CSE 1400 Applied Discrete Mathematics Sets

Transcription

1 CSE 1400 Applie Disrete Mthemtis Sets Deprtment of Computer Sienes College of Engineering Flori Teh Fll 2011 Set Bsis 1 Common Sets 3 Opertions On Sets 3 Preeene of Set Opertions 4 Crtesin Prouts 4 Suset of Set 5 Crinlity of Set 6 Power Set of Set 6 Binomil Coeffiients 7 Counting Bit Strings 8 Boolen Lws 8 Prtition of Set 11 Venn n Euler Digrms 12 Prolems on Sets 15 Astrt Finite n ountle sets re funmentl primitives of isrete mthemtis. Opertions n e efine on sets reting n lger. Counting the numer of elements in set n ounting susets with ertin property re funmentl in omputing proilities n sttistis. Prtitioning set esries equivlenes mong its elementss. Set Bsis A set is n unorere olletion of things. The things in set A re si to e elements or memers of A. The nturl If is n element in A write A. Of ourse, it n our tht prtiulr element is not memer of set A, in whih se write A.

2 se 1400 pplie isrete mthemtis sets 2 numer 7 is memer of the set O of otl numerls 7 O = {0, 1, 2, 3, 4, 5, 6, 7} The nturl numer 8 is not memer of the set O of otl numerls 8 O = {0, 1, 2, 3, 4, 5, 6, 7} A n e esrie y listing its memers s omm seprte list enlose in urly res {}. No element is uplite in set: A thing in set is liste one n only one. A n e lso esrie y omprehension. For instne, For instne, B = {0, 1}, is the set of its. B = { : is it} More generlly, A n e omprehene y esription A = { : p() is True} where p() is proposition out vrile. Even more generlly, A n e omprehene y esription A = { f () : p() is True} where f is funtion n p() is proposition out the vrile. When ontext emns it, the set of ll possile things, lle the universl set n enote, n e nme. Sets n e represente y igrms, for instne, single set is rwn s irle insie retngle. For instne, the set of even nturl numers is omprehene y the esription 2N = { : = 2n n N} For instne, the set of even nturl numers is omprehene y the esription 2N = {2 : N} In omputing prtie, set omprehension requires the proposition p() to e omputle, tht is, there must e n lgorithm tht returns True when A n Flse when A. For instne, the set of nturl numers is the universl set for mny omputing prolems. Strings over n lphet oul e in other pplitions. Two sets n n e rwn in severl reltionships, lle Euler igrms. For instne, when no memers of re in the sets re isjoint. Contrpositively, no memers of re in. Similrly, when some in re in then some memers of re in n the sets interset. Lstly, when ll memers of re in, is si to e suset of. n re isjoint. n interset. is suset of. Chrteristi funtions esrie these three senrios.

3 se 1400 pplie isrete mthemtis sets 3 Definition 1 (Chrteristi Funtion). The hrteristi funtion of A is enote χ A n ompute y the onitionl sttement Flse if A χ A () = True if A When n re isjoint, the vlues χ () n χ () nnot oth e True simultneously. Tht is, ( )(χ () χ () = Flse) When n interset, their is some element where χ () n χ () re simultneously True. Tht is, ( )(χ () χ () = True) An when is suset of, every element in lso elongs to. Tht is, χ () χ () Common Sets In mthemtis, the nme of set is usully written in font lle lkor ol. Thus, for instne, we hve The its or Boolen vlues B = {0, 1} = {1, 0} The igits D = {0, 1,..., 9} The hexeiml igits H = {0, 1,..., 9, A,..., F} The nturl numers N = {0, 1, 2,...} The integers Z = {0, ±1, ±2,...} The integers mo n Z n = {0, 1, 2,..., (n 1)} The rtionl numers Q = {/ :, Z, = 0} The English lphet A = {,,,..., x, y, z} The nioe hrter set = { : 0 (10FFFF) 16 } Opertions On Sets Sets n e omine in simple wys to rete omplex expressions. Let n nme two sets. The union opertor It oul e tht =, so tht two nmes refer to the sme vlue.

