Basic Structures: Sets, Functions, Sequences, and Sums
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1 ac Structure: Set, Fucto, Sequece, ad Sum CSC-9 Dcrete Structure Kotat uch - LSU Set et a uordered collecto o object Eglh alphabet vowel: V { a, e,, o, u} a V b V Odd potve teger le tha : elemet o et member o et O {,,,7,9} Kotat uch - LSU
2 Other et repreetato Set o potve teger le tha : {,,,,99} Odd potve teger le tha : O {,,,7,9} O { omtted elemet a odd potve teger le tha } O { Z odd ad } Kotat uch - LSU Ve Dagram Uvere U O 7 9 U { O { a potve teger le tha } a odd potve teger le tha } Kotat uch - LSU
3 Ueul et N {,,,, } Natural umber Z {,,,,,, } Iteger Z {,,, } Potve teger Q { p / q p Z, q Z, q } Ratoal umber R {et o Real umber} Real umber Kotat uch - LSU Empty et {} {} Kotat uch - LSU
4 Cardalty ze o et Fte et S { a, e,, o, } u S { a, b, c,, z} S {,,,,99} Number o elemet S S S 99 {} { } Ite et N {,,,, } te ze Kotat uch - LSU 7 Equal et Eample: {,,} {,, } {,,} {,,,,,,,} {,,,7,9} { Z odd ad } Kotat uch - LSU
5 Subet Eample: {,,} {,,, } N Z For ay et S : S S S Kotat uch - LSU 9 Proper Subet y y y y Eample: {,,} {,,, } N Z Kotat uch - LSU
6 equvalet to Kotat uch - LSU Power et The power et o S cota all poble ubet o S ad the empty et S {,,} Power et P S {,{},{},{},{,},{,},{,},{,,}} P S S Sze o power et Specal cae P { } P{ } {,{ }} Kotat uch - LSU
7 Ordered tuple relato Ordered -tuple a, a,, a ordered lt o elemet a, a,, a b, b,, b a b Eample:,, Kotat uch - LSU Cartea product Cartea product o two et { a, b a b }, Eample: {, } { a, b, c} {, a,, b,, c,, a,, b,, c} { a,, b,, c,, a,, b,, c,} For th cae: Sze: Kotat uch - LSU 7
8 Cartea product o et,,, { a, a,, a a } Eample: {, } { a, b, c} C {, y} C {, a,,, b,,, c,,, a,,, b,,, c,,, a, y,, b, y,, c, y,, a, y,, b, y,, c, y} Sze: C C Kotat uch - LSU Set ad propoto S P S P horthad or horthad or S P S P Truth et o propoto P { Doma P } all elemet o the doma whch aty P Kotat uch - LSU
9 Set operato Uo { } U {,,} {,, } {,,, } Kotat uch - LSU 7 Iterecto { } U {,,} {,, } {, } Kotat uch - LSU 9
10 Djot et, U {,,} {,9} Kotat uch - LSU 9 Set derece { } U {,,} {,, } {} Kotat uch - LSU
11 Complemet { } U {,,} U {,,,, } {,} Kotat uch - LSU Sze o uo U {,,} {,, } {,,,} {,} Kotat uch - LSU
12 Kotat uch - LSU De Morga law Show that ad Kotat uch - LSU Theorem: Proo: Part : De Morga law rom logc
13 Kotat uch - LSU Part : Ed o Proo De Morga law rom logc Kotat uch - LSU Set dette U Idetty law U U Domato law Idempotet law Complemetato law Complemet law U De Morga law
14 Kotat uch - LSU 7 Commutatve law C C C C ocatve law borpto law C C C C Dtrbutve law Kotat uch - LSU Geeralzed uo ad terecto
15 Eample: {,,, } {,,, } {,,, } {,,, } {,,, } Kotat uch - LSU 9 Computer repreetato o et Repreet et a bary trg U {,,,,,,7,,9,} {,,,7,9} {,,,,} Kotat uch - LSU
