2015年電大離散數(shù)學期末綜合復習資料小抄

1 電大 離散數(shù)學 期末 綜合復習資料 小抄 一、判斷題 1. （ ）命題聯(lián)結(jié)詞 ， ， 是最小聯(lián)結(jié)詞組。 2. （ ）（ PQ） P 為矛盾式。 3. （ ）（ PQ） （ QR） （ PR）為重言式。 4. （ ） A、 B、 C 是任意命題公式，如果 ACBC，一定有 AB。 5. （ ）若集合 A上的二元關(guān)系 R是對稱的， RC一定是對稱的。 6. （ ） R 是 A 上的二元關(guān)系， R 是自反的，當且僅當 r(R)=R。 7. （ ）集合 A上的等價關(guān)系確定了 A的一個劃分。 8. （ ） 有理 數(shù)集是可數(shù)的。 9. （ ）若函數(shù) f， g 為入射則其復合函數(shù)也為入射 。 10. （ ） R 是集合 A 上的關(guān)系， R 有傳遞性的充要條件是 RoRR。 11. （ ）設(shè) 是一個代數(shù)系統(tǒng)，且集合 A 中元素的個數(shù)大于 1。如果該代數(shù)系統(tǒng)中存在幺元 e 和零元 ，則 e。 12. （ ） 交換群必是循環(huán)群。 13. （ ）一個群可以有多個等冪元。 14. （ ）模格一定是分配格。 15. （ ）每個有向圖中，結(jié)點入度數(shù)總和等于結(jié)點出度總和。 16. （ ） 圖 G 的鄰接矩陣 A， Al中的 i 行 j 列表示結(jié)點 vi到 vj 長度為 l 路的數(shù)目。 17. （ ）任何圖中必有偶數(shù)個度數(shù)為奇數(shù)的結(jié)點。 18. （ ）有向圖中，它的每一個結(jié)點位于且只位于一個單側(cè)分圖中。 19. （ ）任意平面圖最多是四色的。 20. （ ）不存在既有歐拉回路又有漢密爾頓回路的圖。 二、填空題 1 設(shè) P：“天下雨”， Q：“他騎自行車上班”， R：“他乘公共汽車上班”。則命題“除非下雨，否則他就騎自行車上班”可符號化為 ?！八蛘唑T自行車，或者乘公共汽車上班”可符號化為 2 設(shè) N(x)： x 是自然數(shù)； J(x)： x 是奇數(shù)； Q(x)： x 是偶數(shù)，用謂詞公式符號化命題“任何自然數(shù)不是偶數(shù)就是奇數(shù)”。 3 設(shè) P(x)： x 是運動員， Q(x)： x 是教練。則命題“不是所有運動員都是教練”可符號化為 。 4 設(shè) D=a,b； P(a,a)=P(b,b)=T； P(a,b)=P(b,a)=F。則公式 (x)(y)(P(x,y)P(y,x)的真值是 。 5 集合 A=,的冪集 P(A)為 6 集合 A=1,2， B=a,b,c,d， C=c,d,e，則 A(B-C)為 7 試用空集 構(gòu)成集合 A（ A） = 和 B= ，使得 AB 且 AB 都成立。并且 AB= 。 8 設(shè) A=1,2,3， R=,，傳遞閉包 t(R)為 。 9 設(shè) A=1,2,3， B=x,y， f： AB，則不同的函數(shù)個數(shù)為 個。 10 Q 為有理數(shù)集， Q 上定義運算 *為 a*b=a+b-ab，則 的幺元為 。 11 代數(shù)系統(tǒng) ，其中 Sk=x|xZx=K， +為普通加法，則 是一個半群的必要條件是 。 12 設(shè) G 為 v 個結(jié)點 e 條邊的連通平面圖，則面 r 等于 。 13 一棵樹有 n2 個結(jié)點度數(shù)為 2， n3個結(jié)點度數(shù)為 3， nk個結(jié)點度數(shù)為 k，則度數(shù)為 1 的結(jié)點的個數(shù)為 。 14 設(shè) T 為根樹，若每個結(jié)點的出度都小 于等于 m，則 T 稱為 樹，若除 外，每個結(jié)點的出度都等于 m，則 T 稱為完全 m 叉樹。 2 15 設(shè) 是偏序集，如果 A 中任意兩個元素都有 和 ，則稱 為格。 三 、解答題 1. 將公式 (PQ) (QR)(PR)化成與之等價且僅含 、 、 的公式。 2. 將下列命題符號化 ： （ 1）他雖聰明但不用功。 （ 2）除非你努力否則你將失敗。 （ 3）我們不能既劃船又跑步 （ 4）僅當你走我才留下。 3. 用謂詞表達式符號化下列命題： （ 1）所有老的國家選手都是運動員。 （ 2）某些 教練是年老的，但是健壯的。 （ 3）任何自然數(shù)不是偶數(shù)就是奇數(shù) 。 （ 4）不是所有運動員都是教練 。 4. 求命題公式 (PQ)的主合取范式。 5. 求命題公式 P(PQ)的主析取范式。 6. 設(shè)集合 A 1, 2, 3， A 上的關(guān)系 R , （ 1） 畫出 R 的關(guān)系圖； （ 2） 寫出 R 的關(guān)系矩陣 ； （ 2）問 R 具有關(guān)系的哪幾種性質(zhì) (自反、反自反、對稱、反對稱、傳遞 )。 7. 構(gòu)造一非空偏序集，它存在一子集有上界，但沒有最小上界。