4 se 1400 pplie isrete mthemtis sets 4 returns the set of ll elements in either set: or. = {z : z is in or z is in } The intersetion opertor returns only the set of elements tht re in oth sets n. = {z : z is in n z is in } is set union. Set omplement,, opertes on single set n returns the elements not in. = {z : z is not in } For instne, let = {1, 5, 9}, = {2, 3, 5, 7}, n V = {4, 6, 8, 9} is the intersetion opertor. e three susets of the universe of eiml igits. Then D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} V = {9} = {0, 2, 3, 4, 6, 7, 8} = {1, 2, 3, 5, 7, 9} ( ) V = {0, 1, 2, 3, 5, 7} Preeene of Set Opertions Preeene n ssoitivity etermine the orer in whih opertions re performe. For sets, omplement is ompute efore intersetion, whih is efore union. All opertions re ompute leftto-right unless prenthesis or other rkets re use to speify orer. Crtesin Prouts Crtesin prouts provies the fountion for uiling reltions n funtions. The Crtesin prout of n is the set of orere (x, y) tht reltes every vlue x with every vlue y. The Crtesin prout of n is written is the set omplement opertor. In rithmeti you lern tht exponentition is performe efore multiplition, whih re performe efore ition. Furthermore, the ssoitivity of ition n multiplition is left-to-right, ut exponentition is right-to-left. For instne, = = For instne, when A = {0}, B = {0, 1} n C = {1, 2} ompute A B C = {0} {1} = {0, 1} (A B) C = {0, 1} {1, 2} = {1} = {(x, y) : x y }

5 se 1400 pplie isrete mthemtis sets 5 The Crtesin prout is two imensionl: It n e represente s set of orere pirs, for instne, B D = {(0, 0), (0, 1),..., (0, 9), (1, 0), (1, 1),..., (1, 9)} The Crtesin prout n e represente s tle, for instne B D is the tle where eh entry is True, represente y 1. B D College lger tehes how to rw funtions n reltions on the rel numers. y = x This is the inestuous reltion, x 2 + (ywhere 1) 2 = 1 eh element B is relte to every element D. y = x y x Susets of B D re lle reltions from B to D. For instne, the the even-o reltion n e represente y the tle where row 0 piks out the even igits s True row 1 piks out the o igits s True Even-O Reltion A Crtesin prout n e represente s noe-ege grph. Su-grphs of omplete grph rise in omputing prtie. For instne, the even-o reltion Suset of Set Often it is useful to tlk out olletion of some elements, ut perhps not ll elements, of set. Suh olletion is lle suset. The set of omposite igits C = {4, 6, 8, 9} is suset of the igits D. The suset reltion etween two sets is very muh like the less thn reltion etween two integers. A proper suset is stritly smller thn its super-set, just s 5 is stritly less thn 7. C = {4, 6, 8, 9} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} = D To llow for the possiility of equlity, write for integers n A for sets.

6 se 1400 pplie isrete mthemtis sets 6 If there is some element in A tht is not in, then A is not suset of. Stte ontr-positively, if A, then every memer of A is memer of. There is one n only one smllest set. It is the set without ny elements, n it is lle the empty set. The empty set is enote y the symol. Interestingly, the empty set is suset of ny set A. If it were not, there woul hve to e n element in tht is not in A, ut there n e no suh element sine ontins no elements. The vuous proof tht A is tht every element in is n element in A. If A is not suset of, write A Crinlity of Set Informlly, the rinlity of set A is the ount of elements in A. If this ount n e nme y nturl numer, then A is finite set. If A is not finite set, it is n infinite set. There re t lest two types of infinite sets. Those tht re ountle n those tht re not. The nturl numers is the efinition of ountle, infinite set. Any other set A is ountle if there is one-to-one n onto funtion f mpping N to A. For instne, the set of frtions A = is ountle euse the funtion { 1 1, 1 2, 1 } 3,... f (n) = 1 n + 1 estlishes one-to-one n onto orresponene etween the two sets. Set A is finite with rinlity n when the elements of A n e put into one-to-one n onto orresponene with the set {0, 1,..., n 1}. A finite set is lso ountle. The ie ehin ountility is tht given ny nturl numer n we n (eventully) ount to it. It is not iffiult to imgine exmples of ountle sets. There re exmples tht re ounter-intuitive. Power Set of Set The olletion of ll susets of set A is lle A s power set. If A hs 3 elements then A s power set ontins 2 3 = 8 The power set of A is enote y 2 A. elements, eh of whih is suset of A. Think of onstruting suset. For eh element there re two hoies: inlue it in the suset or leve it out. Consier the eision tree for onstruting susets of A = {,, }. When the left rnh is followe, the element is not inlue in the suset. When the right rnh is followe, the element is inlue in the suset.

