电大离散数学期末复习要点与重点考试资料小抄.doc

电大离散数学期末复习要点与重点考试资料小抄离散数学是中央广播电视大学开放教育本科电气信息类计算机科学与技术专业的一门统设必修学位课程，共72学时，开设一学期该课程的主要内容包括：集合论、图论、数理逻辑等下面按章给出复习要点与重点第1章 集合及其运算复习要点1理解集合、元素、集合的包含、子集、相等，以及全集、空集和幂集等概念，熟练掌握集合的表示方法具有确定的，可以区分的若干事物的全体称为集合，其中的事物叫元素.集合的表示方法：列举法和描述法. 注意：集合的表示中元素不能重复出现，集合中的元素无顺序之分掌握集合包含(子集)、真子集、集合相等等概念注意：元素与集合，集合与子集，子集与幂集，Î与Ì(Í),空集Æ与所有集合等的关系.空集Æ，是惟一的，它是任何集合的子集集合A的幂集P(A)， A的所有子集构成的集合若½A½n，则½P(A)½=2n2熟练掌握集合A和B的并AÈB，交AÇB，补集A(A补集总相对于一个全集).差集AB，对称差Å，AÅB(AB)È(BA)，或AÅB(AÈB)（AÇB）等运算，并会用文氏图表示掌握集合运算律(见教材第911页)(运算的性质).3掌握用集合运算基本规律证明集合恒等式的方法集合的运算问题：其一是进行集合运算；其二是运算式的化简；其三是恒等式证明证明方法有二：(1)要证明AB，只需证明AÍB，又AÊB；(2)通过运算律进行等式推导重点：集合概念，集合的运算，集合恒等式的证明第2章 关系与函数复习要点1了解有序对和笛卡儿积的概念，掌握笛卡儿积的运算有序对就是有顺序二元组，如<x, y>，x, y 的位置是确定的，不能随意放置 注意：有序对<a，b>¹<b, a>，以a, b为元素的集合a, b=b, a；有序对(a, a)有意义，而集合a, a是单元素集合，应记作a集合A，B的笛卡儿积A×B是一个集合，规定A×B<x,y>½xÎA,yÎB，是有序对的集合.笛卡儿积也可以多个集合合成，A1×A2××An 2理解关系的概念：二元关系、空关系、全关系、恒等关系.掌握关系的集合表示、关系矩阵和关系图，掌握关系的集合运算和求复合关系、逆关系的方法二元关系是一个有序对集合，记作xRy 关系的表示方法有三种：集合表示法， 关系矩阵：RÍA×B，R的矩阵. 关系图：R是集合上的二元关系，若<ai, bj>ÎR，由结点ai画有向弧到bj构成的图形空关系Æ是唯一、是任何关系的子集的关系；全关系；恒等关系，恒等关系的矩阵MI是单位矩阵关系的集合运算有并、交、补、差和对称差复合关系；复合关系矩阵：(按布尔运算)； 有结合律：(R·S)·TR·(S·T)，一般不可交换逆关系；逆关系矩阵满足：；复合关系与逆关系存在：(R·S)1=S1·R13理解关系的性质(自反性和反自反性、对称性和反对称性、传递性的定义以及矩阵表示或关系图表示)，掌握其判别方法(利用定义、矩阵或图，充分条件)，知道关系闭包的定义和求法注：(1)关系性质的充分必要条件： R是自反的ÛIAÍR；R是反自反的ÛIAÇRÆ；R是对称的 ÛRR1；R是反对称的ÛRÇR1ÍIA；R是传递的ÛR·RÍR. (2)IA具有自反性，对称性、反对称性和传递性EA具有自反性，对称性和传递性故IA，EA是等价关系Æ具有反自反性、对称性、反对称性和传递性IA也是偏序关系4理解等价关系和偏序关系概念，掌握等价类的求法和作偏序集哈斯图的方法知道极大(小)元，最大(小)元的概念，会求极大(小)元、最大(小)元、最小上界和最大下界等价关系和偏序关系是具有不同性质的两个关系. 知道等价关系图的特点和等价类定义，会求等价类一个子集的极大(小)元可以有多个，而最大(小)元若有，则惟一.且极元、最元只在该子集内；而上界与下界可以在子集之外由哈斯图便于确定任一子集的最大(小)元，极大(小)元5理解函数概念：函数(映射)，函数相等，复合函数和反函数理解单射、满射和双射等概念，掌握其判别方法设f是集合A到B的二元关系，"aÎA，存在惟一bÎB，使得<a, b>Îf，且Dom(f)=A，f是一个函数(映射)函数是一种特殊的关系集合A×B的任何子集都是关系，但不一定是函数函数要求对于定义域A中每一个元素a，B中有且仅有一个元素与a对应，而关系没有这个限制 二函数相等是指：定义域相同，对应关系相同，而且定义域内的每个元素的对应值都相同 函数有：单射若；满射f(A)=B或使得y=f(x)；双射单射且满射 复合函数 即复合成立的条件是：一般，但.反函数若f：A®B是双射，则有反函数f1：B®A，重点：关系概念与其性质，等价关系和偏序关系，函数. 