电大离散数学作业3答案资料小抄(集合论部分).doc

形成性考核作业 姓 名： 学 号： 得 分： 教师签名： 离散数学作业3离散数学集合论部分形成性考核书面作业本课程形成性考核书面作业共3次，内容主要分别是集合论部分、图论部分、数理逻辑部分的综合练习，基本上是按照考试的题型（除单项选择题外）安排练习题目，目的是通过综合性书面作业，使同学自己检验学习成果，找出掌握的薄弱知识点，重点复习，争取尽快掌握。本次形考书面作业是第一次作业，大家要认真及时地完成集合论部分的综合练习作业。要求：将此作业用A4纸打印出来，手工书写答题，字迹工整，解答题要有解答过程，要求2010年11月7日前完成并上交任课教师（不收电子稿）。并在03任务界面下方点击“保存”和“交卷”按钮，完成并上交任课教师。一、填空题 1设集合，则P(A)-P(B )= 1,2,2,3,1,3,1,2,3 ，A´ B= <1,1>,<1,2>,<2,1>,<2,2>,<3,1>,<3,2> 2设集合A有10个元素，那么A的幂集合P(A)的元素个数为 1024 3设集合A=0, 1, 2, 3，B=2, 3, 4, 5，R是A到B的二元关系，则R的有序对集合为 <2,2>,<2,3>,<3,2>,<3,3>4设集合A=1, 2, 3, 4 ，B=6, 8, 12， A到B的二元关系R那么R1 <6,3>,<8,4> 5设集合A=a, b, c, d，A上的二元关系R=<a, b>, <b, a>, <b, c>, <c, d>，则R具有的性质是反自反性6设集合A=a, b, c, d，A上的二元关系R=<a, a >, <b, b>, <b, c>, <c, d>，若在R中再增加两个元素<c, b>, <d, c>，则新得到的关系就具有对称性7如果R1和R2是A上的自反关系，则R1R2，R1R2，R1-R2中自反关系有 2 个8设A=1, 2上的二元关系为R=<x, y>|xÎA，yÎA, x+y =10，则R的自反闭包为 <1,1>,<2,2> 9设R是集合A上的等价关系，且1 , 2 , 3是A中的元素，则R中至少包含 <1,1>,<2,2>,<3,3> 等元素10设集合A=1, 2，B=a, b，那么集合A到B的双射函数是 <1,a>,<2,b>或<1,b>,<2,a> 二、判断说明题（判断下列各题，并说明理由）1若集合A = 1，2，3上的二元关系R=<1, 1>，<2, 2>，<1, 2>，则(1) R是自反的关系； (2) R是对称的关系 解：(1) 结论不成立 因为关系R要成为自反的，其中缺少元素<3, 3> (2) 结论不成立 因为关系R中缺少元素<2, 1> 2如果R1和R2是A上的自反关系，判断结论：“R-11、R1R2、R1R2是自反的” 是否成立？并说明理由 解：结论成立 因为R1和R2是A上的自反关系，即IAÍR1，IAÍR2 由逆关系定义和IAÍR1，得IAÍ R1-1； 由IAÍR1，IAÍR2，得IAÍ R1R2，IAÍ R1ÇR2所以，R1-1、R1R2、R1ÇR2是自反的ooooabcd图一ooogefho3若偏序集<A，R>的哈斯图如图一所示，则集合A的最大元为a，最小元不存在 错误，按照定义，图中不存在最大元和最小元。 4设集合A=1, 2, 3, 4，B=2, 4, 6, 8，判断下列关系f是否构成函数f：，并说明理由(1) f=<1, 4>, <2, 2,>, <4, 6>, <1, 8>； (2)f=<1, 6>, <3, 4>, <2, 2>；(3) f=<1, 8>, <2, 6>, <3, 4>, <4, 2,> (1) 不构成函数，因为它的定义域Dom(f)A(2) 也不构成函数，因为它的定义域Dom(f)A(3) 构成函数，首先它的定义域Dom(f) =1, 2, 3, 4= A，其次对于A中的每一个元素a，在B中都有一个唯一的元素b，使<a,b>Îf三、计算题1设，求：(1) (AÇB)ÈC； (2) (AÈB)- (BÇA) (3) P(A)P(C)； (4) AÅB解：(1) (AÇB)ÈC=1È1,3,5=1,3,5(2) (AÈB)- (BÇA)=1,2,4,5-1=2,4,5(3) P(A) =,1,4,1,4P(C)= ,2,4,2,4P(A)P(C)=1,1,4(4) AÅB= (AÈB)- (BÇA)= 2,4,52设A=1,2,1,2，B=1,2,1,2，试计算（1）（A-B）； （2）（AB）； （3）A×B解：（1）（A-B）=1,2（2）（AB）=1,2(3) A×B <1,1>,<1,2>,<1,1,2 >,<2,1>,<2,2>,<2,1,2 >,<1,1>,<1,2>,<1,1,2 >,<2,1>,<2,2>,<2,1,2 >3设A=1，2，3，4，5，R=<x，y>|xÎA，yÎA且x+y£4，S=<x，y>|xÎA，yÎA且x+y<0，试求R，S，R·S，S·R，R-1，S-1，r(S)，s(R) 解： R=<1,1>,<1,2>,<1,3>,<2,1>,<2,2>,<3,1>S=R·S=S·R=R-1=<1,1>,<2,1>,<3,1>,<1,2>,<2,2>,<1,3>S-1=r(S)= <1,1>,<2,2>,<3,3>,<4,4>,<5,5>s(R)= <1,1>,<1,2>,<1,3>,<2,1>,<2,2>,<3,1> 4设A=1, 2, 3, 4, 5, 6, 7, 8，R是A上的整除关系，B=2, 4, 6(1) 写出关系R的表示式； (2 )画出关系R的哈斯图； (3) 求出集合B的最大元、最小元 解：(1) R=<1,1>,<1,2>,<1,3>,<1,4>,<1,5>,<1,6>,<1,7>,<1,8>,<2,2>,<2,4>,<2,6>,<2,8>,<3,3>,<3,6>,<4,4>,<4,8>,<5,5>,<6,6>,<7,7>,<8,8>12346578关系R的哈斯图(2) (3) 集合B没有最大元，最小元是2四、证明题 1试证明集合等式：AÈ (BÇC)=(AÈB) Ç (AÈC)证：设，若xAÈ (BÇC)，则xA或xBÇC，即 xA或xB 且 xA或xC即xAÈB 且 xAÈC ，即 xT=(AÈB) Ç (AÈC)，所以AÈ (BÇC)Í (AÈB) Ç (AÈC) 反之，若x(AÈB) Ç (AÈC)，则xAÈB 且 xAÈC， 即xA或xB 且 xA或xC，即xA或xBÇC，即xAÈ (BÇC)，所以(AÈB) Ç (AÈC)Í AÈ (BÇC) 因此AÈ (BÇC)=(AÈB) Ç (AÈC)2试证明集合等式AÇ (BÈC)=(AÇB) È (AÇC)证明：设S=A(BC)，T=(AB)(AC)， 若xS，则xA且xBC，即 xA且xB 或 xA且xC， 也即xAB 或 xAC ，即 xT，所以SÍT 反之，若xT，则xAB 或 xAC， 即xA且xB 或 xA且xC 也即xA且xBC，即xS，所以TÍS 因此T=S 3对任意三个集合A, B和C，试证明：若AB = AC，且A，则B = C 证明：设xÎA，yÎB，则<x，y>ÎA´B， 因为A´B = A´C，故<x，y>Î A´C，则有yÎC， 所以B Í C 设xÎA，zÎC，则<x，z>Î A´C， 因为A´B = A´C，故<x，z>ÎA´B，则有zÎB，所以CÍB 故得A=B 4试证明：若R与S是集合A上的自反关系，则RS也是集合A上的自反关系R1和R2是自反的，"x ÎA，<x, x> Î R1，<x, x> ÎR2，则<x, x> Î R1R2， 所以R1R2是自反的请您删除一下内容，O(_)O谢谢！