16 Set operato become bary trg operato {,,,,} {,,,7,9} {,,,,,7,9} {,,} twe OR twe ND Kotat uch - LSU Poweret PS o S a, a, a,, a, a } { PS bt elemet : { a }: { a }: S : combato P S S Kotat uch - LSU
17 Fucto dam Chou Goodred Rodrguez Steve Name Grade C D F Chou C Rodrguez Kotat uch - LSU : Doma a Codoma a b map to Every elemet o doma ha eactly oe mage Image o a Kotat uch - LSU 7
18 Doma dam Chou Goodred Rodrguez Steve Codoma C D F Doma {dam,chou,goodred,rodrguez,steve} Codoma {,,C, D,F} Rage {,,C,F} et o all mage Kotat uch - LSU : Z Z Doma Z Codoma Z Rage {,,,9, } Kotat uch - LSU
19 Equal ucto : g : C D g D ame doma ame codoma, g ame mappg Kotat uch - LSU 7 I ome programmg laguage, doma ad codoma are eplctly deed t t a { retur a*a; } Kotat uch - LSU 9
20 Kotat uch - LSU 9 dd ad multply ucto R : Real umber R : Eample: Kotat uch - LSU Image o et } { } { S t S t S Eample: {,,9},,} { Set S
21 Oe-to-oe jecto ucto For every, y doma y mple y a Eample: b c d Each elemet o rage mage o oe elemet o doma oe-to-oe g ot oe-to-oe: g g Kotat uch - LSU Icreag ucto: y y Strctly creag: y y Strctly creag ucto are oe-to-oe Kotat uch - LSU
22 Oto urjecto ucto For every uch that Eample: y there y a b c d oto g ot oto: Z, g : Rage = Codoma Kotat uch - LSU Oe-to-oe correpodece bjecto ucto a ucto whch oe-to-oe ad oto a b c d Eample: bjecto Idetty ucto g ot bjecto bjecto Kotat uch - LSU
23 a b oe-to-oe ot oto c ot oe-to-oe oto a b c d a b c d oe-to-oe oto a b c d ot oe-to-oe ot oto ot a ucto a b c Kotat uch - LSU Ivere o a bjecto ucto y y whe doma a b c d codoma Eample: codoma doma a b c d vertble ucto y y Kotat uch - LSU
24 b a b b a a Kotat uch - LSU 7 Compoto o ucto : C g : C g : g g Eample: g g g g g g Kotat uch - LSU
25 detty ucto Suppoe y y y y y Kotat uch - LSU 9 Floor ad Celg Let be real Floor ucto: larget teger le or equal to Celg ucto: mallet teger greater or equal to Eample: Kotat uch - LSU
26 Factoral ucto : N Z!!!!! 7! 9,,9,,7,, Strlg ormula:! e Kotat uch - LSU Sequece Sequece: ucto rom a ubet o teger to a et S Fte equece Ite equece,,,,,,9,7,, a, a, a, a, a lterate repreetato a a a k a, k { a } a, a, a, a, a,,,9,7,, Kotat uch - LSU
27 te equece: a, a,, a Strg: a a a a all elemet o equece cocateated Legth o trg: a a a Empty trg ull: Kotat uch - LSU rthmetc progreo a, a d, a d,, a d, Ital term a Commo derece d Eample: { } tart wth,,7,, Kotat uch - LSU 7
28 Geometrc progreo a, ar, ar Ital term a Commo rato r,, ar, Eample: { c } tart wth,,,,, Kotat uch - LSU Summato Sequece: a, a m, am, am, Sum: a m a m a m a m a Eample: Kotat uch - LSU
29 9 Kotat uch - LSU 7 Theorem: Proo: } { } { } { c b a b a S S b a c S Ed o Proo Kotat uch - LSU Theorem: I are real umber ad, the a,r r {,} r a ar ar Proo: ar S Let
30 Kotat uch - LSU 9 a ar S a ar ar ar ar ar r rs k k k k a ar S rs r a ar S Ed o Proo Kotat uch - LSU Ueul Summato Formula, {,}, r r a ar ar
