设[tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]为环[tex=0.786x1.286]yokTf2U2Z7kNGUXMm22GjQ==[/tex]到环[tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex]上的一个映射，对[tex=3.286x1.214]3p9vSbuXy9b35NRjagiE2WHQaM8BVQGNQrcUwhPhw2o=[/tex]满足1）[tex=8.357x1.357]SW9xzMiS3AiisZ62RdoDh+ctXTbsD0OR9h7BQoiFpB0vXQ8Ayud4cPp3ujN/ygjg[/tex]，2）[tex=6.786x1.357]lnEclGf+4P4Ds+dwUy+lbCNjUpTJ/dktRrz6wSM5PbIJdkah2nhthnPuxtU6nbuQ[/tex]或[tex=6.786x1.357]lnEclGf+4P4Ds+dwUy+lbLc7M6GQulIbsou6LSG/zxWcPqXchiHgXVRnXlO10XZz[/tex]，证明[tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]为同态或反同态。
证明：设[tex=2.714x1.214]QZlcT9hsc9pVwSSKjX8aAQ==[/tex]，而[tex=8.5x1.286]yAtoYzbJnkSy8ZpvbAFTzov5h847vzFXSrvHf3hjZaiAu0zQzX5ZbJtWZUjVwvUEtsfzDI8LbQhTwKSm2ZBKww==[/tex]，不妨设[tex=6.786x1.357]lnEclGf+4P4Ds+dwUy+lbCNjUpTJ/dktRrz6wSM5PbIJdkah2nhthnPuxtU6nbuQ[/tex]，对[tex=1.929x1.071]M1yvbtt6U7UTV4sNd7AP5g==[/tex]，若有[tex=6.786x1.357]3yVkRggLKkLbeB0yfMy0E8M2Ke1lOBoAjWhERfsDEQY3dF45CY50OOTTKIv1ZKpC[/tex]，则[tex=12.929x1.357]lakDCnvlMvuQGgcj7IXE+LC4wMbvI72aHuDu4zIK7iI8bURR3QDPTmvKjL58squM0dJ9NDeMuiUta5zcmGAKmA==[/tex]，此外，由[tex=6.429x1.357]oON63qrALEGyakH3tFMXhpAlU6DacZBy5FOHNAeipwY=[/tex]知，故有两种可能：(1) [tex=20.714x1.357]lakDCnvlMvuQGgcj7IXE+AYNAWdmfPZfcODxzlin8aCzui0LF99f2eJtT/cgKpfGeT4Idmh8oaG4OEMDKLnEAc29HGeKh5Q2M49NVPCe4HPxt3fqrxWbldReKYQ/2ci/[/tex]，此时有[tex=11.214x1.357]oi+Ni1kFIUph3e5ZXUu1U1kuQXR8HM7pLowNMQsLNA1CVhySPzxtAdsvQIrRMMzzDuBOuJyTUJx54zuUzSGLKQ==[/tex]。(2)[tex=20.714x1.357]lakDCnvlMvuQGgcj7IXE+Da5W4SM5nipBKRtIGT0zDtXAE72D6kZ2nXn8kSGBOk19Z4ozC1T530LOLWWwij4zWFE8t+9DUOY0WkU245gAX+0GZeOuchZF3dPaKM/YEr0[/tex]，因此[tex=8.143x1.357]0JNBTWjD1SBahD53H1qzCDIGTGtzK7on4i22yQZmiBjCvngPgd4SXk5/RhmNDlxd[/tex]，这与[tex=8.5x1.286]UB0fX0wMfl97ch064FXyChgo8Uy2mXNHsNIXT+yZkl4U9ffcjwC6RkxgRyGpkGRQu4mxe4Tp3ILcjw8C1m164g==[/tex]矛盾，于是情形 (2) 