2015年电大离散数学期末综合复习资料小抄

1 电大 离散数学 期末 综合复习资料 小抄 一、判断题 1. （ ）命题联结词 ， ， 是最小联结词组。 2. （ ）（ PQ） P 为矛盾式。 3. （ ）（ PQ） （ QR） （ PR）为重言式。 4. （ ） A、 B、 C 是任意命题公式，如果 ACBC，一定有 AB。 5. （ ）若集合 A上的二元关系 R是对称的， RC一定是对称的。 6. （ ） R 是 A 上的二元关系， R 是自反的，当且仅当 r(R)=R。 7. （ ）集合 A上的等价关系确定了 A的一个划分。 8. （ ） 有理 数集是可数的。 9. （ ）若函数 f， g 为入射则其复合函数也为入射 。 10. （ ） R 是集合 A 上的关系， R 有传递性的充要条件是 RoRR。 11. （ ）设 是一个代数系统，且集合 A 中元素的个数大于 1。如果该代数系统中存在幺元 e 和零元 ，则 e。 12. （ ） 交换群必是循环群。 13. （ ）一个群可以有多个等幂元。 14. （ ）模格一定是分配格。 15. （ ）每个有向图中，结点入度数总和等于结点出度总和。 16. （ ） 图 G 的邻接矩阵 A， Al中的 i 行 j 列表示结点 vi到 vj 长度为 l 路的数目。 17. （ ）任何图中必有偶数个度数为奇数的结点。 18. （ ）有向图中，它的每一个结点位于且只位于一个单侧分图中。 19. （ ）任意平面图最多是四色的。 20. （ ）不存在既有欧拉回路又有汉密尔顿回路的图。 二、填空题 1 设 P：“天下雨”， Q：“他骑自行车上班”， R：“他乘公共汽车上班”。则命题“除非下雨，否则他就骑自行车上班”可符号化为 。“他或者骑自行车，或者乘公共汽车上班”可符号化为 2 设 N(x)： x 是自然数； J(x)： x 是奇数； Q(x)： x 是偶数，用谓词公式符号化命题“任何自然数不是偶数就是奇数”。 3 设 P(x)： x 是运动员， Q(x)： x 是教练。则命题“不是所有运动员都是教练”可符号化为 。 4 设 D=a,b； P(a,a)=P(b,b)=T； P(a,b)=P(b,a)=F。则公式 (x)(y)(P(x,y)P(y,x)的真值是 。 5 集合 A=,的幂集 P(A)为 6 集合 A=1,2， B=a,b,c,d， C=c,d,e，则 A(B-C)为 7 试用空集 构成集合 A（ A） = 和 B= ，使得 AB 且 AB 都成立。并且 AB= 。 8 设 A=1,2,3， R=,，传递闭包 t(R)为 。 9 设 A=1,2,3， B=x,y， f： AB，则不同的函数个数为 个。 10 Q 为有理数集， Q 上定义运算 *为 a*b=a+b-ab，则 的幺元为 。 11 代数系统 ，其中 Sk=x|xZx=K， +为普通加法，则 是一个半群的必要条件是 。 12 设 G 为 v 个结点 e 条边的连通平面图，则面 r 等于 。 13 一棵树有 n2 个结点度数为 2， n3个结点度数为 3， nk个结点度数为 k，则度数为 1 的结点的个数为 。 14 设 T 为根树，若每个结点的出度都小 于等于 m，则 T 称为 树，若除 外，每个结点的出度都等于 m，则 T 称为完全 m 叉树。 2 15 设 是偏序集，如果 A 中任意两个元素都有 和 ，则称 为格。 三 、解答题 1. 将公式 (PQ) (QR)(PR)化成与之等价且仅含 、 、 的公式。 2. 将下列命题符号化 ： （ 1）他虽聪明但不用功。 （ 2）除非你努力否则你将失败。 （ 3）我们不能既划船又跑步 （ 4）仅当你走我才留下。 3. 用谓词表达式符号化下列命题： （ 1）所有老的国家选手都是运动员。 （ 2）某些 教练是年老的，但是健壮的。 （ 3）任何自然数不是偶数就是奇数 。 （ 4）不是所有运动员都是教练 。 4. 求命题公式 (PQ)的主合取范式。 5. 求命题公式 P(PQ)的主析取范式。 6. 设集合 A 1, 2, 3， A 上的关系 R , （ 1） 画出 R 的关系图； （ 2） 写出 R 的关系矩阵 ； （ 2）问 R 具有关系的哪几种性质 (自反、反自反、对称、反对称、传递 )。 7. 构造一非空偏序集，它存在一子集有上界，但没有最小上界。它还有一子集，存在最大下界但没有最小元。 8. 以下哪些是函数？哪些是入射？哪些是满射？对任意一个双射，写出它们的逆函数。 a) f: ZN, f(x)=x2+1 b) f: NQ, f(x) = 1/x c) f: 1,2,3a,b,c, f=, d) f: NN, f(x)=2x e) f: RRRR, f(x,y)= 9. 设 S=1,2,3,4,6,12， D为 S上的整除关系， （ 1）试写出该关系并画出哈斯图； （ 2）设子集 B=2,3,6，试求 B的最大元、最小元、极大元和极小元； （ 3）试求 B的上界、 上确界、下界和下确界。 10. 设集合 A 有 m 个元素， B 有 n 个元素，则 A 到 B 的关系有多少个？ A 到 B 的函数有多少个？ 11. 判定 下列 代数系统是否为群，请说明原因。 （ 1） ，其中 R为实数集， +为普通加法； （ 2） ，其中 I为整数集， 为普通乘法 12. 设群 的运算表如下： * e a b e e a b a a b e b b e a 试写出 的所有子群，及其相应的左陪集。 13. 设 G=， V=V1,V2,V3,V4的邻接矩阵： A(G)= 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 3 （ 1）试画出该图。 （ 2） V2 的 入度 d-(V2)和出度 d+(V2)是多少？ （ 3）从 V2 到 V4 长度为 2 的路有几条？ 14. 试求下面有向图的强分图、单侧分图和弱分图 15. （ 1）画一个有欧拉回路和一条汉密尔顿回路的图。 （ 2）画一个有欧拉回路，但没有汉密尔顿回路的图。 （ 3）画一个没有欧拉回路，但有汉密尔顿回路的图。 16. 下图给出的赋权图表示五个城市及对应两个城镇间公路的长度。是给出一个最优的设计方案使各城市间有公路连通。 17. 设有一组权 3、 4、 13、 5、 6、 12， （ 1）求相应的最优树（要求构造的过程中，每个分支点的左 儿子的权小于右儿子的权）。 （ 2）设上述权值分别对应英文字母 b、 d、 e、 g、 o、 y，试根据求得的最优树构造前缀码，并对二进制序列0100110110010001011 译码。 