31 Coutable Set Coutable te et: y te et coutable by deault Coutable te et: S te et coutable there a oe-to-oe correpodece rom to S Z Potve teger Kotat uch - LSU Theorem: Proo: Eve potve teger: Eve potve teger are coutable,,,, Oe-to-oe Correpodece: Potve teger:,,,, correpod to Ed o Proo Kotat uch - LSU
32 Theorem: The et o ratoal umber coutable Proo: We eed to d a method to lt all ratoal umber:,, 7, Kotat uch - LSU Naïve pproach Ratoal umber: Oe-to-oe correpodece: Start wth omator=,,, Potve teger:,,, Doe t work: we wll ever lt umber wth omator :, Kotat uch - LSU,,
33 Kotat uch - LSU etter pproach: ca dagoal Nom= Nom= Nom= Nom= Kotat uch - LSU rt dagoal
34 Kotat uch - LSU 7 ecod dagoal Kotat uch - LSU thrd dagoal
35 ourth dagoal Every elemet wll be evetually caed Kotat uch - LSU 9 Ratoal Number: Dagoal ltg,,,,, Oe-to-oe correpodece: Potve Iteger:,,,,, Ed o Proo Kotat uch - LSU 7
36 Kotat uch - LSU 7 Theorem: Set ucoutable S R, Proo: ume that coutable, S the we ca lt t elemet },,, { S Elemet o S Kotat uch - LSU Lt the elemet o, S
37 7 Kotat uch - LSU t Create ew elemet baed o dagoal 7 9 Kotat uch - LSU t 7 9 I dagoal elemet the et dgt to
38 Kotat uch - LSU t 7 9 I dagoal elemet ot the et dgt to Kotat uch - LSU t 7 9 I dagoal elemet the et dgt to
39 9 Kotat uch - LSU t 7 9 I dagoal elemet the et dgt to Kotat uch - LSU t 7 9 I dagoal elemet ot the et dgt to
40 Kotat uch - LSU 79 t 7 9 y repeatg proce we obta ew umber, Kotat uch - LSU t 7 9 t der o rt dgt Obervato:
41 Kotat uch - LSU t 7 9 t der o ecod dgt Obervato: Kotat uch - LSU t 7 9 t der o thrd dgt Obervato:
42 Obervato: t der o dgt or every t S {,, }, Cotradcto! t, Ed o Proo Kotat uch - LSU We have prove:, R ucoutable It ca be prove: Every ubet o a coutable et coutable It ollow that the et o real umber ucoutable R Kotat uch - LSU
43 The prevou proo techque kow a: Cator dagoalzato argumet The ame techque ca be ued other proo Kotat uch - LSU S Theorem: I a te coutable et, the the power et PS ucoutable Proo: Sce S coutable, we ca lt t elemet S {,,, } Elemet o S Kotat uch - LSU
44 Elemet o the power et have the orm: { } {, },, { {, 7, 9, } } PS Kotat uch - LSU 7 We ecode each elemet o the poweret wth a bary trg o ad Poweret elemet { } PS arbtrary order ary ecodg {, } {,, } Kotat uch - LSU
45 Obervato: Every te bary trg correpod to a elemet o the power et Eample: Correpod to:,,,, } P { S Kotat uch - LSU 9 Let aume or cotradcto that the power et PS coutable The: we ca eumerate the elemet o the poweret P S { t, t, t, } Kotat uch - LSU 9
46 Power et elemet PS uppoe that th the repectve ary ecodg t t t t Kotat uch - LSU 9 Take the bary trg whoe bt are the complemet o the dagoal t t t t Complemet o dagoal ary trg: t Kotat uch - LSU 9
47 The bary trg t correpod to a elemet o the power et PS : t,, } P { S Kotat uch - LSU 9 t Thu, mut be equal to ome : t PS t t t However, the -th bt the bary trg o deret tha the -th bt o, thu: t P S { t, t Cotradcto!!! t,, t t t Kotat uch - LSU 9 t } Ed o Proo 7
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