7 se 1400 pplie isrete mthemtis sets 7 {} {} {, } {} {, } {, } {,, } Theorem 1 (Power Set Crinlity). A set with n elements hs 2 n susets, tht is, the power set of A ontins ll 2 n susets of A when A = n. If A = n, then 2 A = 2 A = 2 n. Binomil Coeffiients Given set A with rinlity n, how mny m-element susets oes A hve? The inomil oeffiient ( n m ) enotes this numer. When step of n lgorithm requires the seletion of suset of vlues, the hosen suset is lle omintion. The size (rinlity) n of the smple spe n the size (rinlity) m of the omintion etermine the ount of omintions tht n e onstrute. The exmple ove shows tht three element set hs The symol ( n m ) is lle n hoose m. 1 suset without ny elements, ( 3 0 ) = 1 3 susets with one element, ( 3 1 ) = 3 3 susets with two elements, ( 3 2 ) = 3 1 suset with three elements, ( 3 3 ) = 1 Any given set A hs 1 suset without ny elements, the empty. Also, there is 1 suset of A tht ontins every element in A, the set A itself. For smll sets eh suset n e liste. Let A e set with rinlity A = n. There re 2 n susets of A. Figure 1 shows the 2 3 = 8 susets of A = {0, 1, 2}. The susets of {0, 1, 2, 3} n e onstrute reursively in terms of the susets of {0, 1, 2}. For instne, to uil ll 2-element susets of N 3 = {0, 1, 2, 3} use the one n two-element susets of N 2 = {0, 1, 2}. There re two ses to onsier. There is only one wy to hoose nothing t ll. There is only one wy to hoose everything. The ounry onitions on the inomil oeffiients re ( n 0 ) = (n n ) = 1. It is onvenient to elre tht the empty set is suset of ny given set A. Desriing how to onstrut something often les to metho for ounting those things. 1. Element 3 is in 2-element suset of N Element 3 is not in 2-element suset of N 3. These two ses re hnle y steps elow tht uil 2-element suset of N 3.

8 se 1400 pplie isrete mthemtis sets 8 1. Tke the union of {3} with eh 1-element suset of {0, 1, 2}. 2. Inlue in the olletion of susets eh lrey uilt 2-element suset of N 2 = {0, 1, 2}. sing the inomil oeffiient ( n m ) to nme the ount of 2-elements susets of 4-element set, write ( 4 2 ) = ( 3 1 ) + These six susets re shown in figure 2. ( ) 3 = = 6 2 Counting Bit Strings There re 2 n ifferent its strings of length n. The numer of it strings of length n with m 1 s n n m 0 s is equl to the numer of m-element susets of n n-element set. For instne, there re 6 its strings of length 4 with two its set to 1 n two its Think of the it string s reor tht elements re in the suset (the it is 1) or not in the suset (the it is 0.) set to 0. Reursion n e use to onstrut n n-long it string with {0011, 0101, 0110, 1001, 1010, 1100} m 1 s. There re two ses. 1. Let s n 1 e n n 1-long it string with m 1 1 s. 2. Let t n 1 e n n 1-long it string with m 1 s. Then 1s n 1 is n it string of length n with m 1 s. n 0t n 1 is n it string of length n with m 1 s. The strings 1s n 1 n 0t n 1 re ifferent Let ( n 1 m 1 ) enote the num- 1s er of strings like s n 1 n let ( n 1 m ) enote the numer of strings like t n 1. Then the numer of n-long it strings with m 1 s n e ompute y the reursion eqution ( ) n m = ( ) n 1 m 1 ( ) n 1 + m Tle 1 lists the inomil oeffiients n hoose m for pirs of igits. Boolen Lws The power set 2 of set together with the union, intersetion, n set omplement opertions form Boolen lger. Tht is, the following properties hol for susets, n Z of. Ientity Lws n 1 hs leing 1 while 0t n 1 hs leing 0. This reursion eqution is known s Psl s ientity. Notie how the sum of two vlues in one row etermine the vlue of term in the next row. ( ) ( ) = 56 + = 2 3 ( ) ( ) = 21 + = ( ) n 1 + m 1 ( ) 8 3 ( ) ( ) ( ) n 1 n = m m Boolen logi on propositions using or ( ), n ( ), n not ( ) opertions form similr Boolen lger.

