Foundations of Computer Science Comp109

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1 Reding Foundtions o Computer Siene Comp09 University o Liverpool Boris Konev konev@liverpool..uk Prt. Funtion Comp09 Foundtions o Computer Siene Disrete Mthemtis nd Its Applitions K. Rosen, Setion.. Disrete Mthemtis with Applitions S. Epp, Chpter 7. Prt. Funtion / Prt. Funtion / Contents Funtions Funtions/methods on progrmming Funtions: deinitions nd emples Domin, odomin, nd rnge Injetive, surjetive, nd ijetive untions Invertile untions Compositions o untions Funtions nd rdinlity Pigeon hole priniple Crdinlity o ininite sets 0 Chpter Speking Mthemtilly untion mhine The squring untion rom R to R is deined y the ormul () = or ll rel numers. Thismensthtnomtterwhtrelnumerinputissustitutedor, the output o will e the squre o tht numer. This ide n e represented y writing ( ) =. In other words, sends eh rel numer to,or,symolilly, :.Notethtthevrile is dummy vrile; ny other symol ould reple it, s long s the replement is mde everywhere the ppers. Prt. Funtion The / suessor untion g rom Z to Z is deined y the ormul g(n) = n +. Prt Thus,. Funtion / no mtter wht integer is sustituted or n, theoutputog will e tht numer plus one: g( ) = +. In other words, g sends eh integer n to n +, or, symolilly, Deinition g: n n +. An emple o onstnt untion is the untion h rom Q to Z deined y the ormul h(r) = orllrtionlnumersr. Thisuntionsendsehrtionlnumer A untion rom set A to set B is n ssignment o etly one elementr to. In other words, no mtter wht the input, the output is lwys : h( ) = or h: r. o B to eh element o A. The untions, g,ndh re represented y the untion mhines in Figure... Input Emple.. Funtions Deined y Formuls Figure.. () Output Emples: y = y = sin() irst letter o your nme Jv puli int (int ) { return +; } C/C++ int (int ) { return +; } Python de (int ): return + Prt. Funtion / We write () = i is the unique element o B ssigned y the untion to the element o. n r I is untion rom A to B we write : A B. Figure : A untion : {,, } {,, } squring untion () () = suessor untion () Figure.. g(n) = n + onstnt untion () h(r) = Auntionisnentityinitsownright.Itnethoughtosertinreltionship etween sets or s n input/output mhine tht opertes ording to ertin rule. This is the reson why untion is generlly denoted y single symol Figure or string: o symols, No untion suh s, G, olog,orsin. AreltionissusetoCrtesinprodutnduntionisspeilkindoreltion. Speiilly, i nd g re untions rom set A to set B, then ={(, y) A B y = ()} nd g ={(, y) A B y = g()}. It ollows tht 7 7 Figure : No untion equls g, written = g, i, nd only i, () = g() or ll in A. / Prt. Funtion Prt. Funtion 7 / Prt. Funtion 8 /

2 Domin, odomin, nd rnge Codomin vs rnge omposition o nd g euse n element is ted upon irst y nd then y g. (g )() = g( ()) or ll X, Composition o untions where g is red g irle ndg( ()) is red g o o. The untion is lled the I omposition : X Y nd g : Yo Z nd re untions, g. then their omposition g is untion rom X to Z given y A B (g )() = g(()). This deinition is shown shemtilly elow. Suppose : A B. A is lled the domin o. B is lled the odomin o. The rnge (A) o is (A) = {() A}. (A) Figure : the rnge o X Y () Y' g Z g( ()) = (g )() g Prt. Funtion 9 / Emple Consider the untion : R R given y () = nd the untion g : R R given y g() = +. Clulte g, g, nd g g. Prt. Funtion 0 / Injetive (one-to-one) untions! Emple 7.. Composition o Funtions Deined y Formuls Prt. Funtion / Let : Z Z e the suessor untion nd let g: Z Z e the squring untion Surjetive (or onto) untions (n) = n + orlln Z nd g(n) = n or ll n Z.. Find the ompositions g nd g. Deinition Let : A B e untion. We ll n injetive (or one-to-one) Cution! Be reul not. Is g = g?eplin. untion i to onuse g nd Deinition : A B is surjetive (or onto) i the rnge o oinides with ( ) = ( ) g( = ()): orgll is, the nme A. Solution the odomin o. This mens tht or every B there eists A with o the untion wheres = (). This is logilly equivlent to g( ()) ( is ) the (vlue ) nd o so injetive. The untions g nd g re deined s ollows: untions never repet vlues. In other untion words, t dierent. inputs give Emples (g )(n) = g( (n)) = g(n + ) = (n + ) or ll n Z, dierent outputs. : Z Z given y () = nd is not surjetive. Emples h : Z Z given y h() = is not surjetive. ( g)(n) = (g(n)) = (n ) = n + orlln Z. : Z Z given y () = is not injetive. h : Q Q given y h () = is surjetive. h : Z Z given y h() = is injetive. Prt. Funtion / Copyright 00 Cengge Lerning. All Rights Reserved. My not e opied, snned, or duplited, in whole or in prt. Due to eletroni rights, some third prty ontent my e suppressed rom the ebook nd/or echp Editoril review hs deemed tht ny suppressed ontent does not mterilly et the overll lerning eperiene. Cengge Lerning reserves the right to remove dditionl ontent t ny time i susequent rights restri Prt. Funtion / Prt. Funtion / Clssiy : {,, } {,, } given y Clssiy g : {,, } {,, } given y Clssiy h : {,, } {, } given y Prt. Funtion / Prt. Funtion / Prt. Funtion 7 /