四 、证明题 1. A (BC),(EF)C,B(AS)BE 2. 试 证明 命题 公式 ( ( ) ( ) ) ( )P Q Q R P R 为永真式 。 3. 试证明 ： (PQ) (PR) (QS) SR 4. 用推理规则证明： (x)(P(x)Q(x) (x) P(x)(y)(P(y)Q(y) 5. 对所有集合 A、 B 和 C，有 (AB)C=A(BC)，当且仅当 CA。 6. 若 R和 S是集合 A上的等价关系，试证明 RS也是 A上的等价关系。 7. 证明集合 0,1和 (0,1)是等势的。 8. 设 f: X-Y 和 g: Y-Z 是函数，使得 gf 是一个满射，且 g 是一个入射。证明 f 是满射。 9. 设 ,是两个群，在 G1G2 上定义运算 为： =，证明 是一个群。 10. f是群 到群 的同态映射， e是 G中的幺元则， f的同态核 K=x|xG且 f(x)=e构成的代数系统 是 的子群。 11. 证明在格中，若 abc，则 V1 V2 V3 V4 V5 4 3 2 5 1 1 2 2 v1 v3 v2 v5 v4 4 （ 1） ab=bc （ 2） (ab)(bc)=b=(ab)(ac) 12. 若有 n 个人，每个人恰有三个朋友，证明 n 必为偶数。 13. 证明当且仅当 G 的一条边 e 不包含在 G 的回路中时， e 才是 G 的割边。 14. 画出 K3,3 图，并证明其不是欧拉图，也不是平面图。 15. 设 G 为连通图，证明当且仅当边 e 是 G 的割边时， e 才在 G 的每颗生成树中。 16. 设 T 是非平凡的无向树， T 中度数最大的结点有 2 个，它们的度数为 k（ k=2），证明： T 中至少有 2k-2 片树叶。 17. 设 G=有 11 个结点， m 条边，证明 G 或者其补图 G是非平面图。 部分参考 答案 一、判断题 1. （错误） 2. （正确） 3. （正确） 4. （错误） 5. （正确） 6. （正确） 7. （正确） 8. （正确） 9. （正确） 10. （正确） 11. （正确） 12. （错误） 13. （错误） 14. （错误） 15. （正确） 16. （正确） 17. （正确） 18. （正确） 19. （正确） 20. （错误） 5 请您删除一下内容， O( _ )O 谢谢！ 2015 年中央电大期末复习考试小抄大全，电大期末考试必备小抄，电大考试必过小抄 After earning his spurs in the kitchens of The Westin, The Sheraton, Sens on the Bund, and a sprinkling of other top-notch venues, Simpson Lu fi nally got the chance to become his own boss in November 2010. Sort of. The Shanghai-born chef might not actually own California Pizza Kitchen (CPK) but he is in sole charge of both kitchen and frontof- house at this Sinan Mansionsstalwart. Its certainly a responsibility to be the head chef, and then to have to manage the rest of the restaurant as well, the 31-year-old tells Enjoy Shanghai. In hotels, for example, these jobs are strictly demarcated, so its a great opportunity to learn how a business operates across the board. It was a task that management back in sunny California evidently felt he was ready for, and a vote of confi dence from a company that, to date, has opened 250 outlets in 11 countries. And for added pressure, the Shanghai branch was also CPKs China debut. For sure it was a big step, and unlike all their other Asia operations that are franchises, they decided to manage it directly to begin with, says Simpson. Two years ago a private franchisee took over the lease, but the links to CPK headquarters are still strong, with a mainland-based brand ambassador on hand to ensure the business adheres to its ethos of creating innovative, hearth-baked pizzas, a slice of PR blurb that Simpson insists lives up to the hype. They are very innovative, he says. The problem with most fast food places is that they use the same sauce on every pizza and just change the toppings. Every one of our 16 pizza sauces is a unique recipe that has been formulated to complement the toppings perfectly. The largely local customer base evidently agrees and on Saturday and Sunday, at least, the place is teeming. The