From 749634f41c612b4fa150a86df7df8fcde2181a68 Mon Sep 17 00:00:00 2001 From: Trinkle23897 <463003665@qq.com> Date: Sun, 2 Feb 2020 22:54:53 +0800 Subject: [PATCH] badge and anonym material --- README.md | 2 +- ...15\344\271\240\346\214\207\345\215\227.md" | 257 +++++++ ...46\344\271\240\345\217\202\350\200\203.md" | 327 +++++++++ .../hw/2019/pdf/w1.pdf" | Bin 0 -> 469657 bytes .../hw/2019/pdf/w10.pdf" | Bin 0 -> 321942 bytes .../hw/2019/pdf/w11.pdf" | Bin 0 -> 505626 bytes .../hw/2019/pdf/w12.pdf" | Bin 0 -> 529330 bytes .../hw/2019/pdf/w13.pdf" | Bin 0 -> 373019 bytes .../hw/2019/pdf/w14.pdf" | Bin 0 -> 349365 bytes .../hw/2019/pdf/w2.pdf" | Bin 0 -> 318551 bytes .../hw/2019/pdf/w3.pdf" | Bin 0 -> 379055 bytes .../hw/2019/pdf/w4.pdf" | Bin 0 -> 430471 bytes .../hw/2019/pdf/w5.pdf" | Bin 0 -> 397908 bytes .../hw/2019/pdf/w6.pdf" | Bin 0 -> 317772 bytes .../hw/2019/pdf/w7.pdf" | Bin 0 -> 344850 bytes .../hw/2019/pdf/w8.pdf" | Bin 0 -> 416588 bytes .../hw/2019/pdf/w9.pdf" | Bin 0 -> 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mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p1-9-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p1-9-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p1-9-d.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-1-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-1-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-1-c.png" create mode 100644 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"\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-d-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-d-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-20-d.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-3-a-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-3-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-3-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-3-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-3-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-4-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-4-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-4-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-4-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-4-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-a-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-d-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h1/p2-6-d.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/h10.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-32-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-33-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-34-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-44.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-47.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-51.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-52.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-54.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-55.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-56.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p8-61.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p9-70.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p9-71-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h10/p9-71-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/h11.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-12-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-a-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-a-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-b-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-b-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-c-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-38-c-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-4-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-4-d-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-42-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-a-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-a-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-b-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-b-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-c-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-46-c-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-50-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-50-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-6-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h11/p11-62.png" create mode 100644 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"\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-4-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-4-d.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-77.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-78.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-80.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h12/p12-87.png" create mode 100644 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100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p3-29.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p3-36-1-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p3-36-2-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p3-5.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p3-9.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h2/p4-10.png" create mode 100644 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create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h3/p4-15.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h3/p4-25-sol1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h3/p4-25-sol2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h3/p4-25.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h3/p4-45.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/h4.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-2-3-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-20.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-23.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-28.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-30.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-32-sol.png" create mode 100644 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"\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-5-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-5-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-5-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-5-d-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-5-d.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-51-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-51-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-53.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-8-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-8-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h4/p11-8-c.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/h5.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p11-40.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p11-42.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p11-46.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p11-47.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p7-1-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p7-1-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p7-2-a-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p7-2-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h5/p7-2-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/h6.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-10-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-10.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-11-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-11.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-23.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-6-a-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-6-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-6-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h6/p7-6-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f3-63.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f3-72.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f6-37.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f6-43.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f6-44.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/f6-45.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h7/h7.tex" create mode 100644 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create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-38-b-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-38-c-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-38-d-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-38-e-sol.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-38.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h8/p6-46-sol.png" create mode 100644 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100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h9/h9.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h9/p10-75.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/h9/p10-89.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/A-1-a.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/A-1-b.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/A-4.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-1-a-sim.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-1-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-1-b-sim.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-1-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-4-res.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-4-sim.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/pA-4.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s1/s1.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/cir1-1-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/cir1-2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/cir2-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/cir2-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d1.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d1.ms14 (Security copy)" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d2.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d2.ms14 (Security copy)" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d3.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d3.ms14 (Security copy)" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d4.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/d4.ms14 (Security copy)" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/g1-1.gra" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/g1-2.gra" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/g2-1.gra" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/g2-2.gra" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res1-1-a-1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res1-1-a-2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res1-2-1.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res1-2-2.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res2-a.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/res2-b.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s2/s2.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s3/cir.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s3/design.ms14" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s3/res.png" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/tex/s3/s3.tex" create mode 100644 "\345\244\247\344\272\214\344\270\212/\347\224\265\345\255\220\345\255\246\345\237\272\347\241\200/hw/2019/\344\275\234\344\270\232\345\213\230\350\257\257.txt" diff --git a/README.md b/README.md index 4ab439423..410ffb95f 100644 --- a/README.md +++ b/README.md @@ -6,7 +6,7 @@ [![](https://img.shields.io/github/issues-pr-closed-raw/PKUanonym/REKCARC-TSC-UHT.svg?style=flat)](https://github.com/PKUanonym/REKCARC-TSC-UHT/issues) ![](https://img.shields.io/github/repo-size/PKUanonym/REKCARC-TSC-UHT.svg?style=flat) -Web page: [REKCARC-TSC-UHT.readthedocs.io](https://REKCARC-TSC-UHT.readthedocs.io) +Web page [![Documentation Status](https://readthedocs.org/projects/rekcarc-tsc-uht/badge/?version=latest)](https://rekcarc-tsc-uht.readthedocs.io/en/latest/?badge=latest): [REKCARC-TSC-UHT.readthedocs.io](https://REKCARC-TSC-UHT.readthedocs.io) 受[浙江大学相关项目](https://github.com/QSCTech/zju-icicles)启发,创立了本项目。 diff --git "a/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\244\215\344\271\240\346\214\207\345\215\227.md" "b/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\244\215\344\271\240\346\214\207\345\215\227.md" new file mode 100644 index 000000000..4b248015a --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\244\215\344\271\240\346\214\207\345\215\227.md" @@ -0,0 +1,257 @@ +# 复习指南 + +## 单选 + +1. “社会主义核心价值体系”这一重大命题和战略任务在( )被首次明确提出。 (1分) + - [ ] 十六届四中全会 + - [x] 十六届六中全会 + - [ ] 十八大 + - [ ] 十八届三中全会 +2. 下列关于理想的表述不正确的是( ) (1分) + - [ ] 理想是人们的世界观、人生观和价值观在奋斗目标上的集中体现 + - [ ] 理想会随着时代的发展而发展 + - [x] 理想是人们对于未来的想象,与现实关系不大 + - [ ] 理想在实践中产生,在实践中发展,在实践中实现 +3. 社会主义核心价值观在( )被首次提出 (1分) + - [ ] 中国共产党第十六次全国代表大会 + - [ ] 中国共产党第十七次全国代表大会 + - [x] 中国共产党第十八次全国代表大会 + - [ ] 中国共产党第十九次全国代表大会 +4. 马克思主义指导思想在社会主义核心价值体系中发挥着( )作用。 (1分) + - [x] 灵魂 + - [ ] 主题 + - [ ] 精髓 + - [ ] 基础 +5. 在对待传统道德的问题上,要反对( )两种错误思潮。 (1分) + - [ ] 全盘肯定;全部继承 + - [ ] 全盘否定;全盘西化 + - [ ] 复古论;全盘西化 + - [x] 复古论;虚无论 +6. 中国传统文化十分强调道德修养和道德教化,将“( )”置于“三不朽”(立德、立功、立言)之首,重视人的精神品格的养成。 (1分) + - [x] 立德 + - [ ] 立言 + - [ ] 立功 + - [ ] 立人 +7. ( )是中国特色社会主义最本质的特征。 (1分) + - [x] 中国共产党的领导 + - [ ] 社会主义 + - [ ] 共同富裕 + - [ ] 人民当家作主 +8. ( )是人的本质属性? (1分) + - [ ] 自然属性 + - [ ] 生活属性 + - [x] 社会属性 + - [ ] 劳动属性 +9. 社会主义道德的原则是( ) (1分) + - [ ] 爱国主义 + - [x] 集体主义 + - [ ] 为人民服务 + - [ ] 国际主义 +10. 除“文明礼貌、助人为乐、爱护公物、保护环境”外,社会公德还有一个主要内容是( ) (1分) + - [x] 遵纪守法 + - [ ] 诚实守信 + - [ ] 公平正义 + - [ ] 和谐相处 +11. ( )是道德起源的首要前提。 (1分) + - [x] 劳动 + - [ ] 自我意识 + - [ ] 社会关系 + - [ ] 家庭 +12. ( )在我国思想道德建设中具有特殊重要的作用,它既是中华民族的传统美德,也是我国公民道德建设的重点,还是社会主义核心价值观的一条重要准则。 (1分) + - [ ] 办事公道 + - [x] 诚实守信 + - [ ] 服务群众 + - [ ] 奉献社会 +13. 近年来,一些打着所谓“还原”历史真相的观点借助网络平台大肆喧嚣,民族英雄被诋毁为人为制造和杜撰的产物,而民族败类则被篡改为真正的爱国者。上述种种“翻案风”从本质上来讲反映了哪种错误的思想倾向( ) (1分) + - [x] 历史虚无主义 + - [ ] 民族主义 + - [ ] 西方文明至上论 + - [ ] 复古主义 +14. 马克思主义伦理学对道德起源的科学阐释,不包括以下哪方面( ) (1分) + - [x] 习俗 + - [ ] 劳动 + - [ ] 社会关系 + - [ ] 自我意识 +15. 实现中华民族伟大复兴的中国梦,必须弘扬中国精神,这就是以( )为核心的民族精神和以( )为核心的时代精神。 (1分) + - [x] 爱国主义;改革创新 + - [ ] 爱国主义;改革开放 + - [ ] 集体主义;改革创新 + - [ ] 集体主义;改革开放 +16. 下面不属于我国宪法的基本原则是( ) (1分) + - [ ] 党的领导原则 + - [x] 人民民主专政原则 + - [ ] 人权保障原则 + - [ ] 民主集中制原则 +17. ( )是调整平等主体的自然人、法人和非法人组织之间的人身关系和财产关系的法律规范。 (1分) + - [ ] 刑法 + - [ ] 行政法 + - [ ] 经济法 + - [x] 民法 +18. 法律出现的根本原因是( ) (1分) + - [ ] 习惯法的积累 + - [ ] 法律意识的逐渐觉醒 + - [x] 社会生产力的发展 + - [ ] 氏族领袖的推动 +19. 以下哪个判断是错误的( ) (1分) + - [ ] 党的领导是人民当家作主和依法治国的根本保证 + - [ ] 人民当家作主是党的领导和依法治国的本质要求 + - [x] 坚持党的领导就意味着党比法大 + - [ ] 依法治国是党领导人民治理国家的基本方略 +20. 中国传统伦理道德注重人和人之间的和谐交往,团结互助,是从( )精神出发的。 (1分) + - [ ] 孝悌 + - [x] 仁爱 + - [ ] 爱国 + - [ ] 勤劳 +21. 道德本质上是( ) (1分) + - [ ] 一种社会实践行为 + - [ ] 一种社会心理活动 + - [x] 一种社会意识形式 + - [ ] 一种社会评价标准 +22. 下列关于法律义务特点的表述中,不正确的是( ) (1分) + - [ ] 法律义务是历史的 + - [ ] 法律义务源于现实需要 + - [ ] 法律义务必须依法设定 + - [x] 法律义务是稳定的 +23. 新时期对于爱国主义理解不正确的是( ) (1分) + - [ ] 爱国主义与爱社会主义具有一致性 + - [x] 民族主义就是爱国主义 + - [ ] 爱国主义与拥护祖国统一具有一致性 + - [ ] 爱国主义应同时包含国际观中国情民族魂 +24. 公民社会经济权利的核心是( ) (1分) + - [ ] 劳动权 + - [ ] 休息权 + - [ ] 社会保障权 + - [x] 生存权 + +## 多选 + +25. 下列关于法律权利与法律义务相互关系的表述中,正确的有( ) (1分) + - [x] 法律权利的实现必须以相应法律义务的履行为条件 + - [x] 有些法律权利和法律义务具有复合性 + - [x] 法律权利与法律义务是目的与手段的关系 + - [x] 法律权利和法律义务平等 +26. 下列关于爱国主义的说法,正确的是( ) (1分) + - [x] 要始终坚持爱国主义与爱社会主义相统一 + - [x] 爱国主义不是抽象的,而是具体的 + - [ ] 爱国是一种情怀,可以不体现在行动上 + - [x] 理性爱国要警惕极端狭隘的民族主义倾向 +27. 马克思曾指出:“蜘蛛的活动与织工的活动相似,蜜蜂建筑蜂房的本领使人间的许多建筑师感到惭愧。但是,最蹩脚的建筑师从一开始就比最灵巧的蜜蜂高明的地方,是他在用蜂蜡建筑蜂房以前,已经在自己的头脑中把它建成了。”这句话说明( ) (1分) + - [x] 人的实践活动是有意识的 + - [ ] 人的本质在其现实性上,它是一切社会关系的总和 + - [x] 人能够对自己的存在和活动的内容、方式有所“观”,并作出选择和采取行动 + - [ ] 人类和自然界的其他生物一样,都有生存的需要 +28. 我国的实体法律部门包括( ) (1分) + - [x] 宪法相关法 + - [ ] 宪法 + - [x] 行政法 + - [x] 经济法 +29. 下列关于爱国主义时代价值的表述正确的是( ) (1分) + - [x] 爱国主义是中华民族继往开来的精神支柱 + - [x] 爱国主义是维护祖国统一和民族团结的纽带 + - [x] 爱国主义是实现中华民族伟大复兴的动力 + - [x] 爱国主义是个人实现人生价值的力量源泉 +30. 马克思主义信仰与宗教信仰的区别是( ) (1分) + - [x] 二者的世界观基础不同 + - [x] 二者对待科学的态度不同 + - [x] 二者对待自身理论、观点的态度不同 + - [x] 二者对待人们自身命运的态度不同 +31. 马克思主义信仰的内涵包括哪几方面( ) (1分) + - [x] 唯物主义的世界图景 + - [x] 共产主义的远大理想 + - [x] 为人民服务的根本宗旨 + - [x] 自由而全面的人生追求 +32. 宪法实施的基本要求包括( ) (1分) + - [ ] 巩固宪法地位 + - [x] 在全社会树立宪法意识,弘扬宪法精神 + - [x] 加强宪法实施 + - [x] 坚持依宪执政,自觉在宪法法律范围内活动 +33. 培育和践行社会主义核心价值观,要( ) (1分) + - [x] 以中华民族优秀传统文化为底蕴 + - [x] 立足中国特色社会主义的奋斗实践 + - [x] 以当代中国面临的时代问题为指向 + - [x] 勤学修德明辨笃行,做社会主义核心价值观的积极践行者 +34. 文明是社会进步的重要标志,也是社会主义现代化国家的重要特征。我国追求的文明价值目标是对面向现代化、面向世界、面向未来的,( )的社会主义文化的概括。 (1分) + - [x] 民族的 + - [x] 科学的 + - [x] 大众的 + - [ ] 世界的 +35. 省察克治体现了道德修养的( ) (1分) + - [x] 自觉性 + - [ ] 外在约束 + - [x] 主观能动性 + - [ ] 知行合一 +36. 大学生要树立马克思主义的科学信仰,是因为( ) (1分) + - [x] 马克思主义具有科学性 + - [x] 马克思主义具有持久生命力 + - [x] 马克思主义不仅致力于科学解释世界,而且致力于积极改变世界 + - [x] 马克思主义具有革命性 +37. 关于集体主义,以下论述正确的是( ) (1分) + - [x] 集体主义是我国调节国家利益、社会整体利益和个人利益关系的基本原则 + - [x] 只有个人的正当利益得到保证,集体才能有更强的凝聚力 + - [ ] 集体主义压制了个性,不利于个人的自由发展 + - [x] 集体主义坚持国家利益、社会整体利益高于个人利益的原则 +38. 下列选项中,( )属于宪法所规定的公民的文化权利。 (1分) + - [x] 科学研究自由 + - [ ] 出版自由 + - [x] 文艺创作自由 + - [ ] 结社自由 +39. 和谐蕴含了中国人的生存智慧,体现着中国人的精神基因,也昭示着中国人的社会理想。和谐的中国是( )的社会主义国家。 (1分) + - [x] 民主与法治相统一 + - [x] 公平与效率相统一 + - [x] 活力与秩序相统一 + - [x] 人与自然相统一 +40. 下面哪些属于法治思维的基本内容( ) (1分) + - [x] 法律至上 + - [x] 权力制约 + - [x] 公平正义 + - [x] 人权保障 +41. 信念是人的( )的有机统一体 (1分) + - [ ] 兴趣 + - [x] 认知 + - [x] 情感 + - [x] 意志 +42. 中华民族在历史发展中逐步形成了优良的爱国传统。下面哪些名言体现了深厚爱国情怀( ) (1分) + - [x] 苟利国家生死以,岂因祸福避趋之 + - [x] 位卑未敢忘忧国 + - [x] 匈奴未灭何以家为 + - [x] 一年三百六十日,多是横戈马上行 +43. 个人品德是通过社会道德教育和个人自觉的道德修养所形成的稳定的心理状态和行为习惯。它是个体对某种道德要求认同和践履的结果,集中体现了道德认知、( )的内在统一。 (1分) + - [x] 道德情感 + - [x] 道德意志 + - [x] 道德信念 + - [ ] 道德认同 +44. 在经济全球化形势下,下列表述正确的是( ) (1分) + - [ ] 经济全球化时代,爱国主义已经过时了 + - [x] 经济全球化时代民族国家的界限和利益差别仍然存在 + - [ ] 要维护国家民族的利益必须抗拒全球化的趋势 + - [x] 经济全球化不等于全球政治、文化一体化 +45. 我国选举法规定的选举原则和制度主要包括( ) (1分) + - [x] 平等选举原则 + - [x] 直接选举和间接选举相结合的原则 + - [x] 秘密投票原则 + - [x] 自由投票原则 +46. 以下( )属于革命道德的精神范畴? (1分) + - [x] 红船精神 + - [x] 井冈山精神 + - [x] 苏区精神 + - [x] 长征精神 +47. 人生观主要从( )方面体现出来。 (1分) + - [ ] 人生感悟 + - [x] 人生目的 + - [x] 人生态度 + - [x] 人生价值 +48. 爱国主义的基本内涵包括( ) (1分) + - [x] 爱祖国的大好河山 + - [x] 爱自己的骨肉同胞 + - [x] 爱祖国的灿烂文化 + - [x] 爱自己的国家 +49. 社会主义核心价值体系主要包括( ) (1分) + - [x] 马克思主义指导思想 + - [x] 中国特色社会主义共同理想 + - [x] 以爱国主义为核心的民族精神和以改革创新为核心的时代精神 + - [x] 社会主义荣辱观 +50. 下列选项中,对于道德的理解,正确的有( ) (1分) + - [x] 道德通过社会舆论传统习俗和人们的内心信念来维系 + - [x] 道德属于上层建筑的范畴,是一种特殊的社会意识形态 + - [ ] 道德是阶级社会特有的社会意识形态,原始社会没有道德 + - [x] 道德是对人们的行为进行善恶评价的心理意识原则规范和行为活动的总和 \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\255\246\344\271\240\345\217\202\350\200\203.md" "b/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\255\246\344\271\240\345\217\202\350\200\203.md" new file mode 100644 index 000000000..c63cd3ef3 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\346\200\235\346\203\263\351\201\223\345\276\267\344\277\256\345\205\273\344\270\216\346\263\225\345\276\213\345\237\272\347\241\200/hw/mooc/\345\255\246\344\271\240\345\217\202\350\200\203.md" @@ -0,0 +1,327 @@ +# 思想道德修养与法律基础慕课学习参考 + + - [第一章 人生的青春之问](#第一章-人生的青春之问) + - [第二章 坚定理想信念](#第二章-坚定理想信念) + - [第三章 弘扬中国精神](#第三章-弘扬中国精神) + - [第四章 践行社会主义核心价值观](#第四章-践行社会主义核心价值观) + - [第五章 明大德守公德严私德](#第五章-明大德守公德严私德) + - [第六章 尊法学法守法用法](#第六章-尊法学法守法用法) + + ## 第一章 人生的青春之问 + + 1. 人生价值的最基本内容是什么? + - [ ] 个人的社会存在 + - [ ] 个人对自身物质和精神满足的需要 + - [x] 人的社会价值 + - [ ] 社会对个人的尊重和满足 + +2. 在马克思主义看来,人的本质在其现实性上是什么? + - [ ] 有生命的自然存在 + - [ ] 饥食渴饮,衣食住行 + - [x] 一切社会关系的总和 + - [ ] 单个人所固有的抽象物 + +3. 以下关于人生目的和意义的说法,哪些是正确的? + - [x] 人生目的和意义无时不在支配着我们的言行 + - [ ] 遭受苦难是寻找人生意义的必要方式 + - [ ] 人生价值是向社会或他人索取 + - [x] 高尚的人生目的是在服务人民和奉献社会的实践中实现自己有意义的人生 + +4. 下面关于人生价值评价标准的观点哪一个是**错误**的? + - [ ] 坚持能力有大小与贡献须尽力相统一 + - [ ] 坚持完善自身与贡献社会相统一 + - [x] 坚持否定自我价值与肯定社会价值相统一 + - [ ] 坚持物质贡献与精神贡献相统一 + +5. 人生观主要由________体现出来。 + - [ ] 人生价值 人生经验 人生志向 + - [ ] 人生志趣 人生哲学 人生价值 + - [x] 人生目的 人生态度 人生价值 + - [ ] 人生目的 人生志向 人生经验 + + + ## 第二章 坚定理想信念 + + 1. 理想作为一种精神现象,是人类社会实践的产物,以下关于理想的表述**不正确**的是 + - [ ] 理想是人们的世界观、人生观和价值观在奋斗目标上的集中体现 + - [ ] 理想受到社会关系影响,会随着时代的发展而变化 + - [x] 理想是人们对于未来的想象,与现实关系不大 + - [ ] 理想是多方面和多类型的,有科学理想和非科学理想之分 + +2. 下列关于理想与信念的说法,正确的是 + - [x] 理想产生于实践,又指引着实践 + - [x] 理想与信念都是人类特有的精神现象 + - [ ] 人只要有对未来的向往就可以,信念无关紧要 + - [x] 理想是信念所指的对象,信念是理想实现的保障 + +3. 如何看待个人理想与社会理想的关系? + - [x] 社会理想与个人理想相互联系,相互影响,相互制约 + - [x] 个人理想以社会理想为指引 + - [x] 社会理想归根到底到底要靠全体社会成员的共同努力来实现 + - [ ] 当社会理想与个人理想有冲突的时候,要坚持自己的个人理想不动摇 + +4. 马克思主义信仰与宗教信仰是完全不同的信仰体系,以下关于二者的区别的描述恰当的有哪些? + - [x] 两者的世界观基础是不一样的 + - [x] 两者对待科学的态度不同 + - [x] 两者对待自身理论、观点的态度不同 + - [x] 两者对待人们自身命运的态度也是不一样的 + +5. 辩证唯物主义和历史唯物主义是马克思主义信仰的 + - [ ] 价值目标 + - [x] 世界观基础 + - [ ] 科学态度 + - [ ] 具体体现 + +6. 马克思主义信仰的内涵包括哪几方面? + - [x] 唯物主义的世界图景 + - [x] 共产主义的远大理想 + - [x] 为人民服务的根本宗旨 + - [ ] 个人至上的价值倾向 + - [x] 自由而全面的人生追求 + +7. 为什么说马克思主义是科学的信仰? +- [x] 坚持科学的世界观和方法论 +- [x] 坚持彻底的唯物主义 +- [x] 坚持无产阶级的党性原则 +- [x] 坚持全心全意为人民服务的精神 +- [ ] 坚持以变革世界为己任 + + +## 第三章 弘扬中国精神 + +1. 实现中国梦,必须弘扬中国精神。下面关于中国精神基本内容的说法正确的是 + - [ ] 以爱国主义为核心的民族精神和以实事求是为核心的时代精神 + - [ ] 以团结统一为核心的民族精神和以改革创新为核心的时代精神 + - [x] 以爱国主义为核心的民族精神和以改革创新为核心的时代精神 + - [ ] 以团结统一为核心的民族精神和以实事求是为核心的时代精神 + +2. 爱国主义的科学内涵包括 + - [x] 爱祖国的大好河山 + - [x] 爱自己的骨肉同胞 + - [x] 爱祖国的灿烂文化 + - [x] 爱自己的国家 + +3. 下列关于新时代的爱国主义基本要求**不正确**的是 + - [ ] 坚持爱国主义和社会主义相统一 + - [ ] 维护祖国统一和民族团结 + - [x] 摒弃古老的传统文化赞颂世界流行文化 + - [ ] 坚持立足民族又面向世界 + +4. 爱国既需要情感的基础,也需要理性的认识,更需要实际的行动。做忠诚的爱国者,应该做到以下哪些方面? + - [x] 维护和推进祖国统一 + - [x] 促进民族团结 + - [x] 增强国家安全意识 + - [ ] 在网上抹黑其他国家 + +5. 当代大学生爱国主义有哪些新特点? + - [x] 细水长流式的爱国主义 + - [x] 爱国激情与理性并存 + - [x] 爱国主义与现实压力的博弈 + - [ ] 经济全球化时代国家不再存在,当代大学生不需要爱国主义了 + +6. 下列关于民族精神和时代精神表述正确的是 + - [x] 都是一个民族赖以生存和发展的精神支撑 + - [x] 民族精神是时代精神的重要基础和依托 + - [x] 时代精神引领民族精神的发展方向 + - [ ] 时代精神终将取代民族精神 + +7. 爱国不仅体现为对祖国的深厚情感,更体现为现实的责任和义务。做忠诚的爱国者,要做到以下哪些? + - [x] 自觉维护国家利益 + - [x] 促进民族团结 + - [x] 维护祖国统一 + - [x] 增强国防观念和国家安全意识 + +8. 以改革创新为核心的时代精神体现在 + - [x] 求真务实、坚韧不拔、锐意进取的精神状态 + - [x] 甘于奉献、奋勇争先、追求进步的责任使命 + - [x] 解放思想、突破陈规、敢于创造的思想观念 + - [ ] 团结统一、爱好和平、勤劳勇敢的民族性格 + + +## 第四章 践行社会主义核心价值观 + +1. 党的十八大提出,要倡导________,倡导________,倡导________,积极培育和践行社会主义核心价值观。 + - [ ] 富强、民主、文明、和谐、自由、平等、正义、法治、爱国、敬业、诚信、友爱 + - [ ] 富强、民主、文明、和平、自由、平等、公正、法治、爱国、敬业、诚信、友善 + - [ ] 富强、民主、文明、和谐、自由、平等、公道、法制、爱国、敬业、诚信、友善 + - [x] 富强、民主、文明、和谐、自由、平等、公正、法治、爱国、敬业、诚信、友善 + +2. 社会主义核心价值体系主要包括 + - [x] 马克思主义指导思想 + - [x] 中国特色社会主义共同理想 + - [x] 以爱国主义为核心的民族精神和以改革创新为核心的时代精神 + - [x] 社会主义荣辱观 + +3. 下列关于社会主义核心价值体系与社会主义核心价值观的关系的说法正确的是 + - [x] 社会主义核心价值观是社会主义核心价值体系的精神内核 + - [x] 社会主义核心价值观体现着社会主义核心价值体系的根本性质和基本特征 + - [x] 社会主义核心价值观是社会主义核心价值体系的高度凝练和集中表达 + - [x] 社会主义核心价值观反映着社会主义核心价值体系的丰富内涵和实践要求 + +4. “富强、民主、文明、和谐”的价值追求回答了我们要建设什么样的国家的重大问题,揭示了当代中国在经济发展、政治文明、文化繁荣、社会进步等方面的价值目标,从________层面标注了社会主义核心价值观的时代刻度。 + - [ ] 社会 + - [ ] 集体 + - [x] 国家 + - [ ] 公民 + +5. 把握网络文化建设主动权,唱响网络空间主旋律,要将________作为宣传思想工作的重中之重。 + - [ ] 新闻舆论工作 + - [ ] 高校思想政治工作 + - [x] 网上舆论工作 + - [ ] 文艺工作 + +6. 当前,软实力已经越来越成为国家间竞争的制高点之一。“软实力”一词是由哈佛大学教授________首次明确提出并详尽阐述的。 + - [ ] 尼古拉斯·欧维纳 + - [x] 约瑟夫·奈 + - [ ] 保罗·肯尼迪 + - [ ] 威廉·欧文斯 + +7. 在《清华大学章程》中,提出了________三位一体的育人理念。 + - [x] 价值塑造 + - [ ] 思想引领 + - [x] 能力培养 + - [x] 知识传授 + +8. 关于社会主义核心价值观的表述,正确的是 + - [x] 社会主义核心价值观指明了中国特色社会主义事业的发展方向 + - [x] 社会主义核心价值观凝聚了我国这一多民族国家的最大公约数 + - [x] 社会主义核心价值观奠定了实现中华民族伟大复兴中国梦的价值支撑 + - [ ] 社会主义核心价值观与资本主义社会倡导的价值观是一致的 + +9. 人类社会发展的历史表明,对一个民族一个国家来说,最持久、最深层的力量就是全社会共同认可的 + - [ ] 思想观念 + - [x] 核心价值观 + - [ ] 价值体系 + - [ ] 意识形态 + +10. 培育和践行社会主义核心价值观,要 + - [x] 以中华民族优秀传统文化为底蕴 + - [x] 立足中国特色社会主义的奋斗实践 + - [x] 以当代中国面临的时代问题为指向 + - [x] 勤学修德明辨笃行,做社会主义核心价值观的积极践行者 + + +## 第五章 明大德守公德严私德 + +1. 下列有关道德的叙述**不正确**的是 + - [ ] 社会关系的形成是道德赖以产生的客观条件 + - [ ] 人类最初的道德是以风俗习惯等形式表现出来的 + - [ ] 人类自我意识的形成与发展是道德产生的主观条件 + - [x] 语言是人类道德起源的第一个历史前提 + +2. 下列选项中,对于道德的理解,正确的有 + - [x] 道德通过社会舆论、传统习俗和人们的内心信念来维系 + - [x] 道德属于上层建筑的范畴,是一种特殊的社会意识形态 + - [ ] 道德是阶级社会特有的社会意识形态,原始社会没有道德 + - [x] 道德是对人们的行为进行善恶评价的心理意识、原则规范和行为活动的总和 + +3. 对待传统道德的问题上,下列**错误**的是 + - [x] 坚持文化复古主义,中国的落后就是因为儒家文化的失落 + - [ ] 吸取借鉴优良的道德文明成果 + - [x] 全盘西化,实行历史虚无主义 + - [ ] 坚持以我为主、为我所用的原则 + +4. 红船精神、苏区精神、长征精神、延安精神、西柏坡精神等红色精神中蕴含的革命道德,在当代中国有哪些价值? + - [x] 有利于加强和巩固社会主义和共产主义的理想信念 + - [x] 有利于培育和践行社会主义核心价值观 + - [x] 有利于引导人们树立正确的道德观 + - [x] 有利于培育良好的社会道德风尚 + +5. 社会主义道德的核心是 + - [ ] 爱国主义 + - [ ] 集体主义 + - [x] 为人民服务 + - [ ] 社会主义荣辱观 + +6. 社会主义道德的原则是 + - [ ] 爱国主义 + - [x] 集体主义 + - [ ] 为人民服务 + - [ ] 社会主义荣辱观 + +7. 以下哪些行为符合网络生活中的道德要求? + - [x] 合理安排上网时间 + - [ ] 沉迷网络交友 + - [x] 正确使用网络工具 + - [x] 积极引导网络舆论 + +8. 社会主义集体主义的道德要求**不包括** + - [ ] 办事公道、无私奉献 + - [ ] 先公后私、先人后己 + - [ ] 公私兼顾、不损公肥私 + - [x] 一心为公、放弃私利 + +9. 志愿服务的精神包括 + - [x] 奉献 + - [x] 友爱 + - [x] 互助 + - [ ] 和谐 + +10. 下面关于社会主义社会中集体利益和个人利益的关系表述正确的是 + - [x] 个人正当利益是集体利益不可分割的组成部分 + - [x] 集体利益是集体所有成员共同利益的统一 + - [x] 集体利益高于个人利益 + - [x] 个人利益与集体利益是辩证统一的 + + + ## 第六章 尊法学法守法用法 + +1. 法律体现的是 + - [ ] 人民群众的整体意志 + - [ ] 社会整体的共同意志 + - [x] 统治阶级的整体意志 + - [ ] 统治阶级内部的个别意志 + +2. 以下哪种社会形态中不存在法律? + - [x] 原始社会 + - [ ] 奴隶社会 + - [ ] 封建社会 + - [ ] 资本主义社会 + +3. 法律的运行包括以下哪几个环节? + - [x] 法律制定 + - [x] 法律执行 + - [x] 法律适用 + - [x] 法律遵守 + +4. 下列哪些**不是**我国现有的实体法律部门? + - [ ] 《反分裂国家法》 + - [ ] 《著作权法》 + - [x] 《行政诉讼法》 + - [x] 《引渡法》 + +5. 诉讼法**不包括** + - [ ] 刑事诉讼法 + - [ ] 民事诉讼法 + - [x] 社会诉讼法 + - [ ] 行政诉讼法 + +6. 下列关于依法治国与以德治国相结合的解释**错误**的是 + - [ ] 道德可以滋养法治精神和支撑法治文化 + - [ ] 通过公民道德建设提高社会文明程度,能为法治实施创造良好的人文环境 + - [x] 坚持依法治国和以德治国相结合,主要是强调发挥道德的教化作用 + - [ ] 道德教化可以劝人向善,也可以弘扬公序良俗,培养人们的规则意识 + +7. 全面依法治国的基本格局指的是 + - [x] 科学立法 + - [x] 严格执法 + - [x] 公正司法 + - [x] 全民守法 + +8. 公民的人身权利包括 + - [x] 人格尊严权 + - [ ] 结社自由权 + - [x] 生命健康权 + - [x] 人身自由权 + +9. 下列哪些属于宪法规定的公民的基本义务? + - [x] 维护国家统一和民族团结 + - [x] 遵守宪法和法律 + - [x] 依法纳税 + - [x] 依法服兵役 + +10. 我国公民的权利救济方式主要包括 + - [x] 司法救济 + - [x] 行政救济 + - [ ] 经济救济 + - [x] 自力救济 diff --git 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000000000..be124c9cb --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh1.tex" @@ -0,0 +1,181 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\title{离散数学——第一周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.09.20} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} +\linespread{0.7} +\begin{document} + +\maketitle + +\paragraph{1.1}\label{1.1} +判断下列语句是否是命题,并对命题确定其真值。\\ + +\quad (1) 火星上有声明存在。 \\ + +\quad (3) 香山比华山高。\\ + +\quad (5) 这盆茉莉花真香!\\ + +\quad (7) 这句话是错的。\\ + +\paragraph{答} +(1) 是命题。但真值尚未确定。(3) 是命题,真值为假。(5) 不是命题。感叹句不是命题。(7) 不是命题。悖论不是命题。 + +\paragraph{1.2}\label{1.2} +$P$表示今天很冷,$Q$表示正在下雪。\\ + +\quad (1) 将下列命题符号化:\\ + +\qquad 如果正在下雪,那么今天很冷。\\ + +\qquad 今天很冷当且仅当正在下雪。\\ + +\qquad 正在下雪的必要条件是今天很冷。\\ + +\quad (2) 用自然语句叙述下列公式:\\ + +\qquad $$\lnot(P\lor Q), \lnot P \land \lnot Q, P \to Q, +\lnot P \lor Q, \lnot \lnot P, \lnot P \leftrightarrow Q. $$ + +\paragraph{答} +\subparagraph{(1)} +$Q\to P, P \leftrightarrow Q, Q\to P.$ +\subparagraph{(2)} +今天很冷与正在下雪两件事不同时发生; +今天不冷或没在下雪; +如果今天很冷,则正在下雪; +今天不冷或正在下雪; +今天并非不冷; +今天不冷当且仅当正在下雪。 + +\paragraph{1.3}\label{1.3} +对下列公式直观叙述在什么样的解释下为真,并列写出真值表来验证。\\ + +\quad (1) $ \lnot (P \lor Q), \lnot P \land Q, \lnot (P \land Q). $ \\ + +\quad (3) $ (P \to Q) \land (P \leftrightarrow Q). $ \\ + +\quad (5) $ P \to (Q \to R), P \land Q \to R.$ \\ + +\paragraph{解} +\subparagraph{(1)} +$\lnot (P \lor Q)$和$ \lnot P \land Q$仅在$P$与$Q$均为$0$时真值为$1$。 +$\lnot (P \land Q)$只需$P, Q$中有一个为$0$,真值即为$1$。真值表如表 1.3(1) 所示。 +\begin{table}[!htb] +\centering +\begin{tabular}{ccccc} +$P$ & $Q$ & $\lnot(P \lor Q)$ & $\lnot P \land \lnot Q$ & $\lnot (P \land Q)$ \\ +0 & 0 & 1 & 1 & 1 \\ +0 & 1 & 0 & 0 & 1 \\ +1 & 0 & 0 & 0 & 1 \\ +1 & 1 & 0 & 0 & 0 +\end{tabular} +\caption*{表 1.3 (1)} +\end{table} + +\subparagraph{(3)}仅当$P$假$Q$真时,公式真值为$1$。真值表如表 1.3(3) 所示。 +\begin{table}[!htb] +\centering +\begin{tabular}{ccc} +$P$ & $Q$ & $(P\to Q)\land \lnot (P \leftrightarrow Q)$ \\ +0 & 0 & 0 \\ +0 & 1 & 1 \\ +1 & 0 & 0 \\ +1 & 1 & 0 +\end{tabular} +\caption*{表 1.3 (3)} +\end{table} + +\subparagraph{(5)}上述公式仅在$P=Q=1, R=0$时真值为$0$,否则为$1$。真值表如图 1.3(5) 所示。 +\begin{table}[!htb] +\centering +\begin{tabular}{ccccc} +$P$ & $Q$ & $R$ & $P \to (Q \to R)$ & $P \land Q \to R$ \\ +0 & 0 & 0 & 1 & 1 \\ +0 & 0 & 1 & 1 & 1 \\ +0 & 1 & 0 & 1 & 1 \\ +0 & 1 & 1 & 1 & 1 \\ +1 & 0 & 0 & 1 & 1 \\ +1 & 0 & 1 & 1 & 1 \\ +1 & 1 & 0 & 0 & 0 \\ +1 & 1 & 1 & 1 & 1 +\end{tabular} +\caption*{表 1.3 (5)} +\end{table} + +\paragraph{1.4}\label{1.4} +下列公式哪个是重言式,永假式和可满足的,并用代入规则(对重言式)或真值表来验证。\\ + +\quad (1) $P \to P$. \\ + +\quad (3) $(Q\to R)\to((P\lor Q)\to (P \lor R)).$ \\ + +\quad (5) $(P \to Q) \to (\lnot Q \to \lnot P).$ \\ + +\paragraph{解} + +\subparagraph{(1)} 该公式是重言式。 + +\subparagraph{(3)} 该公式是重言式。真值表如表 1.4(3) 所示。 + +\begin{table}[!htb] +\centering +\begin{tabular}{ccccccccc} +$P$ & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ +$Q$ & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ +$R$ & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ +$P \lor Q$ & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ +$P \lor R$ & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ +$(P \lor Q) \to (P \lor R)$ & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ +$Q \to R$ & 1 & 1 & 0 & 1 & 1 & 1 & 0 & 1 \\ +$(Q \to R) \to ((P \lor Q) \to (P \lor R))$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 +\end{tabular} +\caption*{表 1.4 (3)} +\end{table} + +\subparagraph{(5)} 该公式是重言式。真值表如表 1.4(5) 所示。 + +\begin{table}[!htb] +\centering +\begin{tabular}{ccccc} +$P$ & 0 & 0 & 1 & 1 \\ +$Q$ & 0 & 1 & 0 & 1 \\ +$P \to Q$ & 1 & 1 & 0 & 1 \\ +$\lnot Q \to \lnot P$ & 1 & 1 & 0 & 1 \\ +$(P \to Q) \to (\lnot Q \to \lnot P)$ & 1 & 1 & 1 & 1 +\end{tabular} +\caption*{表 1.4 (5)} +\end{table} + +\paragraph{1.5}\label{1.5} +形式化下列自然语句。\\ + +\quad (1) 他个子高而且很胖。 \\ + +\quad (3) 并非“他个子高或很胖”。\\ + +\quad (5) 他个子高或者他个子矮而很胖。\\ + +\quad (7) 如果水是清的,那么或者张三能见到池底或者他是个近视眼。\\ + +\paragraph{解} +\subparagraph{(1)}$P\land Q. $ 其中$P: $他很高;$Q: $他很胖。 +\subparagraph{(3)}$\lnot (P \lor Q).$ 定义同上。 +\subparagraph{(5)}$P \lor (\lnot P \land Q).$ 定义同上。 +\subparagraph{(7)}$P\to (Q \land R).$ 其中$P: $水是清的;$Q: $张三能看到池底;$R: $张三是近视眼。 + + + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh10.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh10.tex" new file mode 100644 index 000000000..0d19cad01 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh10.tex" @@ -0,0 +1,162 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} +\newcommand{\ran}{{\rm ran}} +\newcommand{\dom}{{\rm dom}} + +\title{离散数学——第十一周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.11.21} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{10.1}\label{10.1} +列出下列关系$R$的元素。 + +(1) $ A = \{ 0,1,2 \} , B= \{ 0,2,4 \} $ ,而 +$$ R = \{ \langle x,y \rangle | x,y \in A \cap B \} $$ + +(2) $ A = \{ 1,2,3,4,5 \}, B = \{1,2,3\}$ ,而 +$$R = \{ \langle x,y \rangle | x \in A \land y \in B \land x = y^2 \} $$ + +\paragraph{解} +(1) $R = \{ \langle 0,0 \rangle , \langle 0,2 \rangle , \langle 2,0 \rangle , \langle 2,2 \rangle \} $ + +(3) $R = \{ \langle 1,1 \rangle , \langle 4,2 \rangle \} $ + +\paragraph{10.2}\label{10.2} +设$A = \{ \langle 1,2 \rangle , \langle 2,4 \rangle , \langle 3,3 \rangle \} , B = \{ \langle 1,3 \rangle , \langle 2,4 \rangle , \langle 4,2 \rangle \} $ + +求$A \cup B, A \cap B, \dom(A), \dom(B), \ran(A), \ran(B), \dom(A \cup B), \ran(A \cap B)$ + +\paragraph{解} +\begin{align*} + A \cup B & = \{ \langle 1,2 \rangle , \langle 1,3 \rangle , \langle 2,4 \rangle , \langle 3,3 \rangle , \langle 4,2 \rangle \} \\ + A \cap B & = \{ \langle 2,4 \rangle \} \\ + \dom(A) & = \{ 1,2,3 \} \\ + \dom(B) & = \{ 1,2,4 \} \\ + \ran(A) & = \{ 2,3,4 \} \\ + \ran(B) & = \{ 2,3,4 \} \\ + \dom(A \cup B) & = \{ 2,3,4 \} \\ + \ran(A \cap B) & = \{ 4 \} +\end{align*} + +\paragraph{10.3}\label{10.3} +证明: +$$\dom(R \cup S) = \dom(R) \cup \dom(S), \dom(R \cap S)\subseteq \dom(R) \cap \dom(S)$$ + +\paragraph{证} +(1) 证$\dom(R \cup S) = \dom(R) \cup \dom(S)$ +\begin{align*} + x \in \dom(R \cup S) & \Longleftrightarrow ( \exists y) \langle x,y \rangle \in R \cup S \\ + & \Longleftrightarrow ( \exists y) (\langle x,y \rangle \in R \lor \langle x,y \rangle \in S) \\ + & \Longleftrightarrow x \in \dom(R) \lor x \in \dom(S) \\ + & \Longleftrightarrow x \in \dom(R) \cup \dom(S) +\end{align*} +$$\therefore \dom(R \cup S) = \dom(R) \cup \dom(S)$$ + +(2) 证$\dom(R \cap S)\subseteq \dom(R) \cap \dom(S)$ +\begin{align*} + x \in \dom(R \cap S) & \Longleftrightarrow ( \exists y) \langle x,y \rangle \in R \cap S \\ + & \Longleftrightarrow ( \exists y)( \langle x,y \rangle \in R \land \langle x,y \rangle \in S) \\ + & \Longrightarrow ( \exists y) \langle x,y \rangle \in R \land ( \exists y) \langle x,y \rangle \in S \\ + & \Longleftrightarrow x \in \dom(R) \land x \in \dom(S) \\ + & \Longleftrightarrow x \in \dom(R) \cap \dom(S) +\end{align*} +$$\therefore \dom(R \cap S)\subseteq \dom(R) \cap \dom(S)$$ + +\paragraph{10.4}\label{10.4} +设:$A=\{1,2,3\}$,在$A$上有多少不同的关系?设$|A|=n$,在$A$上有多少不同的关系? + +\paragraph{答} $A=\{1,2,3\}$上有$2^9$种不同的关系。$|A|=n$时,$A$上有$2^{n^2}$种不同的关系。 + +\paragraph{10.5}\label{10.5} +列出所有从$A=\{a,b,c\}$到$B=\{d\}$的关系。 + +\paragraph{答} +$$\varnothing, \{ \langle a,d \rangle \} , \{ \langle b,d \rangle \} , \{ \langle c,d \rangle \},$$ +$$ \{ \langle a,d \rangle , \langle b,d \rangle \} , \{ \langle a,d \rangle , \langle c,d \rangle \} , \{ \langle b,d \rangle , \langle c,d \rangle \}, $$ +$$ \{ \langle a,d \rangle , \langle b,d \rangle , \langle c,d \rangle \} $$ +共8个关系。 + +\paragraph{10.6}\label{10.6} +对$n \in \snatural$且$n >2$,从二元关系定义$n$元关系。 +\paragraph{答} +$$ \langle x_1, x_2, x_3 \rangle = \langle \langle x_1, x_2 \rangle , x_3 \rangle $$ +$$ \langle x_1, x_2, x_3, x_4 \rangle = \langle \langle x_1, x_2, x_3 \rangle , x_4 \rangle $$ +$$ \cdots $$ +$$ \langle x_1, x_2, \cdots, x_n \rangle = \langle \langle x_1, x_2, \cdots, x_ { n-1 } \rangle , x_n \rangle $$ + +\paragraph{10.7}\label{10.7} +对$A=\{0,1,2,3,4\}$上的下列关系,给出关系图和关系矩阵。 + +(1) $R_1 = \{ \langle x,y \rangle |2 \le x \land y \le 2 \} $ + +(3) $R_3 = \{ \langle x,y \rangle | x\text{和}y\text{是互质的} \}$ + +\paragraph{解} 关系图如图 10.7 所示。 +\begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.221\textwidth} + \centering + \includegraphics[width=1\textwidth]{p7-1.png} + \caption*{(1)} + \end{minipage} + \begin{minipage}[t]{0.221\textwidth} + \centering + \includegraphics[width=1\textwidth]{p7-3.png} + \caption*{(3)} + \end{minipage} + \caption*{图 10.7} +\end{figure} +关系矩阵 +$$A_1 = \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 0 & 0 & 0 \\ + 1 & 1 & 1 & 0 & 0 \\ + 1 & 1 & 1 & 0 & 0 \\ + 1 & 1 & 1 & 0 & 0 +\end{bmatrix}, +A_3 = \begin{bmatrix} + 0 & 0 & 0 & 0 & 0 \\ + 0 & 1 & 1 & 1 & 1 \\ + 0 & 1 & 0 & 1 & 0 \\ + 0 & 1 & 1 & 0 & 1 \\ + 0 & 1 & 0 & 1 & 0 +\end{bmatrix}$$ + +\paragraph{10.10}\label{10.10} +设$R,S,T$是$A$上的关系,证明 +$$R\circ (S \cup T) = (R \circ S) \cup (R \circ T)$$ +\paragraph{证} +\begin{align*} + \langle x,y \rangle \in R \circ (S \cup \true ) & \Longleftrightarrow ( \exists z)( \langle z,y \rangle \in R \land \langle x,z \rangle \in S \cup \true ) \\ + & \Longleftrightarrow ( \exists z)( \langle z,y \rangle \rangle \in R \land ( \langle x,z \rangle \in S \lor \langle x,z \rangle \in \true )) \\ + & \Longleftrightarrow ( \exists z)(( \langle z,y \rangle \in R \land \langle x,z \rangle \in S) \lor ( \langle z,y \rangle \in R \land \langle x,z \rangle \in \true )) \\ + & \Longleftrightarrow ( \exists z)( \langle z,y \rangle \in R \land \langle x,z \rangle \in S) \lor ( \exists z)( \langle z,y \rangle \in R \lor \langle x,z \rangle \in \true ) \\ + & \Longleftrightarrow \langle x,y \rangle \in R \circ S \lor \langle x,y \rangle \in R \circ \true \\ + & \Longleftrightarrow \langle x,y \rangle \in (R \circ S) \cup (R \circ \true ) +\end{align*} +$$\therefore R\circ (S \cup T) = (R \circ S) \cup (R \circ T)$$ + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh11.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh11.tex" new file mode 100644 index 000000000..99116800a --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh11.tex" @@ -0,0 +1,289 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} +\newcommand{\ran}{{\rm ran}} +\newcommand{\dom}{{\rm dom}} + +\title{离散数学——第十二周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.11.29} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{10.13} \label{10.13} + 对$A$到$B$的关系$R$,$a \in A$,定义$B$的一个子集$R(a)=\{b|aRb\}$,在$C=\{-4,-3,-2,-1,0,1,2,3,4\}$上定义$R= \{ \langle x,y \rangle |x \langle y \} $, $S= \{ \langle x,y \rangle |x-1 < y < x+2 \} $, $ \true = \{ \langle x,y \rangle |x^2 \le y \} $,写出集合$R(0), R(1), S(0), S(-1), \true (0), \true (-1).$ + +\paragraph{解} + \begin{align*} + R(0) & = \{ 1,2,3,4 \} \\ + R(1) & = \{ 2,3,4 \} \\ + S(0) & = \{ 0,1 \} \\ + S(-1) & = \{ -1, 0 \} \\ + T(0) & = \{ 0,1,2,3,4 \} \\ + T(1) & = \{ 1,2,3,4 \} + \end{align*} + +\paragraph{10.14} \label{10.14} + 对命题:“集合$A$上的一个关系$R$,如果是对称的和传递的,就一定是自反的。因为$xRy$和$yRx$蕴含$xRx$。”依据定义找出错误。在$\{1,2,3\}$上构造一个关系,它是对称的和传递的,但不是自反的。 + +\paragraph{答} + $R$中可能不存在任何一个$y$使得$xRy,yRx$成立,因而$xRx$不一定成立。 + + 例如空关系是对称且传递的,但不是自反的。 + +\paragraph{10.15} \label{10.15} + 对集合$A=\{1,2,3\}$上,下列8种关系图(图见书第189面,此处省略),说明每个关系具有的性质。 + +\paragraph{解} 如表 10-15 所示。 + \begin{table}[htb] + \centering + \begin{tabular}{ccccc} + 关系 & 自反性 & 反自反性 & 对称性 & 传递性 \\ + $R_1$ & $\times$ & $\times$ & $\times$ & $\times$ \\ + $R_2$ & $\times$ & $\times$ & $\checkmark$ & $\checkmark$ \\ + $R_3$ & $\checkmark$ & $\times$ & $\checkmark$ & $\checkmark$ \\ + $R_4$ & $\checkmark$ & $\times$ & $\times$ & $\checkmark$ \\ + $R_5$ & $\times$ & $\times$ & $\times$ & $\times$ \\ + $R_6$ & $\times$ & $\checkmark$ & $\checkmark$ & $\times$ \\ + $R_7$ & $\times$ & $\checkmark$ & $\checkmark$ & $\times$ \\ + $R_8$ & $\checkmark$ & $\times$ & $\checkmark$ & $\times$ + \end{tabular} + \caption*{表 10-15} + \end{table} + +\paragraph{10.16} \label{10.16} + 对集合$A=\{1,2,3,\cdots,10\}$,$A$上的关系$R$和$S$各有什么性质。 + $$R= \{ \langle x,y \rangle |x+y=10 \}$$ + $$S= \{ \langle x,y \rangle |x+y\text { 是偶数 } \} $$ + +\paragraph{答}$R:$对称性;$S:$自反性、对称性、传递性。 + +\paragraph{10.17} \label{10.17} + 对$A$上的关系$R$,证明: + + (1) $R\text{是自反的} \Longleftrightarrow I_A\subseteq R$ + +\paragraph{证} (1) + \subparagraph{充分性} 若$R$是自反的,则$ ( \forall x)(x \in A \rightarrow (x,x) \in R)$ + $$ \langle x,y \rangle \in I_A \Longleftrightarrow x \in A \land x = y \Longrightarrow \langle x,y \rangle \in R \land x=y$$ + $$\therefore I_A \subseteq R$$ + \subparagraph{必要性} 若$I_A \subseteq R$ + $$x \in A \Longrightarrow \langle x,x \rangle \in I_A \Longrightarrow \langle x,x \rangle \in R$$ + + 则$R$是自反的。 + +\paragraph{10.18} \label{10.18} + 对$A$上的关系$R_1$和$R_2$,判定下列命题的真假。真的证明之,假的举反例。 + + (1) 若$R_1$和$R_2$是自反的,则$R_1 \circ R_2$是自反的。 + + (3) 若$R_1$和$R_2$是对称的,则$R_1 \circ R_2$是对称的。 + +\paragraph{解} + + (1) 真。 + $$x \in A \Longrightarrow \langle x,x \rangle \in R_1 \land \langle x,x \rangle \in R_2 \Longrightarrow \langle x,x \rangle \in R_1 \circ R_2$$ + + (3) 假。反例:$$A= \{ 1,2,3 \} , R_1= \{ \langle 2,3 \rangle , \langle 3,2 \rangle \} , R_2 = \{ \langle 1,2 \rangle , \langle 2,1 \rangle \} , R_1 \circ R_2 = \{ \langle 1,3 \rangle \} $$不是对称的。 + +\paragraph{10.19} \label{10.19} + 对集合$A=\{1,2,3\}$,给出$A$上的关系$R$的例子,使它具有下列性质。 + + (1) 对称的且反对称的且传递的。 + + (3) 不是对称的且不是反对称的且传递的。 + +\paragraph{答} + + (1) $R = I_A = \{ \langle 1,1 \rangle , \langle 2,2 \rangle , \langle 3,3 \rangle \} $ + + (3) $R = \{ \langle 1,1 \rangle , \langle 1,2 \rangle , \langle 2,1 \rangle , \langle 2,2 \rangle , \langle 1,3 \rangle , \langle 2,3 \rangle \} $ + +\paragraph{10.20} \label{10.20} + 对集合$A= \{ 1,2,3,4 \} $,$A$上的关系$R$为 + $$R= \{ \langle 1,2 \rangle , \langle 4,3 \rangle , \langle 2,2 \rangle , \langle 2,1 \rangle , \langle 3,1 \rangle \} $$ + 说明$R$不是传递的。构造$A$上的关系$R_1$,使$R\subseteq R_1$且$R_1$是传递的。 + +\paragraph{解} + $ \langle 1,2 \rangle \in R, \langle 2,1 \rangle \in R, $但$ \langle 1,1 \rangle \notin R$则$R$不是传递的。 + $$R_1 = \{ \langle 1,1 \rangle , \langle 1,2 \rangle , \langle 2,1 \rangle , \langle 2,2 \rangle , \langle 3,1 \rangle , \langle 3,2 \rangle , \langle 4,1 \rangle , \langle 4,2 \rangle , \langle 4,3 \rangle \} $$ + +\paragraph{10.22} \label{10.22} + 对集合$A= \{ a,b,c,d \} $上的两个关系 + \begin{align*} + R_1 & = \{ \langle a,a \rangle , \langle a,b \rangle , \langle b,d \rangle \} \\ + R_2 & = \{ \langle a,d \rangle , \langle b,c \rangle , \langle b,d \rangle , \langle c,b \rangle \} + \end{align*} + 求$R_1 \circ R_2$,$R_2 \circ R_1$,$R_1^2$,$R_2^2$。 + +\paragraph{解} + \begin{align*} + R_1 \circ R_2 & = \{ \langle c,d \rangle \} \\ + R_2 \circ R_1 & = \{ \langle a,d \rangle , \langle a,c \rangle \} \\ + R_1^2 & = \{ \langle a,a \rangle , \langle a,b \rangle , \langle a,d \rangle \} \\ + R_2^2 & = \{ \langle b,b \rangle , \langle c,c \rangle , \langle c,d \rangle \} + \end{align*} + +\paragraph{10.24} \label{10.24} + $A=\{a,b,c,d,e\}$上的关系$R$的关系图如下。给出$r(R),s(R),t(R)$的关系图。 + +\paragraph{解} + 如图10-24所示。 + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.584\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-24-o.png} + \caption*{(a) $R$} + \end{minipage} + \\ + \begin{minipage}[t]{0.591\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-24-r.png} + \caption*{(b) $r(R)$} + \end{minipage} + \\ + \begin{minipage}[t]{0.581\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-24-s.png} + \caption*{(c) $s(R)$} + \end{minipage} + \\ + \begin{minipage}[t]{0.583\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-24-t.png} + \caption*{(d) $t(R)$} + \end{minipage} + \caption*{图 10-24} + \end{figure} + +\paragraph{10.27} \label{10.27} + 对$A=\{a,b,c,d\}$上的关系 + $$R = \{ \langle a,b \rangle , \langle b,a \rangle , \langle b,c \rangle , \langle c,d \rangle \} $$ + + (1) 分别用矩阵运算和作图法求$r(R),s(R),t(R)$。 + + (2) 用Warshall算法求$t(R)$。 + +\paragraph{解} + + (1) 矩阵运算法: + \begin{align*} + M(R) & = \begin{bmatrix} + 0 & 1 & 0 & 0 \\ + 1 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} \\ + M(r(R)) = M(R \cup I_A) & = \begin{bmatrix} + 1 & 1 & 0 & 0 \\ + 1 & 1 & 1 & 0 \\ + 0 & 0 & 1 & 1 \\ + 0 & 0 & 0 & 1 + \end{bmatrix} \\ + M(s(R)) = M(R \cup R^{-1}) & = \begin{bmatrix} + 0 & 1 & 0 & 0 \\ + 1 & 0 & 1 & 0 \\ + 0 & 1 & 0 & 1 \\ + 0 & 0 & 1 & 0 + \end{bmatrix} \\ + M(t(R)) = M(R \cup R^2 \cup R^3 \cup R^4) & = \begin{bmatrix} + 1 & 1 & 1 & 1 \\ + 1 & 1 & 1 & 1 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} + \end{align*} + + 作图法,如图 10-27 所示: + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.453\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-27-o.png} + \caption*{(a) $R$} + \end{minipage} + \\ + \begin{minipage}[t]{0.454\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-27-r.png} + \caption*{(b) $r(R)$} + \end{minipage} + \\ + \begin{minipage}[t]{0.443\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-27-s.png} + \caption*{(c) $s(R)$} + \end{minipage} + \\ + \begin{minipage}[t]{0.458\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-27-t.png} + \caption*{(d) $t(R)$} + \end{minipage} + \caption*{图 10-27} + \end{figure} + + 以上两种方法均可得 + \begin{align*} + r(R) & = \{ \langle a,a \rangle , \langle a,b \rangle , \langle b,a \rangle , \langle b,b \rangle , \langle b,c \rangle , \langle c,c \rangle , \langle c,d \rangle , \langle d,d \rangle \} \\ + s(R) & = \{ \langle a,b \rangle , \langle b,a \rangle , \langle b,c \rangle , \langle c,b \rangle , \langle c,d \rangle , \langle d,c \rangle \} \\ + t(R) & = \{ \langle a,a \rangle , \langle a,b \rangle , \langle a,c \rangle , \langle a,d \rangle , \langle b,a \rangle , \langle b,b \rangle , \langle b,c \rangle , \langle b,d \rangle , \langle c,d \rangle \} + \end{align*} + + (2) + \begin{align*} + B = M(R) & = \begin{bmatrix} + 0 & 1 & 0 & 0 \\ + 1 & 0 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} \\ + B_1 & = \begin{bmatrix} + 0 & 1 & 0 & 0 \\ + 1 & 1 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} \\ + B_2 & = \begin{bmatrix} + 1 & 1 & 1 & 0 \\ + 1 & 1 & 1 & 0 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} \\ + B_3 & = \begin{bmatrix} + 1 & 1 & 1 & 1 \\ + 1 & 1 & 1 & 1 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} \\ + B_4 & = \begin{bmatrix} + 1 & 1 & 1 & 1 \\ + 1 & 1 & 1 & 1 \\ + 0 & 0 & 0 & 1 \\ + 0 & 0 & 0 & 0 + \end{bmatrix} = M(R_+) + \end{align*} + $$\therefore t(R) = \{ \langle a,a \rangle , \langle a,b \rangle , \langle a,c \rangle , \langle a,d \rangle , \langle b,a \rangle , \langle b,b \rangle , \langle b,c \rangle , \langle b,d \rangle , \langle c,d \rangle \} $$ + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh12.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh12.tex" new file mode 100644 index 000000000..7657377f6 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh12.tex" @@ -0,0 +1,332 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} +\newcommand{\ran}{{\rm ran}} +\newcommand{\dom}{{\rm dom}} + +\title{离散数学——第十三周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.12.06} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{10.28} \label{10.28} + 对有限集合$A$,在$A$上给出最多个等价类和最少个等价类的关系各是什么? + +\paragraph{答} + 给出最多个等价类的关系是恒等关系$I_A$,有$|A|$个等价类。给出最少个等价类的关系是全关系$E_A$,仅$1$个等价类。 + +\paragraph{10.29} \label{10.29} + 设$R$是$A$上传递和自反的关系,$T$是$A$上的关系,$aTb \Longleftrightarrow aRb \land bRa$。证明$T$是等价关系。 + +\paragraph{证} + \begin{align*} + R \text{是自反的} & \Longleftrightarrow ( \forall x)(x \in A \longrightarrow \langle x,x \rangle \in R) \\ + & \Longleftrightarrow ( \forall x)(x \in A \longrightarrow \langle x,x \rangle \in R \land \langle x,x \rangle \in R) \\ + & \Longleftrightarrow ( \forall x)( \langle x,x \rangle \in T) \\ + & \Longleftrightarrow T \text{是自反的} + \end{align*} + + $R$是对称的,则对任意$ \langle x,y \rangle $ + \begin{align*} + \langle x,y \rangle \in T & \Longleftrightarrow \langle x,y \rangle \in R \land \langle y,x \rangle \in R \\ + & \Longleftrightarrow \langle y,x \rangle \in R \land \langle x,y \rangle \in R \\ + & \Longleftrightarrow \langle y,x \rangle \in T + \end{align*} + 从而$T$是对称的。 + + $R$是传递的,则对任意$ \langle x,y \rangle, \langle y,z \rangle $ + \begin{align*} + \langle x,y \rangle \in T \land \langle y,z \rangle \in T & \Longleftrightarrow \langle x,y \rangle \in R \land \langle y,x \rangle \in R \land \langle y,z \rangle \in R \land \langle z,y \rangle \in R \\ + & \Longrightarrow \langle x,z \rangle \in R \land \langle z,x \rangle \in R \\ + & \Longleftrightarrow \langle x,z \rangle \in T + \end{align*} + 从而$T$是传递的。 + + 综上,$T$是等价关系。 + +\paragraph{10.30} \label{10.30} + 对$A= \{ a,b,c,d \} $,$R$是$A$上的等价关系,且 + $$R = \{ \langle a,a \rangle , \langle a,b \rangle , \langle b,a \rangle , \langle b,b \rangle , \langle c,c \rangle , \langle c,d \rangle , \langle d,c \rangle , \langle d,d \rangle \} $$ + 画$R$的关系图,求$A$中各元素的等价类。 + +\paragraph{解} + 关系图如图10-30所示,等价类 + $$[a]_R = [b]_R = \{a,b\}$$ + $$[c]_R = [d]_R = \{c,d\}$$ + + \begin{figure}[!htb] + \centering + \includegraphics[width=0.188\textwidth]{10-30.png} + \caption*{图 10-30} + \end{figure} + +\paragraph{10.31} \label{10.31} + 设$\sinteger_+= \{ x | x \in \sinteger \land x > 0 \} $,判定下列集合$\pi$是否构成$\sinteger_+$的划分。 + + (1) $S_1 = \{ x| x \in \sinteger_+ \land x \text{是素数} \}, S_2 = \sinteger_+ - S_1, \pi = \{S_1, S_2 \} $ + + (2) $\pi = \{ \{ x \} |x \in \sinteger_+ \} $ + +\paragraph{答} + (1) 是,$\pi$划分将$\sinteger_+$划分为素数与非素数。 + + (2) 是,$\pi$划分即是恒等关系导出的划分。 + +\paragraph{10.32} \label{10.32} + 对非空集合$A$,$P(A) - \{\varnothing\}$是否构成$A$的划分? + +\paragraph{答} + 若$|A|=1$,则$P(A) - \{\varnothing\} = \{A\}$是$A$的划分,等同于全关系导出的划分。 + + 若$|A|>1$,则$P(A) - \{\varnothing\}$不是$A$的划分,因为$P(A) - \{\varnothing\}$的元素$A$和$\{a\}$(其中$a$为$A$的任意元素)交集不为空。 + +\paragraph{10.33} \label{10.33} + 有$4$个元素的集合上,不同的等价关系的数目是多少? + +\paragraph{答} + 不同等价关系的数目等同于不同划分的数目,而不同的划分有(1)恒等关系导出的划分;(2)二一一型划分;(3)二二型划分;(4)三一型划分;(5)全关系导出的划分,共15种,如下所示。则不同等价关系的数目亦为15。 + + $ (1) \{ \{ a \} , \{ b \} , \{ c \} , \{ d \} \} $ + + $ (2) \{ \{ a,b \} , \{ c \} , \{ d \} \} , \{ \{ a,c \} , \{ b \} , \{ d \} \} , \{ \{ a,d \} , \{ b \} , \{ c \} \} ,$ + + \quad \quad $ \{ \{ b,c \} , \{ a \} , \{ d \} \} , \{ \{ b,d \} , \{ a \} , \{ c \} \} , \{ \{ c,d \} , \{ a \} , \{ b \} \} $ + + $ (3) \{ \{ a,b \} , \{ c,d \} \} , \{ \{ a,c \} , \{ b,d \} \} , \{ \{ a,d \} , \{ b,c \} \} $ + + $ (4) \{ \{ a,b,c \} , \{ d \} \} , \{ \{ a,b,d \} , \{ c \} \} , \{ \{ a,c,d \} , \{ b \} \} , \{ \{ b,c,d \} , \{ a \} \} $ + + $ (5) \{ \{ a,b,c,d \} \} $ + +\paragraph{10.34} \label{10.34} + 设$R,S$是$A$上的关系,且 + $$S = \{ \langle a,b \rangle | ( \exists c) (aRc \land cRb) \} $$ + 证明若$R$是等价关系,则$S$是等价关系。 + +\paragraph{证} + $R$是等价关系,则$R$自反、对称且传递。 + \begin{align*} + a \in A & \Longrightarrow \langle a,a \rangle \in R \\ + & \Longleftrightarrow \langle a,a \rangle \in R \land \langle a,a \rangle \in R \\ + & \Longleftrightarrow \langle a,a \rangle \in S + \end{align*} + 则$S$自反。 + \begin{align*} + \langle a,b \rangle \in S & \Longleftrightarrow ( \exists c)(aRc \land cRb) \\ + & \Longrightarrow ( \exists c)(cRa \land bRc) \\ + & \Longleftrightarrow \langle b,a \rangle \in S + \end{align*} + 则$S$对称。 + \begin{align*} + \langle a,b \rangle \in S \land \langle b,c \rangle \in S & \Longleftrightarrow ( \exists d)(aRd \land dRb) \land ( \exists e)(bRe \land eRc) \\ + & \Longleftrightarrow ( \exists d) ( \exists e)(aRd \land dRb \land bRe \land eRc) \\ + & \Longrightarrow ( \exists d) ( \exists e)(aRd \land dRe \land eRc) \\ + & \Longrightarrow ( \exists d)(a Rd \land dRc) \\ + & \Longleftrightarrow \langle a,c \rangle \in S + \end{align*} + 则$S$传递。 + + 综上,$S$是等价关系。 + +\paragraph{10.35} \label{10.35} + 设$\sinteger_+$是正整数集合,$A = \sinteger_+ \times \sinteger_+$,$A$上的关系 + $$R= \{ \langle \langle x,y \rangle , \langle u,v \rangle \rangle | xv = yu \} $$ + 证明$R$是等价关系。 + +\paragraph{证} + $$ \langle x,y \rangle \in A \Longleftrightarrow xy = xy \Longleftrightarrow \langle \langle x,y \rangle , \langle x,y \rangle \rangle \in R$$ + 则$R$自反。 + \begin{align*} + \langle \langle x,y \rangle , \langle u,v \rangle \rangle \in R & \Longleftrightarrow xy = uv \\ + & \Longleftrightarrow \langle \langle u,v \rangle , \langle x,y \rangle \rangle \in R + \end{align*} + 则$R$对称。 + \begin{align*} + \langle \langle x,y \rangle , \langle u,v \rangle \rangle \in R \land \langle \langle u,v \rangle , \langle z,w \rangle \rangle \in R & \Longleftrightarrow xy = uv \land uv = zw \\ + & \Longrightarrow xy = zw \\ + & \Longrightarrow \langle \langle x,y \rangle , \langle z,w \rangle \rangle \in R + \end{align*} + 则$R$传递。 + + 综上,$R$是等价关系。 + +\paragraph{10.39} \label{10.39} + 对下列集合上的整除关系,画出Hasse图。 + + (1) \{1,2,3,4,6,8,12,24\} + + (2) \{1,2,3,4,5,6,7,8,9\} + +\paragraph{答} + 如图10-39所示。 + + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.153\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-39-1.png} + \caption*{(1)} + \end{minipage} + \begin{minipage}[t]{0.259\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-39-2.png} + \caption*{(2)} + \end{minipage} + \caption*{图 10-39} + \end{figure} + +\paragraph{10.40} \label{10.40} + 写出下列Hasse图(图10-40)的集合和集合上的偏序关系。 + + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.241\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-40-a.png} + \caption*{(a)} + \end{minipage} + \begin{minipage}[t]{0.230\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-40-b.png} + \caption*{(b)} + \end{minipage} + \caption*{图 10-40} + \end{figure} + +\paragraph{答} + (1) + $$A = \{ a,b,c,d,e,f,g \} $$ + \begin{align*} + R = & \{ \langle a,a \rangle , \langle a,b \rangle , \langle a,c \rangle , \langle a,d \rangle , \langle a,e \rangle , \langle a,f \rangle , \langle a,g \rangle , \\ + & \langle b,b \rangle , \langle b,d \rangle , \langle b,e \rangle , \langle c,c \rangle , \langle c,f \rangle , \langle c,g \rangle , \\ + & \langle d,d \rangle , \langle e,e \rangle , \langle f,f \rangle , \langle g,g \rangle \} + \end{align*} + + (2) + $$A = \{ a,b,c,d,e,f \} $$ + \begin{align*} + R = & \{ \langle a,a \rangle , \langle a,b \rangle , \langle a,c \rangle , \langle a,d \rangle , \langle a,e \rangle , \langle a,f \rangle , \\ + & \langle b,b \rangle , \langle c,c \rangle , \langle d,d \rangle , \langle d,f \rangle , \langle e,e \rangle , \langle e,f \rangle , \langle f,f \rangle \} + \end{align*} + +\paragraph{10.41} \label{10.41} + 画出下列偏序集$ \langle A,R \rangle $的Hasse图,并写出$A$的极大元、极小元、最大元、最小元。 + + (1) $$A = \{ a,b,c,d,e \} , R= \{ \langle a,d \rangle , \langle a,c \rangle , \langle a,b \rangle , \langle a,e \rangle , \langle b,e \rangle , \langle c,e \rangle , \langle d,e \rangle \} \cup I_A $$ + + (2) $$A = \{ a,b,c,d \} , R = \{ \langle c,d \rangle \} \cup I_A$$ + +\paragraph{答} + Hasse图如图10-41所示。 + + (1) 极大元:$e$;极小元:$a$;最大元:$e$;最小元:$a$。 + + (2) 极大元:$a,b,d$;极小元:$a,b,c$;无最大、最小元。 + + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.211\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-41-1.png} + \caption*{(1)} + \end{minipage} + \begin{minipage}[t]{0.183\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-41-2.png} + \caption*{(2)} + \end{minipage} + \caption*{图 10-41} + \end{figure} + +\paragraph{10.42} \label{10.42} + 设$\sinteger_+= \{ x| x \in \sinteger \land x \rangle 0 \} $,$D$是$\sinteger_+$上的整除关系,$T = \{ 1,2, \cdots, 10 \} \subseteq \sinteger_+$。在偏序集$ \langle \sinteger_+, D \rangle $中,求$T$的上界、下界、上确界、下确界。 + +\paragraph{解} + $T$的下界、下确界均为$1$;$T$的上界$k$是$T$中所有数字的倍数,则$k = n{\rm LCM}\{1,2,3,4,5,6,7,8,9,10\}=2520n(n \in \sinteger_+)$,其中${\rm LCM}$表示最小公倍数,$T$的上确界即$2520$。 + +\paragraph{10.43} \label{10.43} + 设$R$是$A$上的偏序关系,$B\subseteq A$,证明$R \cap (B \times B)$是$B$上的偏序关系。 + +\paragraph{证} + 记$B \times B = E_B$是$B$上的全关系。 + $$r(R \cap (B \times B)) \upharpoonright_B = r(R) \upharpoonright_B = R \cap (B \times B)$$ + 则$R \cap (B \times B)$自反。 + + $a \neq b$时, + \begin{align*} + \langle a,b \rangle \in R \cap (B \times B) & \Longleftrightarrow \langle a,b \rangle \in R \land \langle a,b \rangle \in E_B \\ + & \Longrightarrow \langle b,a \rangle \notin R \\ + & \Longrightarrow \langle b,a \rangle \notin R \cap (B \times B) + \end{align*} + 则$R \cap (B \times B)$反对称。 + \begin{align*} + \langle a,b \rangle , \langle b,c \rangle \in R \cap (B \times B) & \Longrightarrow \langle a,b \rangle \in R \land \langle b,c \rangle \in R \\ + & \Longrightarrow \langle a,c \rangle \in R \\ + & \Longrightarrow \langle a,c \rangle \in R \cap (B \times B) + \end{align*} + 则$R \cap (B \times B)$传递。 + + 综上,$R \cap (B \times B)$是$B$上的偏序关系。 + +\paragraph{10.45} \label{10.45} + 给出$A=\{0,1,2\}$上所有的偏序关系的Hasse图。 + +\paragraph{解} + 如图10-45所示,共19种关系。 + + \begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.111\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-45-a.png} + \caption*{(a) 空关系的Hasse图} + \end{minipage} + \\ + \begin{minipage}[t]{0.532\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-45-b.png} + \caption*{(b) 含一条边的Hasse图} + \end{minipage} + \\ + \begin{minipage}[t]{0.507\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-45-c.png} + \caption*{(c) 含两条边的V形Hasse图} + \end{minipage} + \\ + \begin{minipage}[t]{0.513\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-45-d.png} + \caption*{(d) 含两条边的$\Lambda$形Hasse图} + \end{minipage} + \\ + \begin{minipage}[t]{0.384\textwidth} + \centering + \includegraphics[width=1\textwidth]{10-45-e.png} + \caption*{(e) 单链形Hasse图} + \end{minipage} + \caption*{图 10-45} + \end{figure} + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh13.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh13.tex" new file mode 100644 index 000000000..cf1b5fea4 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh13.tex" @@ -0,0 +1,150 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} +\newcommand{\ran}{{\rm ran}} +\newcommand{\dom}{{\rm dom}} + +\title{离散数学——第十四周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.12.15} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{11.1} \label{11.1} + 下列关系中哪个是函数? + + (1) $ \{ \langle x,y \rangle |x \in \snatural \land y \in \snatural \land x+y < 10 \} $ + + (2) $ \{ \langle x,y \rangle | x \in \sreal \land y \in \sreal \land x = y^2 \} $ + + (3) $ \{ \langle x,y \rangle | x \in \sreal \land y \in \sreal \land y = x^2 \} $ + +\paragraph{答} + (1) 不是。$ \langle 1,2 \rangle , \langle 1,1 \rangle \in R$ + + (2) 不是。$ \langle 1,-1 \rangle , \langle 1,1 \rangle \in R$ + + (3) 是。 + +\paragraph{11.2} \label{11.2} + 下列集合是函数吗?如果是,写其定义域和值域。 + + (1) $\{ \langle 1, \langle 2,3 \rangle \rangle , \langle 2, \langle 3,2 \rangle \rangle , \langle 3, \langle 4,1 \rangle \rangle \}$ + + (2) $\{ \langle 1, \langle 2,3 \rangle \rangle , \langle 2, \langle 3,4 \rangle \rangle , \langle 1, \langle 3,4 \rangle \rangle \}$ + + (3) $\{ \langle 1, \langle 2,3 \rangle \rangle , \langle 2, \langle 2,3 \rangle \rangle , \langle 3, \langle 2,3 \rangle \rangle \} $ + +\paragraph{答} + (1) 是。$\dom(f) = \{ 1,2,3 \} , \ran(f) = \{ \langle 2,3 \rangle , \langle 3,2 \rangle , \langle 4,1 \rangle \} $ + + (2) 不是。 + + (3) 是。$\dom(f) = \{ 1,2,3 \} , \ran(f) = \{ \langle 2,3 \rangle \} $ + +\paragraph{11.3} \label{11.3} + 设$f,g\in A_B$,且$f \cap g \neq \varnothing$,$f \cap g$和$f \cup g$是函数吗?如果是,证明之;不是则举反例。 + +\paragraph{答} + (1) $f \cap g$不是$A$到$B$的函数,例如 + $$f= \{ \langle 1,1 \rangle , \langle 2,2 \rangle , \langle 3,3 \rangle \} , g= \{ \langle 1,1 \rangle , \langle 2,3 \rangle , \langle 3,2 \rangle \} $$ + $f \cap g = \{ \langle 1,1 \rangle \} $不是$\{1,2,3\}$到$\{1,2,3\}$的函数,仅是部分函数。 + + (2) $f \cup g$不是$A$到$B$的函数,同上例,$f \cup g = \{ \langle 1,1 \rangle , \langle 2,2 \rangle , \langle 2,3 \rangle , \langle 3,2 \rangle , \langle 3,3 \rangle \} $不是函数。 + +\paragraph{11.4} \label{11.4} + 设$f \cap g \neq \varnothing$, + \begin{gather*} + f(x) = \left\{\begin{aligned}1, &\text{当}x\text{是奇数} \\ \frac{x}{2}, & \text{当}x\text{是偶数}\end{aligned}\right. + \end{gather*} + 求$f(0), f[ \{ 0 \} ], f[ \{ 0,2,4,6,\cdots \} ], f[ \{ 1,3,5,\cdots \} ], f^ { -1 } [ \{ 2 \} ], f^ { -1 } [ \{ 3,4 \} ]$. + +\paragraph{解} + \begin{align*} + f(0) & = 0 \\ + f[ \{ 0 \} ] & = \{ 0 \} \\ + f[ \{ 0,2,4,6,\cdots \} ] & = \{ 0,1,2,3,\cdots \} \\ + f[ \{ 1,3,5,\cdots \} ] & = \{ 1 \} \\ + f^{-1}[ \{ 2 \} ] & = \{ 4 \} \\ + f^{-1} \{ \{ 3,4 \} ] & = \{ 6,8 \} \\ + \end{align*} + +\paragraph{11.6} \label{11.6} + 下列函数是否满射的,单射的,双射的? + + (1) $f: \sreal \rightarrow \sreal, f(x) = x^2 - 2x - 15$ + + (2) $f: \snatural - \{ 0 \} \rightarrow \sreal, f(x) = \log_2 x$ + + (3) $f: \snatural \rightarrow \snatural, f(x) = \left \{ \begin{aligned} 1, & x \text{是奇数} \\ 2, & x \text{是偶数}\end{aligned} \right.$ + + (4) $f: \snatural \rightarrow \snatural, f(x) = x \mod 3 $ + +\paragraph{答} + (1) 不是满射也不是单射的,更不是双射的。 + + (2) 不是满射的但是单射的,不是双射的。 + + (3) 是满射的但不是单射的,不是双射的。 + + (4) 是满射的但不是单射的,不是双射的。 + +\paragraph{11.7} \label{11.7} + 设$R$是$A$上的等价关系,$g: A \rightarrow A / R$是自然映射,什么条件下$g$是双射的? + +\paragraph{解} + $g$是双射的$\Longleftrightarrow$ $R$是恒同映射$I_A$ + \subparagraph{充分性} + $g$是双射的$\Longrightarrow$ $g$是单射的$\Longrightarrow$ $ ( \forall x) ( \forall y)(x \in A \land y \in A \land x \neq y \longrightarrow g(x) \neq g(y))$,而$g$是自然映射,则$ ( \forall x) ( \forall y)(x \in A \land y \in A \land x \neq y \longrightarrow g(x) = [x]_R \neq [y]_R = g(y))$,则$g(x) = \{x\}$。即$R$是恒同映射。 + \subparagraph{必要性} + $\forall x,y$,若$x \in A, y \in A$且$x \neq y$,则$g(x) = \{x\} \neq \{y\} = g(y)$,则$g$是单射的。$\forall B \in A/R, \exists x \in B$,则$g(x)=B$,从而$g(x)$是满射的。从而$g$是双射的。 + +\paragraph{11.9} \label{11.9} + 对有限集合$A$和$B$,$|A|=m, |B|=n$,求在下列情况下$m$和$n$应满足的条件。 + + (1) 存在从$A$到$B$的单射函数。 + + (2) 存在从$A$到$B$的满射函数。 + + (3) 存在从$A$到$B$的双射函数。 + +\paragraph{答} + (1) $n \ge m$, (2) $m \ge n$, (3) $m=n$ + +\paragraph{11.10} \label{11.10} + 对下列集合$A$和$B$,构造从$A$到$B$的双射函数。 + + (1) $A = \{ 1,2,3 \} , B = \{ a,b,c \} $ + + (2) $A = (0,1)\subseteq R, B = (1,3)\subseteq R$ + + (3) $A = P(X), B = X_Y$,$X= \{ a,b,c \} , 其中Y= \{ 0,1 \} $ + +\paragraph{答} + (1) $f: A \rightarrow B, f(1) = a, f(2) = b, f(3) = c$ + + (2) $f: A \rightarrow B, f(x) = 1+2x $ + + (3) $f: A \rightarrow B, f = \chi_X$,其中$\chi_X$是集合$X$的特征函数。 + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh14.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh14.tex" new file mode 100644 index 000000000..2fe2b002b --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh14.tex" @@ -0,0 +1,107 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} +\newcommand{\ran}{{\rm ran}} +\newcommand{\dom}{{\rm dom}} + +\title{离散数学——第十五周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.12.20} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{12.2} \label{12.2} + 用等势定义证明$[0,1] \approx [a,b], (a,b \in \sreal, a a$,则$b \in C$,这与$a$是$C$的最大元相矛盾。从而$C$不是有限集,$\card C \ge \aleph_0$ + + 综上,$\card C = \aleph_0$ + + (5) $\aleph_0 \le \card(B \cup D) \le \aleph_0 + \aleph_0 = \aleph_0, \therefore \card(B \cup D) = \aleph_0$ + + (6) $\card(\snatural_\snatural) = \aleph_0^{\aleph_0} = 2^{\aleph_0} = \aleph_1$ + + (7) $\card(\sreal_\sreal) = \aleph_1 ^ {\aleph_1} = 2^{\aleph_1} = \aleph_2$ + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh2.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh2.tex" new file mode 100644 index 000000000..e18ebd3dd --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh2.tex" @@ -0,0 +1,155 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} + +\title{离散数学——第三周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.09.27} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} +\linespread{0.7} +\begin{document} + +\maketitle + +\paragraph{1.6}\label{1.6} +将下列公式写成波兰式和逆波兰式。\\ + +(1) $ P \rightarrow Q \lor R \lor S $ \\ + +(2) $ P \land \lnot R \leftrightarrow P \lor Q $ \\ + +(3) $ \lnot \lnot P \lor (W \land R) \lor \lnot Q $ + +\paragraph{解} + +(1) 波兰式 $ \rightarrow P \lor \lor QRS $ ,逆波兰式 $ PQR \lor S \lor \rightarrow $。\\ + +(2) 波兰式 $ \leftrightarrow \land P \lnot R \lor PQ $ ,逆波兰式 $ PR \lnot \land PQ \lor \leftrightarrow $。\\ + +(3) 波兰式 $ \lor \lor \lnot \lnot P \land WR \lnot Q $ ,逆波兰式 $ P \lnot \lnot WR \land \lor Q \lnot \lor $。 \\ + +\paragraph{2.1}\label{2.1} +证明下列等值公式。\\ + +(1) $ P \rightarrow (Q \land R)=(P \rightarrow Q) \land (P \rightarrow R) $ \\ + +(3) $ ((P \rightarrow \lnot Q) \rightarrow (Q \rightarrow \lnot P)) \land R=R $ \\ + +(5) $ P \rightarrow (Q \rightarrow R)=(P \land Q) \rightarrow R $ + +\paragraph{证} + +(1) +\begin{align*} +P \rightarrow (Q \land R) &= \lnot P \lor (Q \land R) \\ +&=( \lnot P \lor Q) \land ( \lnot P \lor R) \\ &= (P \rightarrow Q) \land (P \rightarrow R) +\end{align*} + +(3) +\begin{align*} +((P \rightarrow \lnot Q) \rightarrow (Q \rightarrow \lnot P)) \land R &=(( \lnot P \lor \lnot Q) \rightarrow ( \lnot Q \lor \lnot P)) \land R \\ &=R +\end{align*} + +(5) +\begin{align*} +P \rightarrow (Q \rightarrow R)&=P \rightarrow ( \lnot Q \lor R) \\ &= \lnot P \lor ( \lnot Q \lor R) \\ &=( \lnot P \lor \lnot Q) \lor R \\ &= \lnot (P \land Q) \lor R \\ &=(P \land Q) \rightarrow R +\end{align*} + +\paragraph{2.2}\label{2.2} +由下列真值表,分别从$\true$和$\false$来列写出$A, B$和$C$的表达式,并分别以符号$m_i$和$M_i$表示。 + +\begin{table}[!htb] +\centering +\begin{tabular}{ccccc} +\hline +$P$ & $Q$ & $A$ & $B$ & $C$ \\ +\hline +$\false$ & $\false$ & $\true$ & $\true$ & $\true$ \\ +$\false$ & $\true$ & $\true$ & $\false$ & $\false$ \\ +$\true$ & $\false$ & $\true$ & $\false$ & $\false$ \\ +$\true$ & $\true$ & $\false$ & $\true$ & $\false$ \\ +\hline +\end{tabular} +\end{table} + +\paragraph{解} + +\begin{align*} +A + & = ( \lnot P \land \lnot Q) \lor ( \lnot P \land Q) \lor (P \land \lnot Q) \\ + & = m_0 \lor m_1 \lor m_2 \\ + & = ( \lnot Q \lor \lnot P) \\ + & = M_0 \\ +B + & = ( \lnot P \land \lnot Q) \lor (P \land Q) \\ + & = m_0 \lor m_3 \\ + & = (P \lor \lnot Q) \land ( \lnot P \lor Q) \\ + & = M_1 \land M_2 \\ +C + & = ( \lnot P \land \lnot Q) \\ + & = m_0 \\ + & = (P \lor \lnot Q) \land ( \lnot P \lor Q) \land ( \lnot P \lor \lnot Q) \\ + & =M_0 \land M_1 \land M_2 +\end{align*} + +\paragraph{2.3}\label{2.3} +用$\uparrow$和$\downarrow$分别表示出$ \lnot , \land , \lor , \rightarrow $和$\leftrightarrow $。 + +\paragraph{解} + +(1) 用$\uparrow$ +\begin{align*} +\lnot P &= \lnot (P \land P) \\&= P \uparrow P \\ +P \land Q &= \lnot ( \lnot (P \land Q)) \\&= \lnot (P \uparrow Q) \\&= (P \uparrow Q) \uparrow (P \uparrow Q) \\ +P \lor Q &= \lnot ( \lnot P \land \lnot Q) \\&= \lnot P \uparrow \lnot Q \\&= (P \uparrow P) \uparrow (Q \uparrow Q) \\ +P \rightarrow Q &= \lnot P \lor Q \\&= ( \lnot P \uparrow \lnot P) \uparrow (Q \uparrow Q) \\&= P \uparrow (Q \uparrow Q) \\ +P \leftrightarrow Q &= (P \land Q) \lor ( \lnot P \land \lnot Q) \\&= \lnot \lnot ((P \land Q) \lor ( \lnot P \land \lnot Q)) \\&= \lnot ( \lnot (P \land Q) \lor \lnot ( \lnot P \land \lnot Q)) \\&= \lnot (P \uparrow Q) \land ( \lnot P \uparrow Q) \\&= (P \uparrow Q) \uparrow ( \lnot P \uparrow \lnot Q) \\&= (P \uparrow Q) \uparrow ((P \uparrow P) \uparrow (Q \uparrow Q)) +\end{align*} +(2) 用$\downarrow$ +\begin{align*} +\lnot P &= \lnot (P \lor P) \\&= P \downarrow P \\ +P \land Q &= \lnot ( \lnot P \lor \lnot Q) \\&= \lnot P \downarrow \lnot Q \\&= (P \downarrow P) \downarrow (Q \downarrow Q) \\ +P \lor Q &= \lnot \lnot (P \lor Q) \\&= \lnot (P \downarrow Q) \\&= (P \downarrow Q) \downarrow (P \downarrow Q) \\ +P \rightarrow Q &= \lnot P \lor Q \\&= ( \lnot P \downarrow Q) \downarrow ( \lnot P \downarrow Q) \\&= =((P \downarrow P) \downarrow Q) \downarrow ((P \downarrow P) \downarrow Q) \\ +P \leftrightarrow Q &= (P \land Q) \lor ( \lnot P \land \lnot Q) \\&= ( \lnot ( \lnot P \lor \lnot Q) \lor ( \lnot (P \lor Q)) \\&= ( \lnot P \downarrow \lnot Q) \lor (P \downarrow Q) \\&= \lnot \lnot (( \lnot P \downarrow Q) \lor (P \downarrow Q)) \\&= \lnot (( \lnot P \downarrow \lnot Q) \downarrow (P \downarrow Q)) \\&= \lnot (((P \downarrow P) \downarrow (Q \downarrow Q)) \downarrow (P \downarrow Q)) \\&= (((P \downarrow P) \downarrow (Q \downarrow Q)) \downarrow (P \downarrow Q)) \downarrow (((P \downarrow P) \downarrow (Q \downarrow Q)) \downarrow (P \downarrow Q)) +\end{align*} + +\paragraph{2.4}\label{2.4} +证明 \\ + +(1) $A\rightarrow B$与$B^*\rightarrow A^*$同永真、同可满足。\\ + +(2) $A\leftrightarrow B$与$A^*\leftrightarrow B^*$同永真、同可满足。\\ + +\paragraph{证} + +(1) +$$ +A\rightarrow B=\lnot B \rightarrow \lnot A = B^{*-} \rightarrow A^{*-} +$$ + +而$B^{*-} \rightarrow A^{*-}$与$B^* \rightarrow A^*$同永真、同可满足。\\ + +$\therefore$$A\rightarrow B$与$B^* \rightarrow A^*$同永真、同可满足。\\ + +(2) +\begin{align*} +A\leftrightarrow B &= \lnot \lnot (A \leftrightarrow B) \\&= \lnot (A \leftrightarrow \lnot B) \\&= ( \lnot A) \leftrightarrow ( \lnot B) \\ &= (A^{*-})\leftrightarrow (B^{*-}) +\end{align*} + +而$(A^{*-})\leftrightarrow (B^{*-})$与$A^* \leftrightarrow B^*$同永真、同可满足。\\ + +$\therefore$$A\leftrightarrow B$与$A^* \leftrightarrow B^*$同永真、同可满足。 + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh3.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh3.tex" new file mode 100644 index 000000000..378bccd20 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh3.tex" @@ -0,0 +1,212 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} + +\title{离散数学——第四周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.09.28} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} +\linespread{0.7} +\begin{document} + +\maketitle + +\paragraph{2.5}\label{2.5} +给出下列公式的合取范式、析取范式、主合取范式和主析取范式。并给出所有使公式为真的解释。\\ + +(3) $ ( \lnot P \lor \lnot Q) \rightarrow (P \leftrightarrow \lnot Q) $ \\ + +(5) $ P \land (Q \lor ( \lnot P \land R)) $ \\ + +(7) $ P \rightarrow (Q \land ( \lnot P \leftrightarrow Q)) $ \\ + +\paragraph{解} + +(3) +\begin{align*} +( \lnot P \lor \lnot Q) \rightarrow (P \leftrightarrow \lnot Q) + &= \lnot ( \lnot P \lor \lnot Q) \lor (P \land \lnot Q) \lor ( \lnot P \land Q) \\ + &= (P \land Q) \lor (P \land \lnot Q) \lor ( \lnot P \land Q) & \text{(析取范式)} \\ + &= \lor_{1;2;3} & \text{(主析取范式)} \\ +( \lnot P \lor \lnot Q) \rightarrow (P \leftrightarrow \lnot Q) + &= \lnot ( \lnot P \lor \lnot Q) \lor ((P \lor Q) \land ( \lnot P \lor \lnot Q)) \\ + &= (P \land Q) \lor (( \lnot P \lor \lnot Q) \land (P \lor Q)) \\ + &= P \lor Q & \text{(合取范式)}\\ + &= \land _3 & \text{(主合取范式)} +\end{align*} + +该式在$P=\true$或$Q=\true$时为真。