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1 S/PM 0 Final Exam 7 May 005 Name: KEY Pledge: Signatue: Thee ae 80 minutes 3 hous fo this exam and 80 oints on the test; don t send too long on any one uestion! Thee is an exam efeence sheet as the last age of this exam feel fee to tea it off. You may use sca ae. Howeve, all answes must be on these twelve exam ages nothing on the exam efeence sheet will be gaded. Pat I: Logic Pat II: Poofs Pat III: ounting Pat IV: Stuctues Pat V: lications /0 /50 /50 /55 /0 Total /80

2 S/PM 0 Final Exam 7 May 005 Pat I: Logic. 5 oints What is the convese and invese of? lealy label which is which! nswe: The convese is, and the invese is.. 5 oints Using logical euivalences, ove that othe wods, show that T Mali/Sen, age 39 is a tautology. In. You must clealy label each ste! nswe: T Oiginal statement T Definition of imlication twice T DeMogan s law twice T DeMogan s law again T DeMogan s law again T T omlement law Note that the last ste is because anything o ed with its comlement hee it was o ed with its comlement is a tautology.

3 S/PM 0 Final Exam 7 May 005 Pat II: Poofs 3. 5 oints Give an English statement that can be oven tivially i.e. by a tivial oof. nswe: ny imlication whee the conseuence is always tue.. 5 oints Give an English statement that can be oven vacuously i.e. by a vacuous oof. nswe: ny imlication whee the antecedent is always false oints Pove, via a oof by contadiction, that if n is an intege, and 3n is even, then n is even. Rosen, section.5, uestion nswe: Rewite as a oosition: if 3n is even, then n is even. Let be 3n is even, and be n is even. ssume that the imlication is false, which only occus when is tue and is false. Thus, we ae assuming that 3n is even and that n is odd. If n is odd, then n fo some intege definition of even numbes. Then 3 n Thus, since 3n is times some intege lus one that intege being 3, it cannot be even, which contadicts ou oiginal assumtion that 3n is even. Thus, 3n must be even. 3

4 S/PM 0 Final Exam 7 May oints Given the thee oositions,,, can we conclude? Show this by using ules of infeence. You must label all you stes! Mali/Sen, age 5, execise 5 nswe:. Fist hyothesis. Second hyothesis 3. Disjunctive syllogism on stes and. Thid hyothesis 5. Disjunctive syllogism on stes 3 and Theefoe, we cannot conclude that is tue in fact, we conclude that is false. Note that thee ae othe ways to conclude fom the given hyothesis.

5 S/PM 0 Final Exam 7 May oints Using wea mathematical induction, show that n n fo all 0 n. You must label all you stes! Mali/Sen, age 3, execise nswe: Let : n n n P. ase case: 0 0 Inductive hyothesis: Inductive ste: We can elace the at of the inductive ste with the ight side of the inductive hyothesis, namely, to yield:

6 S/PM 0 Final Exam 7 May 005 Pat III: ounting 8. 5 oints How many functions ae thee fom the set {,,, n}, whee n is a ositive intege, to the set {0, }? Rosen, section., uestion 3 nswe: Fo each of the n elements in the domain, the function can ma it to eithe of the values. Thus, thee ae n total ossible functions oints What is the coefficient of x 7 in x 3? Leave you answe in combinatoial fom. vaiant of Rosen, section., uestion 6 nswe: The coefficient is *8 9, oints What must be shown in a combinatoial oof? In othe wods, what must you do in ode to ove a fomula via a combinatoial oof? nswe: You must show that both sides of the euation manage to count the same thing. 6

7 S/PM 0 Final Exam 7 May oints What is the obability of being dealt a staight in a 5-cad oe hand? Leave you answe in combinatoial fom. Recall that a staight is a seies of 5 cads in a seuence. Fo examle,,, 3,, 5 is a staight, as is 0, J, Q, K, note that the ace can be high o low. Suit does not matte in a staight and we ae ignoing staight flushes and oyal flushes. nswe: Total numbe of 5-cad stud oe hands is 5 5. To ic a staight, we fist ic the lowest numbe, of which thee ae 0 choices though 0. We then ic the suit fo each of the 5 cads. This yields 0. Thus, the total obability is ,598,

