Math 10 Discrete Mathematics

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1 Math 0 Dsrete Mathemats T. Heso REVIEW EXERCISES FOR EXM II Whle these problems are represetatve of the types of problems that I mght put o a exam, they are ot lusve. You should be prepared to work ay type of problem related to the materal studed. NOTES: PROOFS ad COUNTEREXMPLES: ll proofs should be wrtte omplete setees wth reasos gve wheever deftos or theorems are used the proof. You do ot eed to justfy algebra steps or results due to losure. Couterexamples should lude the egato of the orgal statemet ad a thorough, logally orret explaato as to why the outerexample shows the orgal statemet s false. SET THEORY: For proofs set theory, assume that all sets refereed are approprately defed ad are subsets of some uversal set U. REVIEW EXERCISES 4.8. pply the Dvso lgorthm to fd the quotet ad remader whe 348 s dvded by 97. Make a trae table to trae the ato of the algorthm. 2. pply the Euldea lgorthm to fd gd 98,544. Make a trae table to trae the ato of the algorthm. 3. Use the Euldea lgorthm to alulate (by had ot alulator) the greatest ommo dvsor of 680 ad 07.. Use the 4. Two postve tegers a ad b are sad to be relatvely prme f, ad oly f gd ab, Euldea lgorthm to fd gd 544,. re 544 ad relatvely prme? How do you kow? 5. Twelve s a magal umber. It s dvsble by 2, 3, 4 ad 6. There are 2 moths a year, 2 hours o a lok, 2 sgs of the zoda, 2 toes the musal sale, 2 a doze, 2 hes a foot, 2 Kghts of the Roud Table, 2 labors of Herules, 2 members a jury (usually) Twelve must have somethg gog for t. We use the demal system beause there are te fgers o our hads. But we mght as well have used a duodemal (base 2 system). I fat, some ultures do use the duodemal systems. Oe a fd a Dozeal Soety ( that advoates the use of the duodemal system. How would umbers be wrtte a base 2 system? Defe a set of 2 dgts that ould be used to represet a umber base 2. Desrbe a method (algorthm) for overtg from base 0 to base 2. Use your method to overt to base 2. Sprg 204

2 Revew Questos for Exam II Page 2 of 5 Chapter 5 6. Perform the omputatos: a. 7 j j0 3 b. 4 k k k 7. Fd a formula (losed form) for eah sequee: a. The sequee a, a2, a3, a, whose frst fve terms are b. The sequee a, a2, a3, a, defed reursvely by a0 ad, for k, ak k ak ,,,, Smplfy eah fatoral: (a) 8! 23!5! (b) 4! ()!! 2!3! 9. Use ether the Prple of Mathematal Iduto or the Prple of Strog Iduto, as approprate, to prove eah of the followg: a.! 2 2! 33!!! for all tegers. b ! 0. For every teger, 2! d. For all tegers, 5 3 s dvsble by 4. e. For all oegatve tegers, s a teger f. sequee s defed reursvely by: a 2, a2 3, ad, for all tegers 3, a 2a a2. Prove that ths sequee s geerated by the formula a 0. DO YOU BELIEVE IT? THEOREM: For all postve tegers, every set of horses, all horses are the same olor. PROOF: Bass Step: The statemet s true the ase where, se a set osstg of oe horse, all horses the set are the same olor. Math 0 Dsrete Mathemats T. Heso

3 Revew Questos for Exam II Page 3 of 5 Idutve Step: Let k be a teger, wth k, ad assume that every set of horses, wth k, all horses are the same olor. Show that a set of k horses, all horses are the same olor. Cosder a set of k horses. Remove oe horse from the set, obtag a set of k horses. By the duto hypothess, all horses the set are the same olor. Se the hoe of whh horse to remove from was arbtrary, remove some other horse from the set of k horses ad replae t wth the frst horse removed from. The resultg set otas k horses ad hee, by the duto hypothess, all horses ths set are the same olor. By addg bak the seod horse removed, we obta a set of k horses, all of the same olor. Therefore, every set of horses, all horses are the same olor. (WD) Is ths proof orret? If ot, what s the mstake?. reurree relato s defed by a. Compute s, s2, s3, s4, ad s 5. reurree relato: s s 8 k, for k tal odto: s 0 k k b. Fd a formula (losed form) for s, for 0.. Prove (by duto) that your formula satsfes the reurree relato. 2. Gve a reurree relato for the sequee a, for all tegers. Use mathematal duto to prove your reursve defto orretly produes the sequee a. a 3. sequee s defed reursvely by: a ad, for all tegers 2, a. Use terato a to fd a formula for a ad use mathematal duto to prove that your formula orretly produes the sequee. 4. Use mathematal duto to prove that F s the th Fboa umber F0 F F FF for all tegers 0, where Chapter Suppose x R x 0. a. Fd the sets (wrte wth ether lst otato or set-bulder otato, as approprate). B. B. C v. C v. C B v. B C v. C, B x R 3 x 3, ad C x Z x 3 Math 0 Dsrete Mathemats T. Heso

4 Revew Questos for Exam II Page 4 of 5 b. Whh of the followg subset relatoshps are true ad whh are false? Gve a reaso for your aswer eah ase.. B. C. B v. C v. B C v. C B 6. Suppose that the uversal set s Z, the set of tegers. Let 0,, 2,3, 4,5,6 B 2,3,5,7,,3 C 6, 4, 2,0, 2, 4,6 a. Use set-bulder otato to gve a alteratve defto of C. b. Fd C B. (Desrbe or lst the elemets ths set.). Fd B C. (Desrbe or lst the elemets ths set.) d. Is B C C? Gve a reaso for your aswer. e. re the sets ad B C dsjot? Why or why ot? 7. Let xyz,, ad let B 0,. a. Fd P, the power set of. b. Fd B. 8. Suppose the uversal set s R x R x the rule, a. Fd. 0. Defe a olleto of subsets of R by For all tegers, x R x.. b. If,,,, 2 3, s a partto of 9. Wrte a elemet-hasg proof to show that B B R? Gve a reaso for your aswer. Math 0 Dsrete Mathemats T. Heso

5 Revew Questos for Exam II Page 5 of Use the algebra of sets to prove that B B, for ay sets, B ad C. 2. Use the algebra of sets to prove that B C B C C., for ay sets, B ad 22. For ay sets, B, C ad D, prove: a. If B B, the B. (Do ot use the Equvalet Codtos Theorem.) b. If B, the B. B B. d. If B, the. (Do ot use the Equvalet Codtos Theorem.) B. e. C BD BC D. 23. For ay sets, B, ad C, prove or dsprove (gve a outerexample): a. If C B, C or C B. b. If C B, C ad C B.. If B ad B C, the C. d. CBDBC D 24. Let ad B be sets. a. prove or dsprove (gve a outerexample) that P PB P B. b. prove or dsprove (gve a outerexample) that P PB P B.. Based o your results from parts (a) ad (b), a you olude that P P B P B? Why or why ot? Math 0 Dsrete Mathemats T. Heso