第3章 图的基本概念复习要点1理解图的概念：结点、边、有向图，无向图、简单图、完全图、结点的度数、边的重数和平行边等.理解握手定理图是一个有序对<V，E>，V是结点集，E是联结结点的边的集合掌握无向边与无向图，有向边与有向图，混合图，零图，平凡图、自回路(环)，无向平行边，有向平行边等概念简单图，不含平行边和环(自回路)的图、 在无向图中，与结点v(ÎV)关联的边数为结点度数(v)；在有向图中，以v(ÎV)为终点的边的条数为入度(v)，以v(ÎV)为起点的边的条数为出度(v)，deg(v)=deg+(v) +deg(v)无向完全图Kn以其边数；有向完全图以其边数了解子图、真子图、补图和生成子图的概念生成子图设图G<V, E>，若E¢ÍE，则图<V, E¢>是<V, E>的生成子图 知道图的同构概念，更应知道图同构的必要条件，用其判断图不同构.重要定理：(1) 握手定理 设G=<V，E>，有；(2) 在有向图D<V, E>中，；(3) 奇数度结点的个数为偶数个 2了解通路与回路概念：通路(简单通路、基本通路和复杂通路)，回路(简单回路、基本回路和复杂回路)会求通路和回路的长度基本通路(回路)必是简单通路(回路) 了解无向图的连通性，会求无向图的连通分支了解点割集、边割集、割点、割边等概念了解有向图的强连通强性；会判别其类型设图G<V，E>，结点与边的交替序列为通路通路中边的数目就是通路的长度起点和终点重合的通路为回路边不重复的通路(回路)是简单通路(回路)；结点不重复的通路(回路)是基本通路(回路). 无向图G中，结点u, v存在通路，u, v是连通的，G中任意结点u, v连通，G是连通图P(G)表示图G连通分支的个数 在无向图中，结点集V¢ÌV，使得P(GV¢)>P(G)，而任意V²ÌV¢,有P（GV²）P(G)，V¢为点割集. 若V¢是单元集，该结点v叫割点；边集E¢ÌE，使得P(GV¢)>P(G)，而任意E²ÌE¢，有P（GE²）P(G)，E¢为边割集若E¢是单元集，该边e叫割边(桥)要知道：强连通单侧连通弱连通，反之不成立3了解邻接矩阵和可达矩阵的概念，掌握其构造方法及其应用重点：图的概念，握手定理，通路、回路以及图的矩阵表示 第4章 几种特殊图复习要点1理解欧拉通路(回路)、欧拉图的概念，掌握欧拉图的判别方法通过连通图G的每条边一次且仅一次的通路(回路)是欧拉通路(回路)存在欧拉回路的图是欧拉图. 欧拉回路要求边不能重复，结点可以重复笔不离开纸，不重复地走完所有的边，走过所有结点，就是所谓的一笔画欧拉图或通路的判定定理(1) 无向连通图G是欧拉图ÛG不含奇数度结点(即G的所有结点为偶数度)；(2) 非平凡连通图G含有欧拉通路ÛG最多有两个奇数度的结点；(3) 连通有向图D含有有向欧拉回路ÛD中每个结点的入度出度连通有向图D含有有向欧拉通路ÛD中除两个结点外，其余每个结点的入度出度，且此两点满足deg(u)deg(v)±12理解汉密尔顿通路(回路)、汉密尔顿图的概念，会做简单判断通过连通图G的每个结点一次，且仅一次的通路(回路)，是汉密尔顿通路(回路)存在汉密尔顿回路的图是汉密尔顿图. 汉密尔顿图的充分条件和必要条件 (1) 在无向简单图G=<V，E>中，½V½³3，任意不同结点，则G是汉密尔顿图(充分条件)(2) 有向完全图D<V，E>, 若，则图D是汉密尔顿图. (充分条件)(3) 设无向图G=<V，E>，任意V1ÌV，则W(GV1)£½V1½(必要条件)若此条件不满足，即存在V1ÌV，使得P(GV!)>½V1½,则G一定不是汉密尔顿图(非汉密尔顿图的充分条件)3了解平面图概念，平面图、面、边界、面的次数和非平面图掌握欧拉公式的应用平面图是指一个图能画在平面上，除结点之外，再没有边与边相交 面、边界和面的次数等概念重要结论：(1)平面图(2)欧拉公式：平面图 面数为r，则(结点数与面数之和边数2)(3)平面图 会用定义判定一个图是不是平面图 4理解平面图与对偶图的关系、对偶图在图着色中的作用，掌握求对偶图的方法给定平面图GV，E，它有面F1，F2，Fn，若有图G*V*，E*满足下述条件： 对于图G的任一个面Fi，内部有且仅有一个结点vi*V*；对于图G的面Fi，Fj的公共边ek，存在且仅存在一条边ek*E*，使ek*(vi*，vj*)，且ek*和ek相交； 当且仅当ek只是一个面Fi的边界时，vi*存在一个环ek*和ek相交；则图G*是图G的对偶图若G*是G的对偶图，则G也是G*的对偶图一个连通平面图的对偶图也必是平面图5掌握图论中常用的证明方法重点：欧拉图和哈密顿图、平面图的基本概念及判别第5章 