不出现，即有[tex=6.786x1.357]3yVkRggLKkLbeB0yfMy0E0Ivog5tF95rjyNggORyssnJ6wh/zp4Y3iktSxro8vkp[/tex]，[tex=3.429x1.071]DxQ+iNTxWK+yRPH9GvsOnk34t++jUpXi6sHq7KnHXuQ=[/tex][tex=2.214x1.286]0sHdbbGDvDXX+HCt6Muz+Q==[/tex]，同样可以证明[tex=12.571x1.357]pxeYd3fbuRy6EuJM6gE29qS6hU26NnT9aT1h56govlMGXzQMln1+pslA0oxLCHlX7Mg+givKclh7DzfeTffcwfUPHioTcO2zol5qpCgQVqw=[/tex]，如果存在[tex=3.429x1.214]p4GY1Dv7TODi4M7cHG8Nuh07Wz1/zjd3fF4ZkLtipRM=[/tex],，使得[tex=18.286x1.286]jM3ETXsfVQ91liJLEXBWX+07Tl4s1smPYWD9RPNSeTTTV/qrwJUhUyXwfGI9cqErlMFkSHSLXLyZ+rQUOZSryjaA3m3MQFw8S2tDNqS/6jbL9PihHPBe+xhd6CKq/qkodf8og+JE5Y0VxB4989lgDn6h/PQjT2at8GYkUIRegCgFcnI0v7gaDYQYCxwpIRmURQ1s/L6zkGQOOn0VyJLBgQ==[/tex]，那么用上面的方法可以证明[tex=2.429x1.071]IQMlWTzYn5GguYydsdowkVLVDt3JRcWRdfoggsu7TxQ=[/tex]有[tex=8.0x1.357]3h0Pq9tzr2v60/mEMRZBsRkjw7rd9Tl0wu6L63uwmKMfnjq6L6nLaSnsZ3k7Ixq5DCvN1CZXcPMdRmFlV0BlvQ==[/tex]，[tex=8.929x1.357]/Ku0sZYr5IrAJLu10XlOVyaepX/zb975Mfry7+Le7zruLzqGI51OxNHy8//QXP8AeXCsc+1QVBp4nDqNkdaCOTACLiLTGXjBx+EiZYofRsU=[/tex]，由此得到[tex=13.286x1.357]B9LIlWwWFmm14GzJrHYF5qToeBBenH/8LElknlX2KzlYDDLD7kI2caHEvUHShjgT/qqwgom3XN1Tf6a2V7ii2kBpwGWzHd1TEDVdvDrOd2TX7x3Sm345mONP8Iidmwt9en8vz/6YExP6GASeak/7Lg==[/tex]，[tex=14.286x1.357]/Ku0sZYr5IrAJLu10XlOV4WC484WCB/Zush+sk0SkfXzh+geZmTm08EBbvX+cWciUSUT9/tJUPQ29EWZjmvyUH66ttnHvEuexh5X/5Fsk1jdRMSau76GvAeLMUOxkeuaf5wwj0ihx6mnaqQIMT2gvA==[/tex]，于是[tex=17.5x1.357]cAILRcDjMW88Q28WoRO21RTuOBn6ccdqP0ohrQHHkF9FyV/ZBWxTm3Xwgqqal+L9bqADzjIiZ4VJk6shxAb94mzc7h0DpheQVY01L1O5HQjdVrTOojqsvW3PSauRenuYZJ/Coytif6lWnlAaItFHCp8r1D3yqOE/l6oiNLrpUjA=[/tex][tex=20.357x1.357]zYEjco6mH0Yh1BjLh9L6Qdv7iBeZ6rSM+IDxDHHdc9kSfT2NStyiykZ6e1HiUb+kMM2K1RP6X+ZK0p4NevvpSvtCQsI/m6r35JfIwOS5/D7s5TwgbVU4TzqKqMUGI84U2nZfwhZS17GHcKUeJjU2NTMT48g6qpFwg9+6LHhiVEv6nyc+dkx/u0clKTKKjlLc[/tex]，而[tex=15.786x1.357]cAILRcDjMW88Q28WoRO21RTuOBn6ccdqP0ohrQHHkF9FyV/ZBWxTm3Xwgqqal+L9bqADzjIiZ4VJk6shxAb94gUTTImzQjiJ2fSzBDH+CcD1Slqw9P8qMI4FrAxdAzoslDSYj7x7CJdf4RN2Re5adp/ggTH15vgd7cYDY5CpOz4=[/tex]或 [tex=7.429x1.357]8v5yZbnDNppGsmPibnPmnyDaB1tE1mJs9UclzyFfYyLpc4SRV2G50Q7EZBpFhIAVNkuQP6JmcdmVt51KoRJuAw==[/tex][tex=2.214x1.286]+ceeOp9yeS64/gwd48/VQw==[/tex]，若式(5)为前者，则有[tex=15.786x1.357]cAILRcDjMW88Q28WoRO21RTuOBn6ccdqP0ohrQHHkF9FyV/ZBWxTm3Xwgqqal+L9bqADzjIiZ4VJk6shxAb94gUTTImzQjiJ2fSzBDH+CcD1Slqw9P8qMI4FrAxdAzoslDSYj7x7CJdf4RN2Re5adp/ggTH15vgd7cYDY5CpOz4=[/tex][tex=20.357x1.357]zYEjco6mH0Yh1BjLh9L6Qdv7iBeZ6rSM+IDxDHHdc9kSfT2NStyiykZ6e1HiUb+kkGFp8W80v55jtibUBGC/+Xma0mX9SfgvGCR/PVH42YcivdwgQ1VAVrVNRZ+Ara27Uqyg64XhWFQid3AC7ypdC9hpy1OzmCT+Vqvpt3cO9/AUKCVFZh9HV5OIJAFhqqb7[/tex]，因此[tex=10.714x1.357]3h0Pq9tzr2v60/mEMRZBsWE9JPpIOl8YbZE+Dk53DlJFVkTUHbxjfyeIUkiOZT79p9QZigDxas3yS5WJDPiDXxxPb0iH9vnk/1PRvEyhgAx0ZLKpyoZ58DS4P0POYr38rrem++xyAZL2dtA1lTGEMw==[/tex]，与式(4)矛盾。若式(4)中为后者，则有[tex=15.786x1.357]cAILRcDjMW88Q28WoRO21RTuOBn6ccdqP0ohrQHHkF9FyV/ZBWxTm3Xwgqqal+L9bqADzjIiZ4VJk6shxAb94oEEL0fJBEA5ZP78V9PHb2AS4eIZrSnnAJSCX258/kHLv8HQXWiFhhHriwkLdOpFDyckrJk/9Ogmk2C3orroDdw=[/tex][tex=20.357x1.357]3bc6dxeb2a2xw0mcFNpetLRaA4FYLjRjQzD6P+mnf589qwe/ZT2ekuKHnAALtu15cwWUzfkRDpgI/D1gDfnGvcpKJJtQodX3/aU3Nz8RgrLxcl5Yk6QPxzWRyInFuOGJyZikykA7Vum5EnUa7llc5JyH1FF4P5EXJo+AIIQAjFb/LZwdW9xDGojhOWH0jQxZ[/tex]，因此 [tex=8.143x1.357]oi+Ni1kFIUph3e5ZXUu1UwWwKVDu22xPzQWFKOUqX3AsQwtrth5fu/js6YDzoJAZ[/tex]与[tex=8.5x1.286]UB0fX0wMfl97ch064FXyChgo8Uy2mXNHsNIXT+yZkl4U9ffcjwC6RkxgRyGpkGRQu4mxe4Tp3ILcjw8C1m164g==[/tex]矛盾，因此[tex=9.0x1.357]3h0Pq9tzr2v60/mEMRZBsbRNH3HzUnSWuDiuQJMW+rHTfBuAifBP1SiYrzHIZZWcSRnlBVJsd6xDPIinsedbjsCaDyDYzd6M9xtWNn8ThCtoxlyESojbQ8U2euzwDI/t[/tex]，[tex=4.929x1.214]zfycwiwnf2BECDyFLqVv2BuU7XtIoUwqKRdA5wG+UyWbR9kzy3VTn2y5EGMSAgir[/tex]，所以[tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]为同态。若假定 [tex=6.786x1.357]lnEclGf+4P4Ds+dwUy+lbLc7M6GQulIbsou6LSG/zxXTLckhf2Il90pKWDZSUBgJ[/tex]，用与上面一样的方法讨论可得[tex=9.0x1.357]3h0Pq9tzr2v60/mEMRZBsbRNH3HzUnSWuDiuQJMW+rGpxngL9GlsL6QyIHn1jdxNEXSgmfaO+xtRE+ELAwkw7giBp2NmY1n/CTmwPdgEsaQYK4MBS+S/u+cRAxF6fVFh[/tex]，[tex=4.929x1.214]zfycwiwnf2BECDyFLqVv2BuU7XtIoUwqKRdA5wG+UyWbR9kzy3VTn2y5EGMSAgir[/tex]，所以[tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]为反同态。
举一反三
