电大离散数学(本)复习题（小抄参考） .doc

专业好文档离散数学（本）复习题1设A=1，2，B=2，3，4，求r(AÇB)，r(A)Çr(B)。 2设A=a，b，c，问IA，EA是否具有自反性，反自反性，对称性，反对称性，传递性？3R，S是集合A上的两个关系。试证明下列等式：（1）(RS)-1= S-1R-1（2）(R-1)-1= R（3）(RS)-1= R-1S-1（4）(RS)-1= R-1S-14设R是集合A上的关系，令R+=(x, y)|xÎA，yÎA，并且存在n>0，使得xRny，则称R+是R的传递闭包，证明：R+是包含R的最小具有传递性的关系。5若非空集合上的非空关系R是反自反的，是对称的，试证明R不是传递的。6A=1，2，3，4，5，6，7，8，9，10，R为A上的整除关系，请给出A的Hasse图，并求出所有的极大元素，极小元素，最大元素，最小元素。7设G是含有3个不同原子的命题公式，当G是恒假公式的时候，G的主析取范式中有多少极小项，主合取范式中有多少极大项？8有人说：“等价关系中的反身性可以不要，因为反身性可以从对称性和传递性推出：由对称性，从a b可得b a，再由传递性得a a”。你的意见呢？9若集合A上的关系R，S具有对称性，证明：RS具有对称性的充要条件为RS= SR。 10若R是等价关系，试证明R-1也是等价关系。11给P和Q指派真值1，给R和S指派真值0，求出下面命题的真值：a) (PÙ(QÙR)ÚØ(PÚQ)Ù(RÚS)b) (Ø(PÙQ)ÚØR)Ú(ØPÙQ)ÚØR)ÙS)c) (Ø(PÙQ)ÚØR)Ú(Q«ØP)®(RÚØS)d) (PÚ(Q®(RÙØP)«(QÚØS)12指出下列公式哪些是恒真的哪些是恒假的：(1)PÙ（P® Q）®Q(2)（P® Q）®（ØPÚQ）(3)（P® Q）Ù （Q®R）®（P® R ）(4)（P« Q）«（PÙ QÚØPÙØ Q）13设S=G1，Gn是命题公式集合。试求出在不增加新原子的情况下从S出发演绎出的所有命题公式。14证明下面的等价式：(1) (ØPÙ(ØQÙR)Ú(QÙR)Ú(PÙR)=R(2) P®(Q®P)=ØP®(P®Q)(3) P®(QÚR)=(P®Q)Ú(P®R)(4) (P®Q)Ù(R®Q)=(PÚR)®Q15找出下面公式的Skolem范式：(1)Ø("xP(x)®$y"zQ(y，z)； (2)"x(ØE(x，0)®($y(E(y，g(x)Ù"z(E(z，g(x)®E(y，z)。16G=(P，L)是有限图，设P(G)，L(G)的元数分别为m，n。证明：n£ ，其中 表示m中取2的组合数。17设G是有限图，M，A分别是G的关联矩阵和相邻矩阵，证明：MM和A2的对角线上的元素是G中所有点的度。18设G为图（可能无限），无回路，但若任意外加一边于G后就形成一回路，试证G必为树。19试举出一个连通的(即漠视为图后是连通的)，但无根的有向图。20设G是有向图，其中含一有向路(e1，en)，其中fin(en)=init(e1)，证明：G不是有向树。21设(I，+)为整数加群，（5I，+）为I的子群，请给出mI的所有陪集。22证明：若一个图G的任意两点度数之和³n-1，n=|P(G)|，则该图有Hamilton路。23给出一个具有5个点的边数最多的非Hamilton图。24给出代数格的定义。25设G为有向图，若G具有有向树定义中的1)和2)，并且没有有向回路。问：若G有限，G是否是有向树？若G不是有限的，如何？26设 * 是集合S上的二元代数运算，且满足结合律，设x，y是S中任意元素，如果x * y = y * x，则x = y。试证明 * 满足等幂律。27请给出一个布尔代数。28设R,S是A上的传递关系，证明或者反驳：(1) RÇS是传递关系；(2) RÈS是传递关系。29试用演绎法证明PÚQ，Q®R，P®M，ØM共同蕴涵RÙ(PÚQ)30. 求证G的任意多个子群的交集是G的子群。并且，G的任意多个正规子群的交集仍是G的正规子群。31设H是G的子群。N是G的正规子群。命HN为H的元素乘N的元素所得的所有元素的集合。求证HN是G的子群。32设H是群G的一个有限非空子集，求证只要H中任意两个元素的积仍在H内，则H是G的子群。33求证循环群的子群仍是循环群。34求证若G的元数是一个质数，则G必是循环群。35设K和H都是群G的子群，试证明：若H·K是G的子群，则K·H = H·K。36什么是等价关系？37如果A上的一个等价关系为R，如何求出一个等价类？38给出命题公式P®Q的真值表。39Skolem范式中的母式有什么特点？40有根的有向图，是否一定是强连通的？41最优树是否一定唯一？42什么是体？43什么是代数格？44半序子格与代数子格是什么关系？"If we don't do that it will go on and go on. We have to stop it; we need the courage to do it."His comments came hours after Fifa vice-president Jeffrey Webb - also in London for the FA's celebrations - said he wanted to meet Ivory Coast international Toure to discuss his complaint.CSKA general director Roman Babaev says the matter has been "exaggerated" by the Ivorian and the British media.Blatter, 77, said: "It has been decided by the Fifa congress that it is a nonsense for racism to be dealt with with fines. You can always find money from somebody to pay them."It is a nonsense to have matches played without spectators because it is against the spirit of football and against the visiting team. It is all nonsense."We can do something better to fight racism and discrimination."This is one of the villains we have today in our game. But it is only with harsh sanctions that racism and discrimination can be washed out of football."The (lack of) air up there Watch mCayman