9 se 1400 pplie isrete mthemtis sets 9 Susets {0} {0, 1} {0, 1, 2} Figure 1: The 2 3 {0, 1, 2}. = 8 susets of {1} {0, 2} {2} {1, 2} Susets Figure 2: The 2 4 susets of {0, 1, 2, 3}. {0} {0, 3} {0, 1, 3} {0, 1, 2, 3} {1} {1, 3} {0, 2, 3} {2} {2, 3} {1, 2, 3} {3} {0, 1} {0, 1, 2} {0, 2} {1, 2}

10 se 1400 pplie isrete mthemtis sets 10 Tle 1: Binomil oeffiients ( n m ) ount the numer of ifferent wys to hoose m elements from set of n elements. Binomil Coeffiients ( n m ) Choose m n = 2. = Complement Lws 1. = 2. = Assoitive Lws 1. ( ) Z = ( Z) 2. ( ) Z = ( Z) Commuttive Lws 1. = 2. = Distriutive Lws 1. ( Z) = ( ) ( Z) 2. ( Z) = ( ) ( Z) Mny theorems n e erive from these funmentl lws, for instne De Morgn s lws = = In English, not (x or y) is not x n not y, n not x n y is not x or not y.

11 se 1400 pplie isrete mthemtis sets 11 A proof of theorems suh s De Morgn s lws n e given y exmining ll ses. The se exmintion n e orgnize into truth tle. Cses Results z z z z Cses Results z z z z Prtition of Set To prtition is to seprte into prts. For instne the eveno reltion prtitions the igits into two susets. {0, 2, 4, 6, 8} n {1, 3, 5, 7, 9} A similr instne is the mo 3 reltion whih prtitions the igits into three susets se on their reminer upon ivision y 3. {0, 3, 6, 9}, {1, 4, 7}, n {2, 5, 8} For sets with smll rinlity it is possile to list ll of the prti-

12 se 1400 pplie isrete mthemtis sets 12 tions. Prtitions with Set 1 suset 2 suset 3 suset {0} {{0}} {0, 1} {{0, 1}} {{0}, {1}} {0, 1, 2} {{0, 1, 2}} {{0}, {1, 2}} {{0}, {1}, {2}} {{1}, {0, 2}} {{2}, {0, 1}} {0, 1, 2, 3} {{0, 1, 2, 3}} {{0, 1, 2}, {3}} {{0}, {1, 2}, {3}} {{0, 3}, {1, 2}} {{1}, {0, 2}, {3}} {{0}, {1, 2, 3}} {{2}, {0, 1}, {3}} {{1, 3}, {0, 2}} {{0}, {1}, {2, 3}} {{1}, {0, 2, 3}} {{0}, {1, 3}, {2}} {{2, 3}, {0, 1}} {{0, 3}, {1}, {2}} {{2}, {0, 1, 3}} The numer wys n n-element set n e prtitione into m susets is Stirling numer of the seon kin enote y { n m }. A Stirling numer of the seon kin n e ompute y the reurrene eqution { } n 1 { } n = m { } n 1 + m m 1 with ounry onitions { } { } 0 n = 1, = 0 n 0 0 m { } n = 1, for n > 0 n These ies our in the stuy of equivlene reltions, whih re reltions tht prtition sets into equivlene lsses. Venn n Euler Digrms Venn s igrmming tehnique show reltionships etween sets. When use in logi, Venn igrms show reltionships mong propositions. Venn igrmming egins with n retngle representing the universe of elements. If the retngle is not she it represents the empty set. The empty set

13 se 1400 pplie isrete mthemtis sets 13 Shing region in Venn igrm inites it is not empty; it is full. Mthemtiins she the non-empty region. Logiin she the empty region. Stuents suffer. A non-empty universe sing this shing rule two regions tht n e ientifie: The universe n the empty set. When one suset, ll it, is rwn s irle insie the universe, two itionl regions tht n reognize: The set n its omplement. When two interseting susets, ll them n, re rwn s irles insie the universe, n itionl 12 regions tht n ientifie. One wy to unerstn this is to nme the four regions in the Venn igrm. Cll them,, n, n note they n e expresse s intersetions of n or their omplements. Region Set Expression The set ou n onstrut the empty set from the set of regions {,,, } y not hoosing ny of them. The universe is onstrute y the opposite: Choose every region n ompute their union. These re the ounry onitions. There re interior ses: Choose one region, Choose two regions, or hoose three regions. These ses n e igrmme. : The omplement of