\\ + +(5) +\begin{align*} +P \land (Q \lor ( \lnot P \land R)) + &= P \land (Q \lor \lnot P) \land (Q \lor R) & \text{(合取范式)} \\ + &= (P \lor (Q \land \lnot Q) \lor (R \land \lnot R)) \land (Q \lor \lnot P \lor (R \land \lnot R)) \\ + & \quad \land (Q \lor R \lor (P \land \lnot P)) \\ + &= (P \lor Q \lor R) \land (P \lor Q \lor \lnot R) \land (P \lor \lnot Q \lor R) \\ + & \quad \land (P \lor \lnot Q \lor \lnot R) \land ( \lnot P \lor Q \lor R) \land ( \lnot P \lor Q \lor \lnot R) \\ + & \quad \land (P \lor Q \lor R) \land ( \lnot P \lor Q \lor R) \\ + &= \land _{2;3;4;5;6;7} & \text{(主合取范式)}\\ + &= \lor _{6;7} & \text{(主析取范式)}\\ + &= (P \land Q) & \text{(析取范式)} +\end{align*} + +该式在$P=\true$且$Q=\true$时为真。\\ + +(7) +\begin{align*} +P \rightarrow (Q \land ( \lnot P \leftrightarrow Q)) + &= \lnot P \lor (Q \land (( \lnot P \land Q) \lor (P \land \lnot Q))) \\ + &= \lnot P \lor (Q \land \lnot P \land Q) \lor (Q \land P \land \lnot Q) \\ + &= \lnot P \lor (Q \land \lnot P) \\ + &= \lnot P & \text{(合取范式、析取范式)} \\ + &= \lor _{0;1} & \text{(主析取范式)} \\ + &= \land _{0;1} & \text{(主合取范式)} +\end{align*} + +该式在$P=\false$时为真。\\ + +\paragraph{2.6}\label{2.6} +分别以$A \to B$永真, $A\land \lnot B$永假以及解释法来证明下列各重言蕴含式$A\Rightarrow B$。\\ + +\paragraph{证} + +(2) $ (P \rightarrow (Q \rightarrow R)) \Rightarrow (P \rightarrow Q) \rightarrow (P \rightarrow R) $ \\ + +\quad (a) $A \to B$永真 +$$ (P \rightarrow (Q \rightarrow R)) \rightarrow ((P \rightarrow Q) \rightarrow (P \rightarrow R)) = (P \rightarrow (Q \rightarrow R)) \rightarrow (P \rightarrow (Q \rightarrow R)) = \true $$ + +\quad (b) $A\land \lnot B$永假 \\ +$$ (P \rightarrow (Q \rightarrow R)) \land \lnot ((P \rightarrow Q) \rightarrow (P \rightarrow R)) = (P \rightarrow (Q \rightarrow R)) \land \lnot (P \rightarrow (Q \rightarrow R)) = \false $$ + +\quad (c) 解释法 \\ +$$P \rightarrow (Q \rightarrow R) = \true $$ + +\qquad 若$P=\true$则$P \rightarrow Q = \true , (P \rightarrow R) = \true , (P \rightarrow Q) \rightarrow (P \rightarrow R) = \true $。 + +\qquad 若$P=\false$则$(P \rightarrow Q) \rightarrow (P \rightarrow R)= \true $。 + +\qquad 综上,蕴含式成立。\\ + +(4) $ (P \land Q) \rightarrow R \Rightarrow P \rightarrow (Q \rightarrow R) $ \\ + +\quad (a) $A \to B$永真 +\begin{align*} +((P \land Q) \rightarrow R) \rightarrow (P \rightarrow (Q \rightarrow R)) + &= ( \lnot (P \land Q) \lor R) \rightarrow ( \lnot P \lor ( \lnot Q \lor R)) \\ + &= ( \lnot P \lor \lnot Q \lor R) \rightarrow ( \lnot P \lor \lnot Q \lor R) \\ + &= \true +\end{align*} + +\quad (b) $A\land \lnot B$永假 \\ +\begin{align*} +((P \land Q) \rightarrow R) \land \lnot (P \rightarrow (Q \rightarrow R)) + &= ( \lnot P \lor \lnot Q \lor R) \land \lnot ( \lnot P \lor \lnot Q \lor R) \\ + &= \false +\end{align*} + +\quad (c) 解释法 \\ +\begin{flalign*} +& (P \land Q) \rightarrow R= \true \\ +& \therefore \lnot (P \land Q) \lor R= \true \\ +& \therefore P \rightarrow ( \lnot Q \lor R) = \true \\ +& \therefore P \rightarrow (Q \rightarrow R) = \true +\end{flalign*} + +\paragraph{2.7}\label{2.7} +判断下列推理式是否正确?\\ + +(9) $ (P \land Q) \rightarrow R \Rightarrow (P \rightarrow R) \land (Q \rightarrow R) $ \\ + +(11) $ P \rightarrow Q \Rightarrow (P \rightarrow R) \rightarrow (Q \rightarrow R) $ \\ + +(13) $ \lnot (P \rightarrow Q) \land (Q \rightarrow P) \Rightarrow P \land \lnot Q $ \\ + +(15) $ (P \rightarrow Q) \land (R \rightarrow Q) \land (S \rightarrow Q) \Rightarrow (P \land R \land \lnot S \rightarrow Q) $ \\ + +\paragraph{解} + +(9) +\begin{align*} +((P \land Q) \rightarrow R) \rightarrow ((P \rightarrow R) \land (Q \rightarrow R)) + &= \lnot ( \lnot (P \land Q) \lor R) \lor (( \lnot P \lor R) \land ( \lnot Q \lor R)) \\ + &= \lnot ( \lnot P \lor \lnot Q \lor R) \lor ( \lnot P \land Q) \lor R \\ + &= (P \land Q \land \lnot R) \lor R \lor ( \lnot P \land \lnot Q) +\end{align*} + +当$R=\false, P=\true, Q=\false$时,上式为$\false$,则推理不正确。\\ + +(11) +\begin{align*} +(P \rightarrow Q) \rightarrow ((P \rightarrow R) \rightarrow (Q \rightarrow R)) + &= \lnot ( \lnot P \lor Q) \lor ( \lnot ( \lnot P \lor R) \lor ( \lnot Q \lor R)) \\ + &= (P \land \lnot Q) \lor (P \land \lnot R) \lor ( \lnot Q \lor R) \\ + &= (P \land \lnot Q) \lor \lnot Q \lor R \lor (P \land \lnot R) \\ + &= \lnot Q \lor R \lor (P \land \lnot R) +\end{align*} + +当$P=\false, Q=\true, R=\false$时,上式为$\false$,则推理不正确。\\ + +(13) +设 $\lnot (P \rightarrow Q) \land (Q \rightarrow P)= \true $ + +\qquad 则 $P \rightarrow Q= \false , Q \rightarrow P= \true $ + +\qquad 由 $P \rightarrow Q= \false$ 知,$P= \true , Q= \false $ + +\qquad 则 $P\land \lnot Q=\true$,推理正确。\\ + +(15) 只要验证 $(P \rightarrow Q) \land (R \rightarrow Q) \land (S \rightarrow Q) \land (P \land R \land \lnot S) \Rightarrow Q$ 即可。 + +\qquad $P \rightarrow Q = \true, R \rightarrow Q = \true, S \rightarrow Q = \true, P \land R \land \lnot S = \true$ 即 $P=R=\true, S=\false$ + +\qquad 由 $P \rightarrow Q = \true, P = \true$ 知 $Q = \true$,从而推理正确。 + +\paragraph{2.8}\label{2.8} +使用推理规则证明 \\ + +(4) $ P \lor Q \rightarrow R \land S, S \lor E \rightarrow U \Rightarrow P \rightarrow U $ \\ + +(6) $ \lnot Q \lor S, (E \rightarrow \lnot U) \rightarrow \lnot S \Rightarrow Q \rightarrow E $ \\ + +\paragraph{证} + +(4) +\begin{align*} +(a) & (P \lor Q) \rightarrow (R \land S) & \text{前提引入} \\ +(b) & (S \lor E) \rightarrow U & \text{前提引入} \\ +(c) & P & \text{附加前提引入} \\ +(d) & P \lor Q & (c)\text{置换} \\ +(e) & R \land S & (a)(d)\text{分离} \\ +(f) & S & (e)\text{置换} \\ +(g) & S \lor E & (f)\text{置换} \\ +(h) & U & (b)(g)\text{分离} \\ +(i) & P \rightarrow U & \text{条件证明规则} +\end{align*} + +(6) +\begin{align*} +(a) & \lnot (Q \lor S) & \text{前提引入} \\ +(b) & (E \rightarrow \lnot U) \rightarrow \lnot S & \text{前提引入} \\ +(c) & Q & \text{附加前提引入} \\ +(d) & Q \rightarrow S & (a)\text{置换} \\ +(e) & S & (c)(d)\text{分离} \\ +(f) & S \rightarrow \lnot (E \rightarrow \lnot U) & (b)\text{置换} \\ +(g) & \lnot (E \rightarrow \lnot U) & (e)(f)\text{分离} \\ +(h) & E \land U & (g)\text{置换} \\ +(i) & E & (h) \\ +(j) & Q \rightarrow E & \text{条件证明规则} +\end{align*} + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh4.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh4.tex" new file mode 100644 index 000000000..266869011 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh4.tex" @@ -0,0 +1,196 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\zhyou}{\text{由}} +\newcommand{\zhfenli}{\text{分离}} +\newcommand{\zhguijie}{\text{归结}} +\newcommand{\zhqianti}{\text{前提}} +\newcommand{\zhzhihuan}{\text{置换}} +\newcommand{\zhdairu}{\text{代入}} +\newcommand{\zherqie}{\text{而且}} +\newcommand{\zhaxiom}{\text{公理}} +\newcommand{\zhtheorem}{\text{定理}} +\newcommand{\zhdefinition}{\text{定义}} + +\title{离散数学——第五周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.10.11} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} +\linespread{0.7} +\begin{document} + +\maketitle + +\paragraph{2.9}\label{2.9} +证明下列推理关系:\\ + +(1) 在大城市球赛中,如果北京队第三,那么如果上海队第二,那么天津队第四。沈阳队不是第一或北京队第三。上海队第二。从而知,如果沈阳队第一,那么天津队第四。\\ + +(2) 如果国家不对农产品给与补贴,那么国家就要对农产品进行控制。如果对农产品进行控制,农产品就不会短缺。或者农产品短缺或者农产品过剩。事实上农产品不过剩。从而国家对农产品给与了补贴。 + +\paragraph{解} + +(1) $P$:北京队第三;$Q$:上海队第二;$R$:天津队第四;$S$:沈阳队第一。\\ + +\quad 欲证:$ P \rightarrow (Q \rightarrow R), \lnot S \lor P, Q \Rightarrow S \rightarrow R $ + +\begin{align*} +(a) & \lnot S \lor P & \text{前提} \\ +(b) & S \to P & (a)\text{置换} \\ +(c) & S & \text{附加前提} \\ +(d) & P & (b)(c)\text{分离} \\ +(e) & P \to (Q \to R) & \text{前提} \\ +(f) & Q \to R & (d)(e)\text{分离} \\ +(g) & Q & \text{前提} \\ +(h) & R & (f)(g)\text{分离} \\ +(i) & S \rightarrow R & \text{条件证明规则} +\end{align*} + +(2) $P$:国家对农产品给与补贴;$Q$:国家对农产品进行控制;$R$:农产品短缺;$S$:农产品过剩。\\ + +\quad 欲证:$ \lnot P \rightarrow Q, Q \rightarrow \lnot R, R \leftrightarrow \lnot S, \lnot S \Rightarrow P $ + +\begin{align*} +(a) & \lnot S & \text{前提} \\ +(b) & R \leftrightarrow \lnot S & \text{前提} \\ +(c) & R & (a)(b)\text{分离} \\ +(d) & Q \to \lnot R & \text{前提} \\ +(e) & R \to \lnot Q & (d)\text{置换} \\ +(f) & \lnot Q & (c)(e)\text{分离} \\ +(g) & \lnot P \to Q & \text{前提} \\ +(h) & \lnot Q \to P & (g)\text{置换} \\ +(i) & P & (f)(h)\text{分离} +\end{align*} + +\paragraph{2.10}\label{2.10} +如果合同是有效的,那么张三应受罚。如果张三受罚,他将破产。如果银行给张三贷款,他就不会破产。事实上,合同有效并且银行给张三贷款了。验证这些前提是否有矛盾。 + +\paragraph{解} +$P$:合同有效;$Q$:张三受罚;$R$:张三破产;$S$:银行给张三贷款。\\ + +条件:$ P \rightarrow Q, Q \rightarrow R, S \rightarrow \lnot R, P \land S $ + +\begin{align*} +(a) & P \land S & \text{前提} \\ +(b) & P & \text{由}(a) \\ +(c) & P \to Q & \text{前提} \\ +(d) & Q & (b)(c)\text{分离} \\ +(e) & Q \to R & \text{前提} \\ +(f) & R & (d)(e)\text{分离} \\ +(g) & S & \text{由}(a) \\ +(h) & S \to \lnot R & \text{前提} \\ +(i) & \lnot R & (g)(h)\text{分离} \\ +(j) & \square & \text{由}(f)(i) +\end{align*} + +\paragraph{2.12}\label{2.12} +利用归结法证明:\\ + +(1) $ (P \lor Q) \land (P \rightarrow R) \land (Q \rightarrow R) \Rightarrow R $ \\ + +(3) $ \lnot (P \land \lnot Q) \land ( \lnot Q \lor R) \land \lnot R \Rightarrow \lnot P $ + +\paragraph{证} + +(1) 子句集 $ \{ P \lor Q, P \rightarrow R, Q \rightarrow R, \lnot R \} $ +\begin{align*} +(a) & P \lor Q \\ +(b) & P \to R \\ +(c) & Q \to R \\ +(d) & \lnot R \\ +(e) & \lnot P \lor R & (b)\text{置换} \\ +(f) & Q \lor R & (a)(e)\text{归结} \\ +(g) & \lnot Q \lor R & (c)\text{置换} \\ +(h) & R \lor R & (f)(g)\text{归结} \\ +(i) & R & (h)\text{置换} \\ +(j) & \square & (d)(i)\text{归结} +\end{align*} + +(3)子句集 $ \{ \lnot P \lor Q, \lnot Q \lor R, \lnot R, P \} $ +\begin{align*} +(a) & \lnot P \lor Q \\ +(b) & \lnot Q \lor R \\ +(c) & \lnot R \\ +(d) & P \\ +(e) & \lnot P \lor R & (a)(b)\text{归结} \\ +(f) & \lnot P & (c)(e)\text{归结} \\ +(g) & \square & (d)(f)\text{归结} +\end{align*} + +\paragraph{2.11}\label{2.11} +若 $ P_i \rightarrow Q_i(i=1, \cdots, n) $ 为真,$ P_1 \lor P_2 \lor \cdots \lor P_n $ 和 $ \lnot (Q_i \land Q_j) (i \neq j) $ 也为真。试证明必有 $ Q_i \rightarrow P_i (i=1, \cdots, n) $ 为真。 + +\paragraph{证} +\begin{align*} +(a) & \lnot (Q_i \land Q_j) & \zhqianti \\ +(b) & Q_i \to \lnot Q_j & (a)\zhzhihuan \\ +(c) & P_i \to Q_i & \zhqianti \\ +(d) & P_i \to \lnot Q_j & (b)(c)\zhfenli \\ +(e) & \lnot Q_i \to \lnot P_i & (c)\zhzhihuan \\ +(f) & P_i \to \lnot P_j & (d)(e)\zhfenli \\ +(g) & P_i \to (\lnot P_1 \land \lnot P_2 \cdots \land \lnot P_{i-1} \land \lnot P_{i+1} \cdots \land \lnot P_n) & \zhyou (f) \\ +(h) & \lnot P_i \lor (\lnot P_1 \land \lnot P_2 \cdots \land \lnot P_{i-1} \land \lnot P_{i+1} \cdots \land \lnot P_n) & (g)\zhzhihuan \\ +(i) & P_1 \lor P_2 \lor \cdots \lor P_n & \zhqianti \\ +(j) & P_i \lor (P_1 \lor P_2 \lor \cdots \lor P_{i-1} \lor P_{i+1} \lor \cdots \lor P_n) & (i)\zhzhihuan \\ +(k) & \lnot (P_1 \lor P_2 \lor \cdots \lor P_{i-1} \lor P_{i+1} \lor \cdots \lor P_n) \to P_i & (j)\zhzhihuan \\ +(l) & (\lnot P_1 \land \lnot P_2 \cdots \land \lnot P_{i-1} \land \lnot P_{i+1} \cdots \land \lnot P_n) \to P_i & (k)\zhzhihuan \\ +(m) & Q_i \to (\lnot Q_1 \land \lnot Q_2 \cdots \land \lnot Q_{i-1} \land \lnot Q_{i+1} \land \cdots \land Q_n) & \zhyou (b) \\ +(n) & Q_i \to (\lnot P_1 \land \lnot P_2 \cdots \land \lnot P_{i-1} \land \lnot P_{i+1} \cdots \land \lnot P_n) & \zhyou (e)(m)\text{三段论} \\ +(o) & Q_i \to P_i & \zhyou (l)(m)\text{三段论} +\end{align*} + +\paragraph{3.1}\label{3.1} +依公理系统证明 \\ + +(1) $ \vdash \lnot (P \land Q) \rightarrow ( \lnot P \lor \lnot Q) $ \\ + +(3) $ \vdash P \rightarrow (Q \lor P) $ + +\paragraph{证} + +(1) +\begin{align*} +(a) & \vdash \lnot \lnot P \rightarrow P & \zhtheorem 3.2.5 \\ +(b) & \vdash \lnot \lnot ( \lnot P \lor \lnot Q) \rightarrow ( \lnot P \lor \lnot Q) & \zhdairu \frac{P}{ \lnot P \lor \lnot Q} \\ +(c) & \vdash \lnot (P \land Q) \rightarrow ( \lnot P \lor \lnot Q) & \zhdefinition (2) +\end{align*} + +(3) +\begin{align*} +(a) & \vdash (Q \rightarrow R) \rightarrow ((P \rightarrow Q) \rightarrow (P \rightarrow R)) & \zhtheorem 3.2.1 \\ +(b) & \vdash ((P \lor Q) \rightarrow (Q \lor P)) \rightarrow ((P \rightarrow (P \lor Q) \rightarrow (P \rightarrow (Q \lor P))) & \zhdairu \frac{Q}{(P \lor Q)},\frac{R}{(Q \lor P)} \\ +(c) & \vdash ((P \lor Q)) \rightarrow (Q \lor P)) & \zhaxiom 3 \\ +(d) & \vdash ((P \rightarrow (P \lor Q) \rightarrow (P \rightarrow (Q \lor P)) & (b)(c) \zhfenli \\ +(e) & \vdash (P \rightarrow (P \lor Q)) & \zhaxiom 2 \\ +(f) & \vdash P \rightarrow (Q \lor P) & (d)(e) \zhfenli +\end{align*} + + +\paragraph{3.2}\label{3.2} +依王浩算法判断下述蕴含式是否正确 \\ + +(1) $ \lnot Q \land (P \rightarrow Q) \Rightarrow \lnot P $ + +\paragraph{解} +\begin{align*} +(a) & \lnot Q \land (P \rightarrow Q) \overset{\rm s}{\Rightarrow} \lnot P & (\text{写成相继式}) \\ +(b) & \lnot Q, (P \rightarrow Q) \overset{\rm s}{\Rightarrow} \lnot P & ( \land \Rightarrow ) \\ +(c) & Q, \lnot Q \overset{\rm s}{\Rightarrow} \lnot P \zherqie \lnot Q \overset{\rm s}{\Rightarrow} P, \lnot P & ( \rightarrow \Rightarrow ) \\ +(d) & Q \overset{\rm s}{\Rightarrow} \lnot P, Q \zherqie \overset{\rm s}{\Rightarrow} P, \lnot P, Q & ( \lnot \Rightarrow ) \\ +(e) & P, Q \overset{\rm s}{\Rightarrow} Q \zherqie P \overset{\rm s}{\Rightarrow} P, Q & ( \Rightarrow \lnot ) +\end{align*} + +上式无联结词而且 $ \overset{\rm s}{\Rightarrow} $ 两端均有共同的命题变项,从而是公理,则原蕴含式正确。 + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh5.