- 6个顶点11条边的所有非同构的连通的简单非平面图有[tex=2.143x2.429]iP+B62/T05A6ZTM0eeaWiQ==[/tex]个，其中有[tex=2.143x2.429]ndZSw3zT0QTOVLVdoUto1Q==[/tex]个含子图[tex=1.786x1.286]J+vVZa2YaMpc6mJBbqVvWw==[/tex]，有[tex=2.143x2.429]lmhx48evnQMhi03NovPXig==[/tex]个含与[tex=1.214x1.214]kFXZ1uR8GjycbJx+Ts2kyQ==[/tex]同胚的子图。供选择的答案[tex=3.071x1.214]3KinXFh3SXhZ7nIe1y9KEV6aadxhhJWeEy6Dij1iObdMUZkY6ZA5J2dVVjPSuhEf[/tex]：(1) 1 ;(2) 2 ;(3) 3 ; (4) 4 ;(5) 5 ;(6) 6 ; (7) 7 ; (8) 8 。
- 设 [tex=0.643x1.214]4ssBDc1re7hhNB3dpzYmRg==[/tex] 为环 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 到环 [tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex] 的满同态. 证明: 如果 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 是交换环, 则 [tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex] 也是交换环.
- 设 [tex=0.714x1.214]o1HMeHvTnCSY+cMqAEISgQ==[/tex] 是环 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 到 [tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex] 的同态满射，[tex=0.5x1.0]3EF1VcotinZAjtQqtSWaxw==[/tex] 是 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 的理想，[tex=0.786x1.143]EiNNRHzKTxg7zGjdHFOxvQ==[/tex] 是 [tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex] 的理想，那么(1) [tex=1.714x1.357]2+VR21kpEOyUGEehE2UvdA==[/tex] 是 [tex=1.0x1.143]vL/JscKF18qJf47ozsjQEQ==[/tex] 的理想;(2) [tex=3.286x1.5]/Rybg/8CUfBGAEQhpY7PXIK2RmbrEgHGC3vZadxbYqI=[/tex] 是 [tex=0.786x1.0]AOSTmhvIsOwsdZlGoks7dg==[/tex] 的理想。
- 9判别下列函数是否是周期函数,若是周期函数,求其周期 :(1) [tex=8.357x1.357]jijpvC8Aw74QOOOJh5Va05j3PtA64Pms1Q5qDGlqeN4=[/tex](2) [tex=5.643x1.357]TG5DUF3HrCbhIJWDEcp5Pj9u3e2PUgpbN4NJQ6DZXLw=[/tex](3) [tex=5.714x1.357]SBxtvKszj8+jJcycMEKn5vqfhi5GLWqH4Gac9QRbIHc=[/tex](4) [tex=6.929x1.357]NZ5EVFRfE4pFsgkbEOhFkNg5/qZx8geAT5eL+yzbq1Q=[/tex]
- 对 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex]的不同值，分别求出循环群[tex=1.143x1.214]StMMJ6qThnpokZJIPGrdFyP3vrLnUdltYxmLxjw8za8=[/tex]的所有生成元和所有子群。(1) 7; (2) 8; (3)10 ;(4) 14 ; (5) 15 (6) 18 。
内容
- 0
求函数[tex=3.286x1.429]kdT+eIE7CHPynuN6CaN40g==[/tex]（抛物线）隐函数的导数[tex=1.071x1.429]BUw1BPFU3fsJlAl/vt9M9w==[/tex]当x=2与y=4及当x=2与y=0时,[tex=0.786x1.357]Hq6bf3CacUy07X+VImUMaA==[/tex]等于什么?
