5b83\u5bf9\u5e94\uff0c\u90a3\u4e48\u5c31\u79f0\u6620\u5c04$f:A–>B$\u4e3a\u4ece\u96c6\u5408$A$\u5230\u96c6\u5408$B$\u7684\u4e00\u4e2a\u51fd\u6570\uff0c\u8bb0\u4f5c$y=f(x) x\\in A$
\n\u5176\u4e2d$x$\u53eb\u81ea\u53d8\u91cf\uff0c$y$\u53eb\u505a$x$\u7684\u51fd\u6570\uff0c\u96c6\u5408$A$\u662f\u51fd\u6570\u7684\u5b9a\u4e49\u57df\uff0c\u96c6\u5408$B$\u662f\u503c\u57df\uff0c$f$\u53eb\u505a\u5bf9\u5e94\u6cd5\u5219\u3002\u5176\u4e2d\u5b9a\u4e49\u57df\uff0c\u503c\u57df\uff0c\u5bf9\u5e94\u6cd5\u5219\u662f\u51fd\u6570\u7684\u4e09\u8981\u7d20\u3002<\/p>\n

\u51fd\u6570\u7684\u4e09\u79cd\u8868\u793a\u65b9\u6cd5\uff1a\u89e3\u6790\u6cd5 \u56fe\u50cf\u6cd5 \u5217\u8868\u6cd5<\/p>\n

\u51fd\u6570\u7684\u7279\u6027<\/p>\n

\uff081\uff09\u6709\u754c\u6027<\/p>\n

\u8bbe\u51fd\u6570$f(x)$\u5728\u533a\u95f4$X$\u4e0a\u6709\u5b9a\u4e49\uff0c\u5982\u679c\u5b58\u5728$M > 0$\uff0c\u5bf9\u4e8e\u4e00\u5207\u5c5e\u4e8e\u533a\u95f4$X$\u4e0a\u7684$x$\uff0c\u6052\u6709 $|f(x)| \u2264 M$\uff0c\u5219\u79f0$f(x)$\u5728\u533a\u95f4$X$\u4e0a\u6709\u754c\uff0c\u5426\u5219\u79f0$f(x)$\u5728\u533a\u95f4\u4e0a\u65e0\u754c<\/p>\n

\uff082\uff09\u5355\u8c03\u6027<\/p>\n

\u8bbe\u51fd\u6570$f(x)$\u7684\u5b9a\u4e49\u57df\u4e3a$D$\uff0c\u533a\u95f4$I$\u5305\u542b\u4e8e$D$\u3002\u5982\u679c\u5bf9\u4e8e\u533a\u95f4\u4e0a\u4efb\u610f\u4e24\u70b9$x_1$\u53ca$x_2$\uff0c\u5f53$x_1 < x_2$\u65f6\uff0c\u6052\u6709$f(x_1) < f(x_2)$\uff0c\u5219\u79f0\u51fd\u6570$f(x)$
\n\u5728\u533a\u95f4$I$\u4e0a\u662f\u5355\u8c03\u9012\u589e\u7684\uff1b\u5982\u679c\u5bf9\u4e8e\u533a\u95f4$I$\u4e0a\u4efb\u610f\u4e24\u70b9$x_1$\u53ca$x_2$\uff0c\u5f53$x_1 < x_2$\u65f6\uff0c\u6052\u6709$f(x_1) > f(x_2)$\uff0c\u5219\u79f0\u51fd\u6570$f(x)$
\n\u5728\u533a\u95f4$I$\u4e0a\u662f\u5355\u8c03\u9012\u51cf\u7684\u3002\u5355\u8c03\u9012\u589e\u548c\u5355\u8c03\u9012\u51cf\u7684\u51fd\u6570\u7edf\u79f0\u4e3a\u5355\u8c03\u51fd\u6570\u3002<\/p>\n