树及其应用复习要点1了解树、树叶、分支点、平凡树、生成树和最小生成树等概念，掌握求最小生成树的方法连通无回路的无向图是树树的判别可以用图T是树的充要条件(等价定义)注意：(1) 树T是连通图； (2)树T满足mn1（即边数=顶点数-1）图G的生成子图是树，该树就是生成树每边指定一正数，称为权，每边带权的图称为带权图G的生成树T的所有边的权之和是生成树T的权，记作W(T)最小生成树是带权最小的生成树2了解有向树、根树、有序树、二叉树、二叉完全树、正则二叉树和最优二叉树等概念了解带权二叉树、最优二叉树的概念，掌握用哈夫曼算法求最优二叉树的方法有向图删去边的方向为树，该图为有向树 对非平凡有向树，恰有一个结点的入度为0(该结点为树根)，其余结点的入度为1，该树为根树 每个结点的出度小于或等于2的根树为二叉树；每个结点的出度等于0或2的根树为二叉完全树；每个结点的出度等于2的根树称为正则二叉树有关树的求法：(1)生成树的破圈法和避圈法求法；(2)最小生成树的克鲁斯克尔求法；(3) 最优二叉树的哈夫曼求法重点：树与根树的基本概念，最小生成树与最优二叉树的求法.第6章 命题逻辑复习要点1理解命题概念，会判别语句是不是命题理解五个联结词：否定ØP、析取Ú、合取Ù、条件®、和双条件«及其真值表，会将简单命题符号化具有确定真假意义的陈述句称为命题命题必须具备：其一，语句是陈述句；其二，语句有唯一确定的真假意义.2了解公式的概念(公式、赋值、成真指派和成假指派)和公式真值表的构造方法能熟练地作公式真值表理解永真式和永假式概念，掌握其判别方法判定命题公式类型的方法：其一是真值表法，其二是等价演算法.3了解公式等价概念，掌握公式的重要等价式和判断两个公式是否等价的有效方法：等价演算法、列真值表法和主范式方法4理解析取范式和合取范式、极大项和极小项、主析取范式和主合取范式的概念，熟练掌握它们的求法命题公式的范式不惟一，但主范式是惟一的 命题公式A有n个命题变元，A的主析取范式有k个极小项，有m个极大项，则于是有(1) A是永真式Ûk=2n(m=0)； (2) A是永假式Ûm2n(k=0)；求命题公式A的析取(合取)范式的步骤：见教材第174页求命题公式A的主析取(合取)范式的步骤：见教材第177和178页5了解C是前提集合A1,A2,Am的有效结论或由A1, A2, , Am 逻辑地推出C的概念要理解并掌握推理理论的规则、重言蕴含式和等价式，掌握命题公式的证明方法：真值表法、直接证法、间接证法重点：命题与联结词，公式与解释，真值表，公式的类型及判定，主析取(合取)范式，命题演算的推理理论.第7章 谓词逻辑复习要点1理解谓词、量词、个体词、个体域，会将简单命题符号化原子命题分成个体词和谓词，个体词可以是具体事物或抽象的概念，分个体常项和个体变项谓词用来刻划个体词的性质或之间的关系量词分全称量词"，存在量词$.命题符号化注意：使用全称量词"，特性谓词后用®；使用存在量词$，特性谓词后用Ù2了解原子公式、谓词公式、变元(约束变元和自由变元)与辖域等概念掌握在有限个体域下消去公式的量词和求公式在给定解释下真值的方法由原子公式、联结词和量词构成谓词公式谓词公式具有真值时，才是命题在谓词公式"xA或$xA中，x是指导变元，A是量词的辖域会区分约束变元和自由变元在非空集合D(个体域)上谓词公式A的一个解释或赋值有3个条件在任何解释下，谓词公式A取真值1，A为逻辑有效式(永真式)；公式A取真值0，A为永假式；至少有一个解释使公式A取真值1，A称为可满足式在有限个体域下，消除量词的规则为：设Da1, a2, , an，则会求谓词公式的真值，量词的辖域，自由变元、约束变元，以及换名规则、代入规则等掌握谓词演算的等价式和重言蕴含式并进行谓词公式的等价演算3了解前束范式的概念，会求公式的前束范式的方法.若一个谓词公式F等价地转化成，那么就是F的前束范式，其中Q1，Q2，Qk只能是"或$，而x1, x2, , xk是个体变元，B是不含量词的谓词公式前束范式仍然是谓词公式 4了解谓词逻辑推理的四个规则会给出推理证明谓词演算的推理是命题演算推理的推广和扩充，命题演算中基本等价式，重言蕴含式以及P，T，CP规则在谓词演算中仍然使用谓词逻辑的推理演算引入了US规则(全称量词指定规则)，UG规则(全称量词推广规则)，ES规则(存在量词指定规则)，EG规则(存在量词推广规则)等.重点：谓词与量词，公式与解释，谓词演算请您删除一下内容，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. 6