它還有一子集，存在最大下界但沒有最小元。 8. 以下哪些是函數(shù)？哪些是入射？哪些是滿射？對任意一個雙射，寫出它們的逆函數(shù)。 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. 設(shè) S=1,2,3,4,6,12， D為 S上的整除關(guān)系， （ 1）試寫出該關(guān)系并畫出哈斯圖； （ 2）設(shè)子集 B=2,3,6，試求 B的最大元、最小元、極大元和極小元； （ 3）試求 B的上界、 上確界、下界和下確界。 10. 設(shè)集合 A 有 m 個元素， B 有 n 個元素，則 A 到 B 的關(guān)系有多少個？ A 到 B 的函數(shù)有多少個？ 11. 判定 下列 代數(shù)系統(tǒng)是否為群，請說明原因。 （ 1） ，其中 R為實數(shù)集， +為普通加法； （ 2） ，其中 I為整數(shù)集， 為普通乘法 12. 設(shè)群 的運算表如下： * e a b e e a b a a b e b b e a 試寫出 的所有子群，及其相應的左陪集。 13. 設(shè) G=， V=V1,V2,V3,V4的鄰接矩陣： A(G)= 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 3 （ 1）試畫出該圖。 （ 2） V2 的 入度 d-(V2)和出度 d+(V2)是多少？ （ 3）從 V2 到 V4 長度為 2 的路有幾條？ 14. 試求下面有向圖的強分圖、單側(cè)分圖和弱分圖 15. （ 1）畫一個有歐拉回路和一條漢密爾頓回路的圖。 （ 2）畫一個有歐拉回路，但沒有漢密爾頓回路的圖。 （ 3）畫一個沒有歐拉回路，但有漢密爾頓回路的圖。 16. 下圖給出的賦權(quán)圖表示五個城市及對應兩個城鎮(zhèn)間公路的長度。是給出一個最優(yōu)的設(shè)計方案使各城市間有公路連通。 17. 設(shè)有一組權(quán) 3、 4、 13、 5、 6、 12， （ 1）求相應的最優(yōu)樹（要求構(gòu)造的過程中，每個分支點的左 兒子的權(quán)小于右兒子的權(quán)）。 （ 2）設(shè)上述權(quán)值分別對應英文字母 b、 d、 e、 g、 o、 y，試根據(jù)求得的最優(yōu)樹構(gòu)造前綴碼，并對二進制序列0100110110010001011 譯碼。 四 、證明題 1. A (BC),(EF)C,B(AS)BE 2. 試 證明 命題 公式 ( ( ) ( ) ) ( )P Q Q R P R 為永真式 。 3. 試證明 ： (PQ) (PR) (QS) SR 4. 用推理規(guī)則證明： (x)(P(x)Q(x) (x) P(x)(y)(P(y)Q(y) 5. 對所有集合 A、 B 和 C，有 (AB)C=A(BC)，當且僅當 CA。 6. 若 R和 S是集合 A上的等價關(guān)系，試證明 RS也是 A上的等價關(guān)系。 7. 證明集合 0,1和 (0,1)是等勢的。 8. 設(shè) f: X-Y 和 g: Y-Z 是函數(shù)，使得 gf 是一個滿射，且 g 是一個入射。證明 f 是滿射。 9. 設(shè) ,是兩個群，在 G1G2 上定義運算 為： =，證明 是一個群。 10. f是群 到群 的同態(tài)映射， e是 G中的幺元則， f的同態(tài)核 K=x|xG且 f(x)=e構(gòu)成的代數(shù)系統(tǒng) 是 的子群。 11. 證明在格中，若 abc，則 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 個人，每個人恰有三個朋友，證明 n 必為偶數(shù)。 13. 證明當且僅當 G 的一條邊 e 不包含在 G 的回路中時， e 才是 G 的割邊。 14. 畫出 K3,3 圖，并證明其不是歐拉圖，也不是平面圖。 15. 設(shè) G 為連通圖，證明當且僅當邊 e 是 G 的割邊時， e 才在 G 的每顆生成樹中。 16. 設(shè) T 是非平凡的無向樹， T 中度數(shù)最大的結(jié)點有 2 個，它們的度數(shù)為 k（ k=2），證明： T 中至少有 2k-2 片樹葉。 17. 設(shè) G=有 11 個結(jié)點， m 條邊，證明 G 或者其補圖 G是非平面圖。 部分參考 答案 一、判斷題 1. （錯誤） 2. （正確） 3. （正確） 4. （錯誤） 5. （正確） 6. （正確） 7. （正確） 8. （正確） 9. （正確） 10. （正確） 11. （正確） 12. （錯誤） 13. （錯誤） 14. （錯誤） 15. （正確） 16. （正確） 17. （正確） 18. （正確） 19. （正確） 20. （錯誤） 5 請您刪除一下內(nèi)容， O( _ )O 謝謝！ 2015 年中央電大期末復習考試小抄大全，電大期末考試必備小抄，電大考試必過小抄 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