kids-eat-for-free policy at weekends is undoubtedly a big draw, as well as is the spacious second-fl oor layout overlooked by a canopy of green from Fuxing Park over the road. The company is also focusing on increasing brand recognition and in recent years has taken part in outside events such as the regular California Week. Still, the sta are honest enough to admit that business could be better； as good, in fact, as in CPKs second outlet in the popular Kerry Parkside shopping mall in Pudong. Sinan Mansions has really struggled to get the number of visitors that were envisaged when it first opened, and it hasnt been easy for any of the tenants here, adds Simpson. Were planning a third outlet in the city in 2015, and we will probably choose a shopping mall again because of the better foot traffic. The tearooms once frequented by Coco Chanel and Marcel Proust are upping sticks and coming to Shanghai, Xu Junqian visits the Parisian outpost with sweet treats. One thing the century-old Parisian tearoom Angelina has shown is that legendary fashion designer Coco Chanel not only had style and glamor but also boasted great taste in food, pastries in particular. One of the most popular tearooms in Paris, Angelina is famous for having once been frequented by celebrities such as Chanel and writer Marcel Proust. Now Angelina has packed up its French ambience, efficient service, and beautiful, comforting desserts and flown them to Shanghai. At the flagship dine-in and take-out space in Shanghai, everything mimics the original tearoom designed from the beginning of the 20th century, in Paris, the height of Belle Epoque. The paintings on the wall, for example, are exactly the same as the one that depicts the landscape of southern France, the hometown of the owner； and the small tables are intentional imitations of the ones that Coco Chanel once sat at every afternoon for hot chocolate. The famous hot chocolate, known as LAfricain, is a luxurious mixture of four types of cocoa beans imported from Africa, blended in Paris and then shipped to Shanghai. Its sinfully sweet, rich and thick as if putting a bar of melting chocolate directly on the tongue and the fresh whipped cream on the side makes a light, but equally gratifying contrast. It is also sold in glass bottles as takeaway. The signature Mont-Blanc chestnut cake consists of three parts: the pureed chestnut on top, the vanilla cream like stuffing, and the meringue as base. Get all three layers in one scoop, not only for the different textures but also various flavors of sweetness. The dessert has maintained its popularity for a century, even in a country like France, perhaps the worlds most competitive place for desserts. A much overlooked pairing, is the Paris-New York choux pastry and N226 chocolate flavored