3 Clssiy h : {,, } {,, } given y Bijetions Inverse untions We ll ijetive i is oth injetive nd surjetive. Emples : Q Q given y () = is ijetive. I is ijetion rom set X to set Y, then there is untion rom Y to X tht undoes the tion o ; tht is, it sends eh element o Y k to the element o X tht it me rom. This untion is lled the inverse untion or. Then () = i, nd only i, () =. Prt. Funtion 8 / Prt. Funtion 9 / Prt. Funtion 0 / Emple Emple Bijetions nd representtions k : R R given y k() = + is invertile nd k (y) = (y ). Let A = { R, } nd : A A e given y () =. Show tht is ijetive nd determine the inverse untion. Let S = {,,..., n} nd let B n e the set o it strings o length n. The untion : Pow(S) B n whih ssigns eh suset A o S to its hrteristi vetor is ijetion. Prt. Funtion / Prt. Funtion / Prt. Funtion / Crdinlity o inite sets nd untions The pigeonhole priniple Pigeons nd pigeonholes Rell: The rdinlity o inite set S is the numer o elements in S A ijetion : S {,..., n}. For inite sets A nd B A B i there is surjetive untion rom A to B. A B i there is injetive untion rom A to B. A = B i there is ijetion rom A to B. Let : A B e untion where A nd B re inite sets. The pigeonhole priniple sttes tht i A > B then t lest one vlue o ours more thn one. In other words, we hve () = () or some distint elements, o A. I (N+) pigeons oupy N holes, then some hole must hve t lest pigeons. Prt. Funtion / Prt. Funtion / Prt. Funtion /

4 Emple Emple Emple Prolem. There re people on us. Show tht t lest two o them hve irthdy in the sme month o the yer. Prolem. How mny dierent surnmes must pper in telephone diretory to gurntee tht t lest two o the surnmes egin with the sme letter o the lphet nd end with the sme letter o the lphet? Prolem. Five numers re seleted rom the numers,,,,,, 7 nd 8. Show tht there will lwys e two o the numers tht sum to 9. Prt. Funtion 7 / Prt. Funtion 8 / Prt. Funtion 9 / Etended pigeonhole priniple Emple Emple Consider untion : A B where A nd B re inite sets nd A > k B or some nturl numer k. Then, there is vlue o whih ours t lest k + times. Prolem. How mny dierent surnmes must pper in telephone diretory to gurntee tht t lest ive o the surnmes egin with the sme letter o the lphet nd end with the sme letter o the lphet? Prolem. Show tht in ny group o si people there re either three who ll know eh other or three omplete strngers. Prt. Funtion 0 / Prt. Funtion / Prt. Funtion / Bijetions nd rdinlity Emple: The rdinlity o the power set. Power set nd it vetors Rell tht the rdinlity o inite set is the numer o elements in the set. Sets A nd B hve the sme rdinlity i there is ijetion rom A to B. Deinition The power set Pow(A) o set A is the set o ll susets o A. In other words, Pow(A) = {C C A}. For ll n Z + nd ll sets A: i A = n, then Pow(A) = n. Rell tht i ll elements o set A re drwn rom some ordered sequene S = s,..., s n : the hrteristi vetor o A is the sequene (,..., n ) where i = { i si A 0 i s i A We use the orrespondene etween it vetors nd susets: Pow(A) is the numer o it vetors o length n. Prt. Funtion / Prt. Funtion / Prt. Funtion /