- 1
设[tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]和[tex=0.5x0.786]xdTs2QHMXTpKzI7ZnwCRMQ==[/tex]分别是域[tex=0.786x1.286]BlkXDnmzWHxe4M6E9LlofQ==[/tex]上线性空间[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]到[tex=1.071x1.143]SEwIem1RXUAaU4aCzKG5tQ==[/tex]与[tex=1.071x1.143]SEwIem1RXUAaU4aCzKG5tQ==[/tex]到[tex=1.286x1.143]5e6TFUJHLxbGL39BTJK478PRVrwxa0yFlrmakbRHqtY=[/tex]的一个同构映射。证明：[tex=0.5x0.786]xdTs2QHMXTpKzI7ZnwCRMQ==[/tex][tex=0.571x0.786]G/buLKOLYVDEKMZ76t752w==[/tex]是[tex=0.857x1.286]ZpwhzmyivskaH5M1X7ozaQ==[/tex]到[tex=1.286x1.143]5e6TFUJHLxbGL39BTJK478PRVrwxa0yFlrmakbRHqtY=[/tex]的一个同构映射。
- 2
设h为X上函数,证明下列两个条件等价,（1）h为一单射（2）对任意X上的函数[tex=5.429x1.214]3BrfPgAFe5dbHQTMAYnbS+118W4YAj6CiW06EKMaxNI=[/tex]蕴涵[tex=1.786x1.214]pxzkG5OdsKT9CiCwC5OvPQ==[/tex]
- 3
判断下列命题是否为真：（1）[tex=3.643x1.357]/5abqJjwKZ1qr+6hsVFF5EBvfq3ggOFNlHMClz0h9nk=[/tex]（2）[tex=2.929x1.357]rGJpyjIjJpbcoBTWxP0Jiw==[/tex]（3）[tex=4.5x1.357]2wycHMoqU83MyEp17iBils58bR7YLuCTI2G9NVAdlfY=[/tex]（4）[tex=5.214x1.357]CTz2gu+IIm1GgNmYMGaduCRtA41wnW4WqwRWwEhq6aA=[/tex]（5）[tex=4.857x1.357]1DcE2BMMOaZhTuxR/mjgsboXxfg5ET59Dp4I/jjEDuw=[/tex]（6）[tex=4.643x1.357]BSryrsQYOvTP2hTWRu6t4nAuJwlSs4L9jaq70EpB+Us=[/tex]（7）若[tex=6.0x1.357]y0IZLUnBO88nR8WBZYvd7QXv5S1OMINV5cQNzPyiyAc=[/tex]，则[tex=3.429x1.357]1brfPwTkVVIX4GfoMIUskA==[/tex]（8）若[tex=7.643x1.357]MhLfJXZnhbXiB0x3oNtFzThV4Y1mJxe1VYr7PkJE/T6hmTD3WWp+UxbNwvUQ6DHk[/tex]，则[tex=4.143x1.357]LZUA94ISo1po5HWsOVeBCjo0rMvj7uw3bGw5HiZenrI=[/tex]
- 4
set1 = {x for x in range(10)} print(set1) 以上代码的运行结果为？ A: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} B: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9,10} C: {1, 2, 3, 4, 5, 6, 7, 8, 9} D: {1, 2, 3, 4, 5, 6, 7, 8, 9,10}