【China's 10 must-see animations】The Chinese animation industry has seen considerable growth in the last several years. It went through a golden age in the late 1970s and 1980s when successively brilliant animation work was produced. Here are 10 must-see classics from China's animation outpouring that are not to be missed. Let's recall these colorful images that brought the country great joy. Calabash Brothers Calabash Brothers (Chinese: 葫芦娃) is a Chinese animation TV series produced by Shanghai Animation Film Studio. In the 1980s the series was one of the most popular animations in China. It was released at a point when the Chinese animation industry was in a relatively downed state compared to the rest of the international community. Still, the series was translated into 7 different languages. The episodes were produced with a vast amount of paper-cut animations. Black Cat Detective Black Cat Detective (Chinese: 黑猫警长) is a Chinese animation television series produced by the Shanghai Animation Film Studio. It is sometimes known as Mr. Black. The series was originally aired from 1984 to 1987. In June 2006, a rebroadcasting of the original series was announced. Critics bemoan the series' violence, and lack of suitability for children's education. Proponents of the show claim that it is merely for entertainment. Effendi "Effendi", meaning sir and teacher in Turkish, is the respectful name for people who own wisdom and knowledge. The hero's real name was Nasreddin. He was wise and witty and, more importantly, he had the courage to resist the exploitation of noblemen. He was also full of compassion and tried his best to help poor people. Adventure of Shuke and Beita【舒克与贝塔】 Adventure of Shuke and Beita (Chinese: 舒克和贝塔) is a classic animation by Zheng Yuanjie, who is known as King of Fairy Tales in China. Shuke and Beita are two mice who don't want to steal food like other mice. Shuke became a pilot and Beita became a tank driver, and the pair met accidentally and became good friends. Then they befriended a boy named Pipilu. With the help of PiPilu, they co-founded an airline named Shuke Beita Airlines to help other animals. Although there are only 13 episodes in this series, the content is very compact and attractive. The animation shows the preciousness of friendship and how people should be brave when facing difficulties. Even adults recalling this animation today can still feel touched by some scenes. Secrets of the Heavenly Book Secrets of the Heavenly Book, (Chinese: 天书奇谈) also referred to as "Legend of the Sealed Book" or "Tales about the Heavenly Book", was released in 1983. The film was produced with rigorous dubbing and fluid combination of music and vivid animations. The story is based on the classic literature "Ping Yao Zhuan", meaning "The Suppression of the Demons" by Feng Menglong. Yuangong, the deacon, opened the shrine and exposed the holy book to the human world. He carved the book's contents on the stone wall of a white cloud cave in the mountains. He was then punished with guarding the book for life by the jade emperor for breaking heaven's law. In order to pass this holy book to human beings, he would have to get by the antagonist fox. The whole animation is characterized by charming Chinese painting, including pavilions, ancient architecture, rippling streams and crowded markets, which fully demonstrate the unique beauty of China's natural scenery. Pleasant Goat and Big Big Wolf【喜洋洋与灰太狼】 Pleasant Goat and Big Big Wolf (Chinese:喜羊羊与灰太狼) is a Chinese animated television series. The show is about a group of goats living on the Green