Islands-based Webb, the head of Fifa's anti-racism taskforce, is in London for the Football Association's 150th anniversary celebrations and will attend City's Premier League match at Chelsea on Sunday."I am going to be at the match tomorrow and I have asked to meet Yaya Toure," he told BBC Sport."For me it's about how he felt and I would like to speak to him first to find out what his experience was."Uefa has opened disciplinary proceedings against CSKA for the "racist behaviour of their fans" during City's 2-1 win.Michel Platini, president of European football's governing body, has also ordered an immediate investigation into the referee's actions.CSKA said they were "surprised and disappointed" by Toure's complaint. In a statement the Russian side added: "We found no racist insults from fans of CSKA."Baumgartner the disappointing news: Mission aborted.The supersonic descent could happen as early as Sunda.The weather plays an important role in this mission. Starting at the ground, conditions have to be very calm - winds less than 2 mph, with no precipitation or humidity and limited cloud cover. The balloon, with capsule attached, will move through the lower level of the atmosphere (the troposphere) where our day-to-day weather lives. It will climb higher than the tip of Mount Everest (5.5 miles/8.85 kilometers), drifting even higher than the cruising altitude of commercial airliners (5.6 miles/9.17 kilometers) and into the stratosphere. As he crosses the boundary layer (called the tropopause),e can expect a lot of turbulence.The balloon will slowly drift to the edge of space at 120,000 feet ( Then, I would assume, he will slowly step out onto something resembling an Olympic diving platform.Below, the Earth becomes the concrete bottom of a swimming pool that he wants to land on, but not too hard. Still, he'll be traveling fast, so despite the distance, it will not be like diving into the deep end of a pool. It will be like he is diving into the shallow end.Skydiver preps for the big jumpWhen he jumps, he is expected to reach the speed of sound - 690 mph (1,110 kph) - in less than 40 seconds. Like hitting the top of the water, he will begin to slow as he approaches the more dense air closer to Earth. But this will not be enough to stop him completely.If he goes too fast or spins out of control, he has a stabilization parachute that can be deployed to slow him down. His team hopes it's not needed. Instead, he plans to deploy his 270-square-foot (25-square-meter) main chute at an altitude of around 5,000 feet (1,524 meters).In order to deploy this chute successfully, he will have to slow to 172 mph (277 kph). He will have a reserve parachute that will open automatically if he loses consciousness at mach speeds.Even if everything goes as planned, it won't. Baumgartner still will free fall at a speed that would cause you and me to pass out, and no parachute is guaranteed to work higher than 25,000 feet (7,620 meters).cause there4