\uff083\uff09\u5947\u5076\u6027<\/p>\n

\u8bbe$f(x)$\u4e3a\u4e00\u4e2a\u5b9e\u53d8\u91cf\u5b9e\u503c\u51fd\u6570\uff0c\u82e5\u6709$f(-x) = – f(x)$\uff0c\u5219$f(x)$\u4e3a\u5947\u51fd\u6570\u3002
\n\u51e0\u4f55\u4e0a\uff0c\u4e00\u4e2a\u5947\u51fd\u6570\u5173\u4e8e\u539f\u70b9\u5bf9\u79f0\uff0c\u4ea6\u5373\u5176\u56fe\u50cf\u5728\u7ed5\u539f\u70b9\u505a180\u5ea6\u65cb\u8f6c\u540e\u4e0d\u4f1a\u6539\u53d8\u3002
\n\u8bbe$f(x)$\u4e3a\u4e00\u5b9e\u53d8\u91cf\u5b9e\u503c\u51fd\u6570\uff0c\u82e5\u6709$f(-x) = f(x)$\uff0c\u5219$f(x)$\u4e3a\u5076\u51fd\u6570\u3002
\n\u51e0\u4f55\u4e0a\uff0c\u4e00\u4e2a\u5076\u51fd\u6570\u5173\u4e8ey\u8f74\u5bf9\u79f0\uff0c\u4ea6\u5373\u5176\u56fe\u5728\u5bf9y\u8f74\u6620\u5c04\u540e\u4e0d\u4f1a\u6539\u53d8\u3002<\/p>\n

\u51fd\u6570\u7684\u5bfc\u6570\uff1a\u5982\u679c\u51fd\u6570$f(x)$\u5728$(a, b)$\u4e2d\u6bcf\u4e00\u70b9\u5904\u90fd\u53ef\u5bfc\uff0c\u5219\u79f0$f(x)$\u5728$(a, b)$\u4e0a\u53ef\u5bfc\uff0c\u5219\u53ef\u5efa\u7acb$f(x)$\u7684\u5bfc\u51fd\u6570\uff0c\u7b80\u79f0\u5bfc\u6570\uff0c\u8bb0\u4e3a$f'(x)$<\/p>\n

\u51fd\u6570$y=f(x)$\u5728\u70b9$x_1$\u5904\u7684\u5bfc\u6570\u7684\u51e0\u4f55\u610f\u4e49\uff1a
\n\u51fd\u6570$y=f(x)$\u5728\u70b9$x_1$\u5904\u7684\u5bfc\u6570\u662f\u66f2\u7ebf$y=f(x)$\u5728$P(x_1, f(x_1))$\u5904\u7684\u5207\u7ebf\u7684\u659c\u7387$f'(x_1)$
\n\u76f8\u5e94\u7684\u5207\u7ebf\u65b9\u7a0b\u662f$y-y_1 = f'(x_1)(x – x_1)$<\/p>\n

\u590d\u5408\u51fd\u6570\uff1a\u51fd\u6570\u7684\u5d4c\u5957 $y=f(t)$ $t=g(x)$\uff0c\u5373$y=f(g(x))$<\/p>\n

\u5e38\u51fd\u6570\uff1a$y = C$ ($C$\u662f\u5e38\u6570)
\n\u4e00\u6b21\u51fd\u6570\uff1a$y = kx + b$ ($k$\u4e3a\u4e00\u6b21\u9879\u7cfb\u6570\uff0c$b$\u4e3a\u5e38\u6570)
\n\u4e8c\u6b21\u51fd\u6570\uff1a$y = ax_2 + bx +c \\, (a!=0)$<\/p>\n

\u4e8c\u6b21\u51fd\u6570\u662f\u629b\u7269\u7ebf\uff0c\u4f46\u629b\u7269\u7ebf\u4e0d\u4e00\u5b9a\u662f\u4e8c\u6b21\u51fd\u6570\u3002\u5f00\u53e3\u5411\u4e0a\u6216\u8005\u5411\u4e0b\u7684\u629b\u7269\u7ebf\u624d\u662f\u4e8c\u6b21\u51fd\u6570\u3002\u629b\u7269\u7ebf\u662f\u8f74\u5bf9\u79f0\u56fe\u5f62\u3002\u5bf9\u79f0\u8f74\u4e3a\u76f4\u7ebf$x=-b\/2a$\u3002\u5bf9\u79f0\u8f74\u4e0e\u629b\u7269\u7ebf\u552f\u4e00\u7684\u4ea4\u70b9\u4e3a\u629b\u7269\u7ebf\u7684\u9876\u70b9$P$\u3002\u7279\u522b\u5730\uff0c\u5f53$b=0$\u65f6\uff0c\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74\u662f$y$\u8f74\uff08\u5373\u76f4\u7ebf$x=0$\uff09<\/p>\n","protected":false},"excerpt":{"rendered":"

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