14 se 1400 pplie isrete mthemtis sets 14 From the four regions {,,, } hoose No regions One region Two regions Three regions Four regions A funtionl reltionship etween the numer interseting susets n numer of ifferent regions tht n e esrie n e expresse y funtion r(n) = 2 2n where n is the numer of interseting susets n r(n) ounts the totl numer of istint regions tht n e onstrute. Count of Susets Count of Regions Funtion Regions = 2 2 Susets

15 se 1400 pplie isrete mthemtis sets 15 Prolems on Sets 1. Desrie the following sets. List representtive smple of elements estlishing pttern. Give funtion tht omputes the elements. () The set of even integers. () The set of o integers. () The set of integers tht hve reminer of 2 when ivie y 3. () The Mersenne numers. (e) The tringulr numers. (f) The set of solutions to the polynomil eqution x 2 x 1 = 0. (g) The set of solutions to the eqution x 1 = Wht nottion woul e use to stn for the following phrses? () x is n element of. () x is not n element of. () is suset of. () is proper suset of. (e) is not suset of. (f) The union of n. (g) The intersetion of n. (h) The omplement of. (i) The empty set 3. Answer True or Flse. () 2 {{2}}. () 2 {2}. () 2 {2}. () { }. (e) 0. (f) {x}. (g) = { }. (h) {x} {x}. (i) {x} {x}. 4. Below re stnr nmes for sets tht our in omputing. Desrie these sets. () B () H () N () Z (e) Z + (f) Q (g) R (h) 5. Wht is the rinlity of eh of these sets? (). () { }. () D. () { x : x 2 x 1 = 0 }. (e) N. (f) Q. (g) R.

16 se 1400 pplie isrete mthemtis sets Wht is the power set of eh of these sets? () {0}. () { x : x 2 x 1 = 0 }. (). () { }. 7. Wht is the rinlity of eh of these sets? () The power set of {0}. () The power set of { x : x 2 x 1 = 0 }. () The power set of D, the set of igits. () The power set of H, the set of hexeiml numerls. 8. Desrie how to efine the set ifferene opertor: using the stnr union, intersetion, n set omplement opertors. 9. She the Venn igrm to inite the given region is not empty. (). () Z. V Z () V. () Z. Z Z 10. She the Venn igrm to inite the given region is not empty. (). () Z. Z Z () Z. () Z. Z Z

17 se 1400 pplie isrete mthemtis sets Let n e susets of some universl set. Are the following sttements True or Flse? () = () = () = () = (e) = (f) = (g) 12. Let M 10 = {0, 1, 3, 7, 15, 31, 63, 127, 255, 511} e the set of the first 10 Mersenne numers. Let T 10 = {0, 1, 3, 6, 10, 15, 21, 28, 36, 45} e the set of the first 10 tringulr numers. Fin () M 10 T 10. () M 10 T 10. () M 10 T 10. () T 10 M sing the three sets = {1, 2, 3, 4, 5} = {0, 2, 4, 6, 8} Z = {0, 3, 5, 9, } over the universe of igits D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} ompute the following set opertions () () Z () ( ) Z () ( ) Z 14. sing the three sets = {1, 2, 3, 4, 5} = {0, 2, 4, 6, 8} Z = {0, 3, 5, 9, } over the universe of igits D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} write expressions using set opertors union, intersetion n omplement tht re equl to the following sets. () S = {6, 8} () T = {1, 2, 4} () V = {0, 1, 2, 3, 4, 5, 6, 8} () W = {3, 5} 15. Consier the set S = {x : x x}. () Show tht S S is ontrition. () Show tht S S is ontrition. This is known s Russell s prox.

18 se 1400 pplie isrete mthemtis sets Wht is the time tken to ompute the inomil oeffiient using the ftoril formul ( ) n = k n! k!(n k)!? Assume tht time is mesure y the numer of multiplitions in the reue formul ( ) n n(n 1) (n k + 1) = k k(k 1) Wht is the time tken to ompute the inomil oeffiient using the reursive (Psl s) formul ( ) ( ) ( ) n n 1 n 1 = +? k k k 1 Assume tht time is mesure y the miniml numer of itions require to uil Psl s tringle to ( n k ). In prtiulr, you my ssume tht k = min {k, n k} n use the symmetry of the tringle.