Pasture, and the story revolves around a clumsy wolf who wants to eat them. It is a popular domestic animation series and has been adapted into movies. Nezha Conquers the Dragon King（Chinese: 哪吒闹海） is an outstanding animation issued by the Ministry of Culture in 1979 and is based on an episode from the Chinese mythological novel "Fengshen Yanyi". A mother gave birth to a ball of flesh shaped like a lotus bud. The father, Li Jing, chopped open the ball, and beautiful boy, Nezha, sprung out. One day, when Nezha was seven years old, he went to the nearby seashore for a swim and killed the third son of the Dragon King who was persecuting local residents. The story primarily revolves around the Dragon King's feud with Nezha over his son's death. Through bravery and wit, Nezha finally broke into the underwater palace and successfully defeated him. The film shows various kinds of attractive sceneries and the traditional culture of China, such as spectacular mountains, elegant sea waves and exquisite ancient Chinese clothes. It has received a variety of awards. Havoc in Heaven The story of Havoc in Heaven（Chinese: 大闹天宫）is based on the earliest chapters of the classic story Journey to the West. The main character is Sun Wukong, aka the Monkey King, who rebels against the Jade Emperor of heaven. The stylized animation and drums and percussion accompaniment used in this film are heavily influenced by Beijing Opera traditions. The name of the movie became a colloquialism in the Chinese language to describe someone making a mess. Regardless that it was an animated film, it still became one of the most influential films in all of Asia. Countless cartoon adaptations that followed have reused the same classic story Journey to the West, yet many consider this 1964 iteration to be the most original, fitting and memorable, The Golden Monkey Defeats a Demon【金猴降妖】 The Golden Monkey Defeats a Demon (Chinese: 金猴降妖), also referred as "The Monkey King Conquers the Demon", is adapted from chapters of the Chinese classics "Journey to the West," or "Monkey" in the Western world. The five-episode animation series tells the story of Monkey King Sun Wukong, who followed Monk Xuan Zang's trip to the West to take the Buddhistic sutra. They met a white bone evil, and the evil transformed human appearances three times to seduce the monk. Twice Monkey King recognized it and brought it down. The monk was unable to recognize the monster and expelled Sun Wukong. Xuan Zang was then captured by the monster. Fortunately Bajie, another apprentice of Xuan Zang, escaped and persuaded the Monkey King to come rescue the monk. Finally, Sun kills the evil and saves Xuan Zang. The outstanding animation has received a variety of awards, including the 6th Hundred Flowers Festival Award and the Chicago International Children's Film Festival Award in 1989. McDull【麦兜】 McDull is a cartoon pig character that was created in Hong Kong by Alice Mak and Brian Tse. Although McDull made his first appearances as a supporting character in the McMug comics, McDull has since become a central character in his own right, attracting a huge following in Hong Kong. The first McDull movie McMug Story My Life as McDull documented his life and the relationship between him and his mother.The McMug Story My Life as McDull is also being translated into French and shown in France. In this version, Mak Bing is the mother of McDull, not his father. 5