8 S/PM 0 Final Exam 7 May oints Let X { x, x, x }, 00 be a set of 00 distinct ositive integes. If these ositive integes ae divided by 75, then show that at least two of the emaindes must be the same. Mali/Sen,. 35 execise nswe: Poof via igeonhole incile. The numbe of igeons, N, is the numbe of integes 00. The numbe of igeonholes,, is the numbe of ossible emaindes, o 75. We need to fit 00 numbes into 75 ossible emaindes o 00 igeons into 75 igeonholes. y the igeonhole N 00 incile, thee must be at least one igeonhole emainde that has 75 igeons numbes in it oints Fom the set of integes in the set {,,, 30}, what is the least numbe of integes that must be chosen so that at least one of them is divisible by eithe 3 o 5? Exlain you answe. Mali/Sen,. 35 execise 3 nswe: Thee ae numbes in the ange that ae divisible by 3 o 5: {3, 5, 6, 9, 0,, 5, 8, 0,,, 5, 7, 30}. Thus, thee ae 6 numbes that ae not divisible by 3 o 5: {,,, 7, 8,, 3,, 6, 7, 9,, 3, 6, 8, 9}. The least numbe that must be chosen is moe than 6 wost case is that you ic all 6 fist, then one that is divisible by 3 o 5. Thus, the answe is 7. 8

9 S/PM 0 Final Exam 7 May 005 Pat IV: Stuctues. 5 oints What is the cadinality of a owe set of a set of n elements? déjà vu nswe: n 5. 5 oints Descibe, in English, what -to- and onto mean fo functions. nswe: -to- means that fo evey element in the co-domain, thee will be one and only one element in the domain that mas to it. Onto means that evey element in the co-domain has something maed to it oints What is the diffeence between a seuence that has an aithmetic ogession, and one that has a geometic ogession? nswe: n aithmetic ogession means each tem is a constant amount geate o less than the last tem. geometic ogession means that each tem is a constant facto geate o less than the last tem oints Descibe, in English, what tansitive closue means. nswe: Tansitive closue means that when thee is a ath between any two nodes a and b, then thee is an edge fom a to b. 9

10 S/PM 0 Final Exam 7 May oints What is the diffeence between asymmety and antisymmety? nswe: Ieflexivity. n asymmetic elation such as < must be ieflexive, as any element cannot be elated to itself. n antisymmetic elation such as is allowed to have elements elated to themselves, so it does not have to be ieflexive oints Which oeties ae euied fo an euivalence elation and which ae euied fo a atial odeing? nswe: oth must be eflexive and tansitive. n euivalence elation must be symmetic, and a atial odeing must be antisymmetic oints Given a elation R, what is the diffeence between R*, R -, and R? lealy descibe what each means. nswe: R* is the tansitive closue of R if thee is a ath between a and b, then thee is an edge between a and b. R - is the invese elation, whee all the edges ae evesed fomally, R { b, a a, b R}. R is the comlementay elation, which contains all the edges that ae not in R fomally, R { a, b a, b R}. 0

11 S/PM 0 Final Exam 7 May oints Given sets and, ove that. You can eithe use set builde notation o set euivalences, but you cannot use membeshi tables. vaiant of Mali/Sen, age, execise 7 nswe: Oiginal statement Definition of diffeence DeMogan s law Distibutive law Distibutive law again omlement law Identity law Definition of diffeence

12 S/PM 0 Final Exam 7 May 005 Pat V: lications. 0 oints State a eal wold alication of each of the following discete mathematical concets. oolean logic nswe: PUs aithmetic alications ae done via oolean logic Mathematical induction nswe: Veifying ogam coectness. Relations nswe: Relational databases o MaQuest. Pime numbes nswe: The RS algoithm, o encytion. lgoithms nswe: omute ogams! Fibonacci seuence nswe: Reoducing abbits, conch shell adii, etc.

13 S/PM 0 Final Exam 7 May S/PM 0 Final Exam Refeence Sheet Set and logical identities Sets Rosen,. 89 Name oolean logic Rosen,. U Identity laws F T U U Domination laws F F T T Idemotent laws omlementation law ommutative laws ssociative laws Distibutive laws DeMogan s laws bsotion laws U omlement laws F T Rules of infeence Rosen,. 58 Rule of Infeence Tautology Name ddition Simlification onjunction [ ] Modus onens [ ] Modus tollens [ ] Hyothetical syllogism [ ] Disjunctive syllogism [ ] Resolution

14 S/PM 0 Final Exam 7 May 005 This age intentionally left blan Do not include any wo on this age. It will not be gaded