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh5.tex" new file mode 100644 index 000000000..0656babc7 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh5.tex" @@ -0,0 +1,176 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\zhyou}{\text{由}} +\newcommand{\zhfenli}{\text{分离}} +\newcommand{\zhguijie}{\text{归结}} +\newcommand{\zhqianti}{\text{前提}} +\newcommand{\zhzhihuan}{\text{置换}} +\newcommand{\zhdairu}{\text{代入}} +\newcommand{\zherqie}{\text{而且}} +\newcommand{\zhaxiom}{\text{公理}} +\newcommand{\zhtheorem}{\text{定理}} +\newcommand{\zhdefinition}{\text{定义}} + +\title{离散数学——第六周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.10.18} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{3.3}\label{3.3} +依自然演绎系统证明 + +(3) $ A \rightarrow B, A \rightarrow \lnot B \vdash \lnot A $ + +\paragraph{证} (3) +\begin{align*} +(a) & A \rightarrow B, A \rightarrow \lnot B, A \vdash A & (\text{肯定前提}) \\ +(b) & A \rightarrow B, A \rightarrow \lnot B, A \vdash A \rightarrow B & (\text{肯定前提}) \\ +(c) & A \rightarrow B, A \rightarrow \lnot B, A \vdash A \rightarrow \lnot B & (\text{肯定前提}) \\ +(d) & A \rightarrow B, A \vdash B & (\text{分离规则}) \\ +(e) & A \rightarrow \lnot B, A \vdash \lnot B & (\text{分离规则}) \\ +(f) & A \rightarrow B, A \rightarrow \lnot B, A \vdash B & ((a)(b)(d)\text{传递律}) \\ +(g) & A \rightarrow B, A \rightarrow \lnot B, A \vdash \lnot B & ((a)(c)(e)\text{传递律}) \\ +(h) & A \rightarrow B, A \rightarrow \lnot B \vdash \lnot A & ((f)(g)\text{反证律}) +\end{align*} + +\paragraph{4.5}\label{4.5} +将下列语句符号化 + +(2) 凡有理数都可写为分数。 + +(5) 过平面上两个点,有且仅有一条直线通过。 + +(6) 凡实数都能比较大小。 + +(7) 在北京工作的人未必都是北京人。 + +(8) 只有一个北京。 + +(10) 如果明天天气好,有些学生将去香山。 + +\paragraph{答} + +(2) $P(x): $ $x$是有理数;$Q(x): $ $x$可写成分数。 + +语句可化为$ ( \forall x)(P(x) \rightarrow Q(x))$。 + +(5) $P(x): $ $x$是平面上的点;$Q(x): $ $x$是直线;$R(x,y): $ $x$通过$y$。 + +语句可化为$ ( \forall x) ( \forall y)((P(x) \land P(y) \land \lnot E(x,y))$ + +(6) $P(x): $ $x$为实数;$Q(x,y): $ $x,y$能比较大小。 + +语句可化为$ ( \forall x) ( \forall y)((P(x) \land P(y)) \rightarrow Q(x,y))$ + +(7) $P(x): $ $x$是人;$Q(x): $ $x$在北京工作;$R(x): $ $x$是北京人。 + +语句可化为$ \lnot ( \forall x)(P(x) \land Q(x) \rightarrow R(x))$ + +(8) $P(x):$ $x$是北京;$E(x,y): $ $x,y$相同。 + +语句可化为$ ( \exists x)(P(x) \land ( \forall y)(P(x) \rightarrow E(x,y)))$ + +(10) $P: $ 明天天气好;$Q(x): $ $x$是学生;$R(x): $ $x$去香山。 + +语句可化为$ P \rightarrow ( \exists x)(Q(x) \land R(x))$ + +\paragraph{4.6}\label{4.6} +设$P(x)$表示x是有理数,$Q(x)$表示$x$是实数,$R(x)$表示$x$是无理数,$L(x)$表示$x$是正整数,$S(x)$表示$x$是偶数,$W(x)$表示$x$是奇数,试将下列公式翻译成自然语句。 + +(3) $ \lnot ( \forall x )(Q(x) \rightarrow P(x)) $ + +(6) $( \forall x ) (L(x) \rightarrow S(x) \bar \lor W(x)))$ + +(9) $ ( \forall x ) (L(x) \rightarrow P(x)) \land \lnot ( \forall x) (P(x) \rightarrow I(x)) $ + +\paragraph{答} +(3) 不是所有实数都是有理数。 + +(6) 任意正整数不是偶数就是奇数。 + +(9) 任意正整数都是有理数,但并非所有有理数都是正整数。 + +\paragraph{4.7}\label{4.7} +设个体域为$ \{ a, b, c \} $,试将下列公式写成命题逻辑公式 + +(3) $( \forall x )P(x) \land ( \exists x ) Q(x) $ + +(4) $( \forall x ) (P(x) \rightarrow Q(x))$ + +(5) $( \forall x ) \lnot P(x) \lor ( \forall x ) P(x)$ + +\paragraph{答} +(3) $ P(a) \land P(b) \land P(c) \land (Q(a) \lor Q(b) \lor Q(c)) $ + +(4) $ (P(a) \rightarrow Q(a)) \land (P(b) \rightarrow Q(b)) \land (P(c) \rightarrow Q(c)) $ + +(5) $ ( \lnot P(a) \lor P(a)) \land ( \lnot P(b) \lor P(b)) \land ( \lnot P(c) \lor P(c)) $ + +\paragraph{4.8}\label{4.8} +判断下列公式是普遍有效的,不可满足的还是可满足的? + +(1) $ ( \forall x)P(x) \rightarrow P(y) $ + +(2) $ ( \exists x)(P(x) \land Q(x)) \rightarrow ( ( \exists x)P(x) \land ( \exists x)Q(x)) $ + +(3) $ ( \forall x)P(x) $ + +(4) $ ( \exists x)(P(x) \land \lnot P(x)) $ + +(5) $ ( \forall x)(P(x) \rightarrow Q(x)) $ + +(6) $ ( \forall x)(P(x) \lor \lnot P(x)) $ + +(7) $ ( ( \exists x)P(x) \land ( \exists x)Q(x)) \rightarrow ( \exists x)(P(x) \land Q(x)) $ + +\paragraph{答} (1) 普遍有效; (2) 普遍有效; (3) 可满足; (4) 不可满足; (5) 可满足; (6) 普遍有效; (7) 可满足。 + +\paragraph{4.9}\label{4.9} +给出一个公式,使其在$ \{ 1, 2 \} $上是可满足的,而在$ \{ 1 \} $上是不可满足的。 + +\paragraph{答} $P(x): x=2$, 公式: $(\exists x)P(x)$ + +\paragraph{4.10}\label{4.10} +设个体域为$ \{ a,b \} $,并对$P(x,y)$设定为$P(a,a)=\true$, $P(a,b)=\false$, $P(b,a)=\false$, $P(b,b)=\true$,计算下列公式的真值。 + +(3) $ ( \forall x) ( \forall y)P(x,y) $ + +(5) $ ( \exists y) \lnot P(a,y) $ + +(7) $ ( \forall x) ( \forall y)(P(x,y) \rightarrow P(y,x)) $ + +\paragraph{解} +(3) $x=a, y=b, P(x,y)=\false$ + +$ \therefore (\forall x)(\forall y)P(x,y)=\false$ + +(5) $ y=b, \lnot P(a,y)=\true $ + +$\therefore (\exists y)\lnot P(a,y)=\true $ + +(7) +\begin{gather*} +x=a, y=a, P(x,y)=P(y,x)= \true , P(x,y) \rightarrow P(y,x)= \true \\ +x=a, y=b, P(x,y)=P(y,x)= \false , P(x,y) \rightarrow P(y,x)= \true \\ +x=b, y=a, P(x,y)=P(y,x)= \false , P(x,y) \rightarrow P(y,x)= \true \\ +x=b, y=b, P(x,y)=P(y,x)= \true , P(x,y) \rightarrow P(y,x)= \true \\ +\therefore ( \forall x) ( \forall y)(P(x,y) \rightarrow P(y,x))= \true +\end{gather*} + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh6.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh6.tex" new file mode 100644 index 000000000..eeb16e622 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh6.tex" @@ -0,0 +1,110 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\zhyou}{\text{由}} +\newcommand{\zhfenli}{\text{分离}} +\newcommand{\zhguijie}{\text{归结}} +\newcommand{\zhqianti}{\text{前提}} +\newcommand{\zhzhihuan}{\text{置换}} +\newcommand{\zhdairu}{\text{代入}} +\newcommand{\zherqie}{\text{而且}} +\newcommand{\zhaxiom}{\text{公理}} +\newcommand{\zhtheorem}{\text{定理}} +\newcommand{\zhdefinition}{\text{定义}} + +\title{离散数学——第七周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.10.23} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{5.1}\label{5.1} +证明下列等值式和蕴含式 + +(6) $ ( \exists x)(P(x) \rightarrow Q(x)) = (al x)P(x) \rightarrow ( \exists x)Q(x) $ + +(8) $ ( \exists x)P(x) \land ( \forall x)Q(x) \Rightarrow ( \exists x)(P(x) \lor Q(x)) $ + +(10) $ ( \exists z) ( \exists y) ( \exists x)((P(x,z) \rightarrow Q(x,z)) \lor ((R(y,z) \rightarrow S(y,z)) $ + +\qquad $ = ( ( \forall z) ( \forall x)P(x,z) \rightarrow ( \exists z) ( \exists x)Q(x,y)) \lor ( ( \forall z) ( \forall y)R(x,y) \rightarrow ( \exists z) ( \exists y)S(y,z)) $ + +\paragraph{证} +(6) +\begin{align*} +( \exists x)(P(x) \rightarrow Q(x)) +&= ( \exists x)(\lnot P(x) \lor Q(x)) +\\ &= ( \exists x)(\lnot P(x)) \lor ( \exists x)(Q(x)) +\\ &= \lnot ( \forall x)P(x) \lor ( \exists x)Q(x) +\\ &= ( \forall x)P(x) \rightarrow ( \exists x)Q(x) +\end{align*} + +(8) +\begin{align*} +( \exists x)P(x) \land ( \forall x)Q(x) +&= ( \exists x)P(x) \land ( \forall y)Q(y) +\\ &= ( \exists x)(P(x) \land ( \forall y)Q(y)) +\\ &\Rightarrow ( \exists x)(P(x) \land Q(x)) +\end{align*} + +(10) +\begin{align*} +& ( \exists z) ( \exists y) ( \exists x)((P(x,z) \rightarrow Q(x,z)) \lor ((R(y,z) \rightarrow S(y,z)) +\\ & = ( \exists z) ( \exists y) ( \exists x)((P(x,z) \rightarrow Q(x,z)) \lor ( \exists z) ( \exists y) ( \exists x)(R(y,z) \rightarrow S(y,z)) +\\ & = ( \exists z) ( \exists x)(P(x,z) \rightarrow Q(x,z)) \lor ( \exists z) ( \exists y)(R(y,z) \rightarrow S(y,z)) +\\ & = ( \exists z) ( \exists x)(\lnot P(x,z) \lor Q(x,z)) \lor ( \exists z) ( \exists y)(\lnot R(y,z) \lor S(y,z)) +\\ & = ( ( \exists z) ( \exists x) \lnot P(x,z) \lor ( \exists z) ( \exists x) Q(x,z)) \lor ( ( \exists z) ( \exists y) \lnot R(y,z) \lor ( \exists z) ( \exists y)S(y,z)) +\\ & = (\lnot ( \forall z) ( \forall x)P(x,z) \lor ( \exists z) ( \exists x)Q(x,z)) \lor (\lnot ( \forall z) ( \forall y) R(y,z) \lor ( \exists z) ( \exists y)S(y,z)) +\\ & = ( ( \forall z) ( \forall x)P(x,z) \rightarrow ( \exists z) ( \exists x)Q(x,y)) \lor ( ( \forall z) ( \forall y)R(x,y) \rightarrow ( \exists z) ( \exists y)S(y,z)) +\end{align*} + +\paragraph{5.2}\label{5.2} +判断下列各公式哪些是普遍有效的并给出证明,不是普遍有效的举出反例。 + +(3) $( ( \exists x)P(x) \rightarrow ( \forall x)Q(x)) \rightarrow )(\forall x)P(x) \rightarrow ( \forall x)Q(x)) $ + +(5) $ ( ( \exists x)P(x) \rightarrow ( \exists x)Q(x)) \rightarrow (\exists x)(P(x) \rightarrow Q(x)) $ + +(7) $ ( \exists x)P(x) \land ( \exists x)Q(x) \rightarrow ( \exists x)(P(x) \land Q(x)) $ + +\paragraph{解} +(3) +\begin{align*} +& ( ( \exists x)P(x) \rightarrow ( \forall x)Q(x)) \rightarrow ( \forall x)(P(x) \rightarrow Q(x)) +\\ &= (\lnot ( \exists x)P(x) \lor ( \forall x)(Q(x)) \rightarrow ( \forall x)(P(x) \rightarrow Q(x)) +\\ &= ( ( \forall x) \lnot P(x) \lor ( \forall x)Q(x)) \rightarrow ( \forall x)(P(x) \rightarrow Q(x)) +\\ &= ( ( \forall x) \lnot P(x) \lor ( \forall x)Q(x)) \rightarrow ( \forall x)(\lnot P(x) \lor Q(x)) +\\ &= \true +\end{align*} + +则原公式普遍有效。 + +(5) +\begin{align*} +& ( ( \forall x)P(x) \rightarrow ( \exists x)Q(x)) \rightarrow ( \exists x)(P(x) \rightarrow Q(x)) +\\ &= (\lnot ( \exists x)P(x) \lor ( \exists x)Q(x)) \rightarrow ( \exists x)(\lnot P(x) \lor Q(x)) +\\ &= ( ( \forall x) \lnot P(x) \lor ( \exists x)Q(x)) \rightarrow ( \exists x)(\lnot P(x) \lor Q(x)) +\\ &= ( ( \exists x) \lnot P(x) \rightarrow ( \exists x) \lnot P(x)) \lor ( \exists x)Q(x) +\\ &= \true \lor ( \exists x)Q(x) +\\ &= \true +\end{align*} + +则原公式普遍有效。 + +(7) 原公式并非普遍有效。例如在论域$\{1,2\}$上,令$P(1)=Q(2)=\true$, $P(2)=Q(1)=\false$, 则原公式真值为$\false$。 + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh7.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh7.tex" new file mode 100644 index 000000000..4845d3a29 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh7.tex" @@ -0,0 +1,114 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[hyperref, UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} + +\title{离散数学——第八周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.11.01} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{5.4}\label{5.4} +求下列(1)到(5)的前束范式,(6)到(8)的$\exists$前束范式,(9)(10)的Skolem范式(只含$\forall$)。 + +(4) $( \lnot ( \exists x)P(x) \lor ( \forall y)Q(y)) \rightarrow ( \forall z)R(z) $ + +(8) $ ( \forall x)(P(x) \rightarrow Q(x)) \rightarrow ( ( \exists x)P(x) \rightarrow ( \exists x)Q(x)) $ + +(9) $ ( \forall x)(P(x) \rightarrow ( \exists y)Q(x,y)) \lor ( \forall z)R(z) $ + +(10) $ ( \exists y) ( \forall y) ( \forall z) ( \exists u) ( \forall v)P(x,y,z,u,v) $ + +\paragraph{解} +(4) +\begin{align*} + & ( \lnot ( \exists x)P(x) \lor ( \forall y)Q(y)) \rightarrow ( \forall z)R(z) \\ + & = \lnot (\lnot ( \exists x)P(x) \lor ( \forall y)Q(y)) \lor ( \forall z)R(z) \\ + & = ( ( \exists x)P(x) \land ( \exists y) \lnot Q(y)) \lor ( \forall z)R(z) \\ + &= ( \exists x) ( \exists y) ( \forall z)(((P(x) \land \lnot Q(y)) \lor R(z)) +\end{align*} + +(8) 在普遍有效的意义下 +\begin{align*} + & ( \forall x)(P(x) \rightarrow Q(x)) \rightarrow ( ( \exists x)P(x) \rightarrow ( \exists x)Q(x)) \\ + &= ( \forall x)(\lnot P(x) \lor Q(x)) \rightarrow (\lnot ( \exists x)P(x) \lor ( \exists x)Q(x)) \\ + &= ( \exists x)(P(x) \land \lnot Q(x)) \lor ( \forall x) \lnot P(x) \lor ( \exists x)Q(x) \\ + &= ( \exists x) ( \exists z) ( \forall y)((P(x) \land \lnot Q(x)) \lor \lnot P(y) \lor Q(z)) +\end{align*} + +(9) 在不可满足的意义下 +\begin{align*} + & ( \forall x)(P(x) \rightarrow ( \exists y)Q(x,y)) \lor ( \forall z)R(z) \\ + &= ( \forall x) ( \exists y) ( \forall z)(\lnot P(x) \lor Q(x,y) \lor R(z)) \\ + &= ( \forall x) ( \forall z)(\lnot P(x) \lor Q(x,f(x)) \lor R(z)) +\end{align*} + +(10) 在不可满足的意义下 +\begin{align*} + & ( \exists y) ( \forall y) ( \forall z) ( \exists u) ( \forall v)P(x,y,z,u,v) \\ + &= ( \forall x) ( \forall z) ( \exists u) ( \forall v)P(x,a,z,u,v) \\ + &= ( \forall x) ( \forall z) ( \forall v)P(x,a,z,f(x,z),v) +\end{align*} + +\paragraph{5.5}\label{5.5} +使用推理规则和归结法作推理演算 + +(1) $ ( \forall x)(P(x) \lor Q(x)) \land ( \forall x)(Q(x) \rightarrow \lnot R(x)) \Rightarrow ( \exists x)(R(x) \rightarrow P(x)) $ + +(4) 大学里的学生不是本科生就是研究生,有的学生是高材生,John不是研究生但是高材生,从而如果John是学生必是本科生。 + +\paragraph{解} +(1) +\begin{align*} + (a) & ( \forall x)(P(x) \lor Q(x)) \land ( \forall x)(Q(x) \rightarrow \lnot R(x)) & \text{Premise} \\ + (b) & ( \forall x)(P(x) \lor Q(x)) & \text{Simplification of } (a) \\ + (c) & ( \forall x)(Q(x) \rightarrow \lnot R(x)) & \text{Simplification of } (a) \\ + (d) & ( \forall x)(R(x) \rightarrow \lnot Q(x)) & \text{Transposition of }(c) \\ + (e) & ( \forall x)(\lnot Q(x) \rightarrow P(x)) & \text{Material implication of } (b) \\ + (f) & ( \forall x)(R(x) \rightarrow P(x)) & \text{Hypothetical syllogism of } (d),(e) \\ + (g) & R(c) \rightarrow P(c) & \text{ Elimination of } \forall \text{ from } (f)\\ + (h) & ( \exists x)(R(x) \rightarrow P(x)) & \text{ Introduction of } \exists \text{ to } (g) \\ +\end{align*} + +(4) $P(x): $ $x$ 是学生,$Q(x): $ $x$ 是本科生,$R(x): $ $x$ 是研究生,$S(x): $ $x$ 是高材生,$a: $ John。 + +前提: +\begin{align*} + ( \forall x)(P(x) \rightarrow (Q(x) \land \lnot R(x)) \lor (\lnot Q(x) \land R(x))) \\ + ( \exists x)S(x) \\ + \lnot R(a) \\ + S(a) +\end{align*} + +结论:$P(a) \rightarrow Q(a)$ + +证明: +\begin{align*} + (a) & ( \forall x)(P(x) \rightarrow (Q(x) \land \lnot R(x)) \lor (\lnot Q(x) \land R(x))) & \text{Premise} \\ + (b) & P(a) \rightarrow (Q(a) \land \lnot R(a)) \lor (\lnot Q(a) \land R(a)) & \text{Elimination of } \forall \text{ from } (a) \\ + (c) & P(a) & \text{Assumption} \\ + (d) & (Q(a) \land \lnot R(a)) \lor (\lnot Q(a) \land R(a)) & \text{Modus ponens of } (b)(c) \\ + (e) & \lnot (R(a) \land \lnot Q(a)) \rightarrow (Q(a) \land \lnot R(a)) & \text{Material implication of } (d) \\ + (f) & \lnot R(a) & \text{Premise} \\ + (g) & \lnot R(a) \lor Q(a) & \text{Addition of } (g) \\ + (h) & \lnot (R(a) \land \lnot Q(a)) & \text{DeMorgan's Law of } (g) \\ + (i) & Q(a) \land \lnot R(a) & \text{Modus ponens of } (e)(h) \\ + (j) & Q(a) & \text{Simplification of } (i) \\ + (k) & P(a) \rightarrow Q(a) & \text{Conditional proof} +\end{align*} + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh8.