tea. The choux pastry is a mouthful of airy pecan-flavored whipped cream, while the tea, a blend of black teas from China and Ceylon, cocoa and rose petals, offers a more subtle fragrance of flowers and chocolate. Ordering these two items, featuring a muted sweetness, makes it easier for you to fit into your little black dress. Breakfast, brunch, lunch and light supper are also served at the tearoom, a hub of many cultures and takes in a mix of different styles of French cuisines, according to the management team. The semi-cooked foie gras terrine, is seductive and deceptive. Its generously served at the size and shape of a toast, while the actual brioche toast is baked into a curved slice dipped with fig chutney. The flavor, however, is honest: strong, smooth and sublime. And you dont actually need the toast for crunchiness. This is the season for high teas, with dainty cups of fine china and little pastries that appeal to both visual and physical appetites. But there is one high tea with a difference, and Pauline D. Loh finds out just exactly why it is special. Earl Grey tea and macarons are all very well for the crucial recuperative break in-between intensive bouts of holiday season shopping. And for those who prefer savory to sweet, there is still the selection of classic Chinese snacks called dim sum to satisfy and satiate. High tea is a meal to eat with eye and mouth, an in-between indulgence that should be light enough not to spoil dinner, but sufficiently robust to take the edge off the hunger that strikes hours after lunch. The afternoon tea special at Shang-Xi at the Four Seasons Hotel Pudong has just the right elements. It is a pampering meal, with touches of luxury that make the high tea session a treat in itself. Whole baby abalones are braised and then topped on a shortcrust pastry shell, a sort of Chinese version of the Western vol-au-vent, but classier. Even classier is the dim sum staple shrimp dumpling or hargow, upgraded with the addition of slivers of midnight dark truffles. This is a master touch, and chef Simon Choi, who presides unchallenged at Shang-Xi, has scored a winner again. Sweet prawns and aromatic truffles whats not to love? His masterful craftsmanship is exhibited in yet another pastry a sweet pastry that is shaped to look like a walnut, but which you can put straight into the mouth. It crumbles immediately, and the slightly sweet, nutty morsel is so easy to eat youll probably reach straight for another. My favorite is the dessert that goes by the name yangzhi ganlu, or ambrosia from the gods. The hotel calls it chilled mango cream with sago, pomelo and birds nest made with ingredients that resonate with every female soul. It does taste like ambrosia, with the sweet-sour fragrance of the mango forming the first layer o