5 The numer o n-it vetors is n The numer o n-it vetors is n Ininite sets Sets A nd B hve the sme rdinlity i there is ijetion rom A to B. Emples: We prove the sttement y indution. Bse Cse: Tke n =. There re two it vetors o length : (0) nd (). Indutive Step: Assume tht the property holds or n = m, so the numer o m-it vetors is m. Now onsider the set B o ll (m + )-it vetors. We must show tht B = m+. Every (,,..., m+ ) B strts with n m-it vetor (,,..., m ) ollowed y m+, whih n e either 0 or. Thus B = m + m = m+. Z nd even integers onsider (n) = n { R 0 < < } nd R + onsider g() = { R 0 < < } nd R 7. Crdinlity with Applitions to Comput Deine untion F: S R s ollows: Drw numer line nd ple the intervl, S, somewhtenlrgednde irle, tngent to the line ove the point 0. This is shown elow. L Numer line F() 0 Prt. Funtion / Countle sets Prt. Funtion 7 / Countle Sets: Q For eh point on the irle representing S, drw stright line L through most point o the irle nd. LetF() e the point o intersetion o L nd the line. (F() is lled the projetion o onto the numer line.) It Prt is ler. Funtion rom the geometry o the sitution tht distint points on 8 the / ir distint points on the numer line, so F is one-to-one. In ddition, given ny p Unountle sets the numer line, line n e drwn through y nd the top-most point o the ir line must interset the irle t some point, nd,ydeinition,y = F(). T onto. Hene F is one-to-one orrespondene rom S to R, nd so S nd R sme rdinlity. A set tht is either inite or hs the sme rdinlity s N is lled ountle. Z Prt. Funtion 9 / Cntor s digonl rgument Prt. Funtion 0 /... You know tht every positive integer is rel numer, so putting Em together with Cntor s theorem (Theorem 7..) shows tht the ininity o the relnumers is greter thn theininity othesetollpositive integers. Inee you re sked to show tht ny set nd its power set hve dierent rdinlities. there is one-to-one untion rom ny set to its power set (the untion tht t element to the singleton set {}), this implies tht the rdinlity o ny se thn the rdinlity o its power set. As result, you n rete n ininite seq lrger nd lrger ininities! For emple, you ould egin with Z, the set o ll nd tke Z, P(Z), P(P(Z)), P(P(P(Z))),ndsoorth. A set tht is not ountle is lled unountle. Applition: Crdinlity nd Computility S = { R 0 < < } is unountle Knowledge o the ountility nd unountility o ertin sets n e used to question o omputility. We egin y showing tht ertin set is ountle. Emple 7.. Countility o the Set o Computer Progrms in Computer Lnguge Show tht the set o ll omputer progrms in given omputer lnguge is ou Solution This result is onsequene o the t tht ny omputer progrm lnguge n e regrded s inite string o symols in the (inite) lphet o guge. Given ny omputer lnguge, let P e the set o ll omputer progrms in guge. Either P is inite or P is ininite. I P is inite, then P is ountle n done. I P is ininite, set up inry ode to trnslte the symols o the lp the lnguge into strings o 0 s nd s. (For instne, either the seven-it A Stndrd Code or Inormtion Interhnge, known s ASCII, or the eight-it E Binry-Coded Prt. Funtion Deiml Interhnge Code, known s EBCDIC, might e used.) / For eh progrm in P, usetheodetotrnsltellthesymolsintheprog 0 s nd s. Order these strings y length, putting shorter eore longer, nd Suppose S is ountle. Then the deiml representtions o these numers n e written s list Copyright 00 Cengge Lerning. All Rights Reserved. My not e opied, snned, or duplited, in whole or in prt. Due to eletroni rights, some third prty ontent my e suppressed rom the ebook nd/or ec Editoril review hs deemed tht ny suppressed ontent does not mterilly et the overll lerning eperiene. Cengge Lerning reserves the right to remove dditionl ontent t ny time i susequent rights re = n... = n... = n.... n = 0. n n n... nn.... Let d = 0.d d d... d n... where {, i ii d i =, i ii = Then d is not in the sequene,, Prt. Funtion /