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh8.tex" new file mode 100644 index 000000000..91204ead6 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh8.tex" @@ -0,0 +1,173 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} + +\title{离散数学——第九周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.11.08} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{9.1}\label{9.1} +列出下列集合所有的元素 + +(4) $A_4 = \{ z|z= \{ x,y \} \land x \in \sinteger \land y \in \sinteger \land 0 \le x \le 2 \land -2 \le 2 \le 1 \} $ + +\paragraph{解} +(4) $A_4 = \{ \{ -2,0 \} , \{ -2,1 \} , \{ -2,2 \} , \{ -1,0 \} , \{ -1,1 \} , \{ -1,2 \} , \{ 0 \} , \{ 0,1 \} , \{ 0,2 \} , \{ 1 \} , \{ 1,2 \} \} $ + +\paragraph{9.2}\label{9.2} +写出下列集合的表达式 + +(4) $\{3,5,7,11,13,17,19,23,29,\cdots\}$ + +\paragraph{解} +(4) $A_4 = \{ x|x \in \snatural \land x>2 \land ( \forall y)(y \in \snatural \land y>1 \rightarrow ( \forall z)(z \in \snatural \land z>1 \rightarrow yz \neq x)) \} $ + +\paragraph{9.3}\label{9.3} +给出集合$A,B,C$的例子,使$A \in B$, $B \in C$但$A \notin C$。 + +\paragraph{答} +$A = \varnothing, B = \{ \varnothing \} , C = \{ \{ \varnothing \} \} $ + +\paragraph{9.4}\label{9.4} +给出集合$A,B,C$的例子,使$A \in B$, $B \in C$且$A \in C$。 + +\paragraph{答} +$A = \varnothing, B = \{ \varnothing \} , C = \{ \varnothing, \{ \varnothing \} \} $ + +\paragraph{9.6}\label{9.6} +对任意的集合$A,B$和$C$,下列命题是否为真,若真则证明之。若假则举反例。 + +(2) 若$A \in B$且$B \subseteq C$,则$A \subseteq C$ + +(4) 若$A \in B$且$B \nsubseteq C$,则$A \notin C$ + +\paragraph{解} +(2) 假。反例:$A = \{ \varnothing \} , B = C = \{ \{ \varnothing \} \} $ + +(4) 假。反例:$A = \varnothing, B = \{ \varnothing \} , C= \{ \varnothing, \{ \{ \varnothing \} \} \} $ + +\paragraph{9.7}\label{9.7} +写出下列集合的幂集和笛卡尔积 + +(1) $\{ a, \{ a \} \} $的幂集 + +(3) $ \{ \varnothing, a, \{ b \} \} $的幂集 + +(5) $ P(P(\varnothing)) \times P(P(\varnothing)) $ + +\paragraph{解} +(1) $P( \{ a, \{ a \} \} ) = \{ \varnothing, \{ a \} , \{ \{ a \} \} , \{ a, \{ a \} \} \} $ + +(3) $P( \{ \varnothing, a, \{ b \} \} ) = \{ \varnothing, \{ \varnothing \} , \{ a \} , \{ \{ b \} \} , \{ \varnothing, a \} , \{ \varnothing, \{ b \} \} , \{ a, \{ b \} \} , \{ \varnothing, a, \{ b \} \} \} $ + +(5) +\begin{align*} + & P(P(\varnothing)) \times P(P(\varnothing)) \\ + & = P( \{ \varnothing \} ) \times P( \{ \varnothing \} ) \\ + & = \{ \varnothing, \{ \varnothing \} \} \times \{ \varnothing, \{ \varnothing \} \} \\ + & = \{ \langle \varnothing, \varnothing \rangle , \langle \varnothing, \{ \varnothing \} \rangle , \langle \{ \varnothing \} , \varnothing \rangle , \langle \{ \varnothing \} , \{ \varnothing \} \rangle \} +\end{align*} + +\paragraph{9.8}\label{9.8} +设$B = P(P(P(\varnothing)))$ + +(1) 是否$\varnothing \in B$?是否$\varnothing \subseteq B$? + +(3) 是否$\{\{\varnothing \}\}\in B$?是否$\{\{\varnothing \}\} \subseteq B$? + +\paragraph{解} +$$ B = P(P(P(\varnothing))) = P(P( \{ \varnothing \} )) = P( \{ \varnothing, \{ \varnothing \} \} ) = \{ \varnothing, \{ \varnothing \} , \{ \{ \varnothing \} \} , \{ \varnothing, \{ \varnothing \} \} \} $$ + +(1) $ \varnothing \in B, \varnothing\subseteq B $ + +(3) $ \{ \{ \varnothing \} \} \in B, \{ \{ \varnothing \} \} \subseteq B $ +\paragraph{9.9}\label{9.9} +画出下列集合的文氏图: + +(1) $(-A) \cap (-B)$ + +(3) $A \oplus (B \cup C)$ + +\paragraph{解} 如图 9.9 所示。 + +\begin{figure}[!htb] + \centering + \begin{minipage}[t]{0.305\textwidth} + \centering + \includegraphics[width=1\textwidth]{9-9-1.png} + \caption*{(1)} + \end{minipage} + \begin{minipage}[t]{0.305\textwidth} + \centering + \includegraphics[width=1\textwidth]{9-9-3.png} + \caption*{(3)} + \end{minipage} + \caption*{图 9.9} +\end{figure} + +\paragraph{9.10}\label{9.10} +用公式表示下列文氏图(题图9.10 (1))中的集合: + +\begin{figure}[!htb] + \centering + \includegraphics[width=0.305\textwidth]{9-10-1.png} + \caption*{题图 9.10 (1)} +\end{figure} + +\paragraph{答} (1) $(B \cap C)-A$ + +\paragraph{9.11}\label{9.11} +化简下列各式: + +(2) $ \{ \varnothing, \{ \varnothing \} \} - \varnothing $ + +(4) $ \{ \varnothing, \{ \varnothing \} \} - \{ \{ \varnothing \} \} $ + +\paragraph{解} +(2) $ \{ \varnothing, \{ \varnothing \} \} - \varnothing = \{ \varnothing, \{ \varnothing \} \} $ + +(4) $ \{ \varnothing, \{ \varnothing \} \} - \{ \{ \varnothing \} \} = \{ \varnothing \} $ + +\paragraph{9.12}\label{9.12} +设全集 $E=\{1,2,3,4,5\}$,集合$A = \{ 1,4 \} $, $B= \{ 1,2,5 \}$, $C= \{ 2,4 \}$。求下列集合: + +(1) $A \cap -B$ + +(3) $ -(A \cap B)$ + +(5) $P(A) - P(B)$ + +\paragraph{解} +(1) $ A \cap -B = \{ 1,4 \} \cap \{ 3,4 \} = \{ 4 \} $ + +(3) $ -(A \land B) = - ( \{ 1,4 \} \cap \{ 1,2,5 \} ) = - \{ 1 \} = \{ 2,3,4,5 \} $ + +(5) +\begin{align*} + & P(A) - P(B) \\ + & = P( \{ 1,4 \} ) - P( \{ 1,2,5 \} ) \\ + & = \{ \varnothing, \{ 1 \} , \{ 4 \} , \{ 1,4 \} \} - \{ \varnothing, \{ 1 \} , \{ 2 \} , \{ 5 \} , \{ 1,2 \} , \{ 1,5 \} , \{ 2,5 \} , \{ 1,2,5 \} \} \\ + & = \{ \{ 4 \} , \{ 1,4 \} \} +\end{align*} + +\end{document} \ No newline at end of file diff --git "a/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh9.tex" "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh9.tex" new file mode 100644 index 000000000..0623ad493 --- /dev/null +++ "b/\345\244\247\344\270\200\344\270\212/\347\246\273\346\225\243\346\225\260\345\255\246(1)/hw/2019/tex/dmh9.tex" @@ -0,0 +1,237 @@ +% UTF-8 encoding +% Compile with latex+dvipdfmx, pdflatex, xelatex or lualatex + +\documentclass[UTF8]{ctexart} +\usepackage{graphicx} +\usepackage{amssymb} +\usepackage{amsmath} +\usepackage{subfigure} +\usepackage{geometry} +\usepackage{caption} + +\newcommand{\true}{{\rm T}} +\newcommand{\false}{{\rm F}} +\newcommand{\snatural}{\mathbb{N}} +\newcommand{\sinteger}{\mathbb{Z}} +\newcommand{\srational}{\mathbb{Q}} +\newcommand{\sreal}{\mathbb{R}} +\newcommand{\card}{\rm{card}} +\newcommand{\modt}{\text{mod }} + +\title{离散数学——第十周作业} +\author{计83 刘轩奇 2018011025} +\date{2019.11.15} + +\geometry{left=2.0cm, right=2.0cm, top=2.5cm, bottom=2.5cm} + +\begin{document} + +\maketitle + +\paragraph{9.13} \label{9.13} +给定$\snatural$的下列子集,$A,B,C,D$为 +$A = \{ 1,2,7,8 \},$ +$B = \{ x | x^2 < 50 \} ,$ +$C = \{ x|0 \le x \le 20 \land x \text {可被} 3\text {整除} ,$ +$D = \{ x|x=2^K \land K \in \snatural \land 0 \le K \le 5 \}.$ +列出下列集合的所有元素。 + +(3) $B-(A \cup C)$ + +\paragraph{解} +(3) $B - (A \cup C) = \{ 0,1,2,3,4,5,6,7 \} - \{ 0,1,2,3,6,7,8,9,12,15,18 \} = \{ 4,5 \} $ + +\paragraph{9.14} \label{9.14} +写出下列集合: + +(1) $\cup \{ \{ 3,4 \} , \{ \{ 3 \} , \{ 4 \} \} , \{ 3, \{ 4 \} \} , \{ \{ 3 \} ,4 \} \}$ + +(2) $\cap \{ \{ 1,2,3 \} , \{ 2,3,4 \} , \{ 3,4,5 \} \} $ + +\paragraph{解} +(1) $\cup \{ \{ 3,4 \} , \{ \{ 3 \} , \{ 4 \} \} , \{ 3, \{ 4 \} \} , \{ \{ 3 \} ,4 \} \} = \{ 3,4, \{ 3 \} , \{ 4 \} \} $ + +(2) $\cap \{ \{ 1,2,3 \} , \{ 2,3,4 \} , \{ 3,4,5 \} \} = \{ 3 \}$ + +\paragraph{9.15} \label{9.15} +写出下列集合,其中:$PP(A)=P(P(A)), PPP(A)=P(P(P(A)))$ + +(1) $ \cup \{ PPP(\varnothing), PP(\varnothing), P(\varnothing), \varnothing \} $ + +(2) $ \cap \{ PPP(\varnothing), PP(\varnothing), P(\varnothing) \} $ + +\paragraph{解} +$$P(\varnothing) = \{ \varnothing \} , P(P(\varnothing)) = \{ \varnothing, \{ \varnothing \} \} , P(P(P(\varnothing))) = \{ \varnothing, \{ \varnothing \} , \{ \{ \varnothing \} \} , \{ \varnothing, \{ \varnothing \} \} \} $$ + +(1) $ \cup \{ PPP(\varnothing), PP(\varnothing), P(\varnothing), \varnothing \} = \{ \varnothing, \{ \varnothing \} , \{ \{ \varnothing \} \} , \{ \varnothing, \{ \varnothing \} \} \} $ + +(2) $ \cap \{ PPP(\varnothing), PP(\varnothing), P(\varnothing) \} = \{ \varnothing \} $ + +\paragraph{9.16} \label{9.16} +设$ A = \{ \{ \varnothing \}, \{ \{ \varnothing \} \} \} $,写出集合: + +(1) $ P(A) $ 和 $ \cup P(A) $ + +(2) $ \cup A $ 和 $ P( \cup A) $ + +\paragraph{解} +(1) $ P(A) = \{ \varnothing, \{ \{ \varnothing \} \} , \{ \{ \{ \varnothing \} \} \} , \{ \{ \varnothing \} , \{ \{ \varnothing \} \} \} \} , \cup P(A) = \{ \{ \varnothing \} , \{ \{ \varnothing \} \} \} $ + +(2) $ \cup (A) = \{ \varnothing, \{ \varnothing \} \} , P( \cup (A)) = \{ \varnothing, \{ \varnothing \} , \{ \{ \varnothing \} \} , \{ \varnothing, \{ \varnothing \} \} \} $ + +\paragraph{9.17} \label{9.17} +设$A,B,C$是任意的集合,证明: + +(2) $(A-B)-C=(A-C)-(B-C)$ + +(4) $A\subseteq C \land B\subseteq C \Leftrightarrow A \cup B\subseteq C$ + +\paragraph{解} +(2) +\begin{align*} + & (A-C) - (B-C) \\ + = & (A \cup -C) \cup -(B \cup -C) \\ + = & (A \cup -C) \cup (-B \cap C) \\ + = & (A \cup -C \cup -B) \cap (A \cup -C \cup C) \\ + = & (A \cup -B \cup -C) \cap (A \cup E) \\ + = & ((A-B) - C) \cap E \\ + = & (A-B)-C +\end{align*} +(4) 充分性: +$$A\subseteq C \land B\subseteq C \Rightarrow (A \cup B)\subseteq (C \cup C) \Rightarrow A \cup B\subseteq C$$ +必要性:若$A \cup B \subseteq C$ + +任意$x$,若$x \in A$,则$x \in A \cup B$,从而$x \in C$,则$A \subseteq C$。 + +任意$x$,若$x \in B$,则$x \in A \cup B$,从而$x \in C$,则$B \subseteq C$。 +$$\therefore A \cup B\subseteq C \Rightarrow A\subseteq C \land B\subseteq C$$ + +\paragraph{9.18} \label{9.18} +满足下列条件的集合$A,B$有什么关系? + +(1) $A-B=B$ + +(3) $A \cap B = A \cup B$ + +\paragraph{答} +(1) $A=B=\varnothing$ + +(3) $A=B$ + +\paragraph{9.19} \label{9.19} +给出下列命题成立的充要条件: + +(2) $ (A-B) \cup (A-C) = \varnothing $ + +(4) $ (A-B) \oplus (A-C) = \varnothing $ + +\paragraph{解} +(2) +\begin{align*} + & (A-B) \cup (A-C) = \varnothing \\ + \Longleftrightarrow & (A \cup -B) \cap (A \cup -C) = \varnothing \\ + \Longleftrightarrow & A \cap -(B \cap C) = \varnothing \\ + \Longleftrightarrow & A\subseteq B \cap C +\end{align*} +(4) +\begin{align*} + & (A-B) \oplus (A-C) = \varnothing \\ + \Longleftrightarrow & ((A-B)-(A-C) \cup ((A-C)-(A-B)) = \varnothing \\ + \Longleftrightarrow & ((A \cap -B) \cap -(A \cap -C)) \cup ((A \cap -C) \cap -(A \cap B)) = \varnothing \\ + \Longleftrightarrow & ((A \cap -B) \cap (-A \cup C)) \cup ((A \cap -C) \cap (-A \cup B)) = \varnothing \\ + \Longleftrightarrow & (A \cap -A \cap -B) \cup (A \cap -B \cap C) \cup (A \cap -A \cap -C) \cup (A \cap B \cap -C) = \varnothing \\ + \Longleftrightarrow & \varnothing \cup (A \cap -B \cap -C) \cup \varnothing \cup (A \cap B \cap -C) = \varnothing \\ + \Longleftrightarrow & A \cap -B \cap C = \varnothing \land A \cap B \cap -C = \varnothing \\ + \Longleftrightarrow & A-B\subseteq -C \land A-C\subseteq -B \\ + \Longleftrightarrow & A-B\subseteq A-C \land A-C\subseteq A-B & (\ast)\text{待后补证} \\ + \Longleftrightarrow & A-B = A-C +\end{align*} +补证(*)$A-B\subseteq -C \Longleftrightarrow A-B\subseteq A-C$ +\begin{align*} + A-B\subseteq -C + \Longleftrightarrow & ( \forall x) (x \in A-B \rightarrow x \in -C) \\ + \Longleftrightarrow & ( \forall x)((x \in A \land x \in -B) \rightarrow x \in C) \\ + \Longleftrightarrow & ( \forall x)((x \in A \land x \in -B) \rightarrow (x \in A \land x \in -C) \\ + \Longleftrightarrow & ( \forall x)(x \in A-B \rightarrow x \in A-C) \\ + \Longleftrightarrow & A-B\subseteq A-C +\end{align*} + +\paragraph{9.26} \label{9.26} +(1) 若 $A \times B = \varnothing$ ,则$A$和$B$应满足什么条件。 + +(2) 对集合 $A$ ,是否可能$A = A \times A$。 + +\paragraph{答} +(1) $A \times B = \varnothing \Longleftrightarrow A=\varnothing \lor B = \varnothing$ + +(2) $A = A \times A \Longleftrightarrow A = \varnothing$ + +\paragraph{9.28} \label{9.28} +求$1$至$250$之间被$2,3,5$中任何一个整除的整数的个数。 + +\paragraph{解} +设 +\begin{align*} + E & = \{x|x \in \snatural \land 1\le x \le 250\} \\ + A & = \{x|x \equiv 0 (\modt 2) \land x \in E\} \\ + B & = \{x|x \equiv 0 (\modt 3) \land x \in E\} \\ + C & = \{x|x \equiv 0 (\modt 5) \land x \in E\} +\end{align*} +$$|A|=\lfloor 250/2 \rfloor = 125, |B|=\lfloor 250/3 \rfloor = 83, |C|=\lfloor 250/5 \rfloor = 50 $$ +$$|A \cap B|=\lfloor 250/6 \rfloor = 83, |A\cap C|=\lfloor 250/6 \rfloor = 25, |B\cap C|=\lfloor 250/15 \rfloor = 16 $$ +$$|A \cap B \cap C|=\lfloor 250/30 \rfloor = 8 $$ +\begin{align*} + \therefore |A \cup B \cup C| & = |A| + |B| + |C| - |A \cup B| - |A \cup C| - |B \cup C| + |A \cup B \cup C| \\ & = 125+83+50-41-25-16+8 = 184 +\end{align*} + +\paragraph{9.30} \label{9.30} +证明不存在集合$A_1, A_2, A_3, A_4$使 +$$A_4 \in A_3 \land A_3 \in A_2 \land A_2 \in A_1 \land A_1 \in A_4$$ + +\paragraph{证} +假设这样的$A_1, A_2, A_3, A_4$存在,则令$B=\{A_1, A_2, A_3, A_4\}$ + +由正则公理,$B$含有极小项,由对称性不妨设该极小项为$A_1$。则$A_1 \cap B = \varnothing$。 + +而$A_2 \in A_1, A_2 \in B$,则$A_2 \in A \cap B$,这与$A_1 \cap B = \varnothing$矛盾。 + +则假设错误,不存在这样的四个集合。 + +\paragraph{9.31} \label{9.31} +证明不存在由所有单元素集合组成的集合。 + +\paragraph{证} +假设存在所有单元素集合组成的集合$A$。 + +则令$B=\{A\}$,$B$是单元素集合,则$B \in A$,而又有$A \in B$,这与定理9.7.7: $\lnot (A_1 \in A_2 \land A_2 \in A_1)$矛盾。 + +故假设错误,不存在由所有单元素集合组成的集合。 + +\paragraph{9.32} \label{9.32} +证明存在所有素数组成的集合。 + +\paragraph{证} +由无穷公理知自然数集$\snatural$存在,设谓词$P(x)$表示$x$为素数,则由子集公理 +$$(\exists A)(\forall x)(x \in A \longleftrightarrow x \in \snatural \land P(x))$$ +即存在$A=\{x|x\text{为素数}\land x \in \snatural\}$为全体素数组成的集合。 + +\paragraph{9.33} \label{9.33} +证明若$A$是传递集合,则$A_+$是传递集合。 + +\paragraph{证} +\begin{align*} + A\text{为传递集} + & \Longleftrightarrow ( \forall x) ( \forall y)((x \in y \land y \in A) \rightarrow x \in A) \\ + A^+ = A \cup \{ A \} & \Longrightarrow ( \forall x)(x \in A \rightarrow x \in A^+) \\ + & \Longleftrightarrow ( ( \forall x) ( \forall y)((x \in y \land y = A) \rightarrow x \in A^+) +\end{align*} +\begin{align*} + & A\text{为传递集} \land A^+ = A \cup \{ A \} \\ + \Longrightarrow & ( ( \forall x) ( \forall y)((x \in y \land y \in A) \rightarrow x \in A^+) \land ( \forall x) ( \forall y)((x \in y \land y \in A) \rightarrow x \in A^+) \\ + \Longrightarrow & ( \forall x) ( \forall y)(((x \in y \land y \in A) \rightarrow x \in A^+) \land (((x \in y \land y \in A) \rightarrow x \in A^+)) \\ + \Longrightarrow & ( \forall x) ( \forall y)((x \in y) \land (y \in A \lor y = A)) \rightarrow x \in A^+) \\ + \Longrightarrow & ( \forall x) ( \forall y)((x \in y \land y \in A^+) \rightarrow x \in A^+) \\ + \Longrightarrow & A^+\text{为传递集} +\end{align*} + +\end{document} \ No 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