We prove that to every positive integer n there exists a positive integer h such that the following holds: If S is a set of h elements and f a mapping of the pow+:r set 13 of S into b such that f(T) E T for all TE '@, then there exists a strictly increasing sequence TI c ---c T,, oi' subsets of S su
A constructive proof for a theorem on contractive mappings in power sets
โ Scribed by Egbert Harzheim
- Publisher
- Elsevier Science
- Year
- 1983
- Tongue
- English
- Weight
- 981 KB
- Volume
- 45
- Category
- Article
- ISSN
- 0012-365X
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โฆ Synopsis
Recently I proved the following theorem: To every positive integer m there exists a positive integer h such that the following holds: If S is a set of h elements and f a mapping of the power set q of S into q such that f(T) E T for all TE 8, then there exists a strictly increasing sequence T, c l l -c T, of subsets of S such that one of the following three possibilities holds: (a) All sets f(q), i= 1,. . . , m, are equal. (b) For all i = 1, . . . , m we have f(Ti) = Tit,. (c) Ti=f(Ti+,)for all i=l,...,m-1. The proof given in [2] was non-constructive. In this paper now we give a constructive proof. By the way, this also yields a solution of a problem of Rado [3, p. 1061. ' ' -* 'IA) = : ((oycs y >((oH J1 (?A ' ll c+wq 'r+wq +wc_ '1-4~)~~ ' * l -'IQ =: (A@ : SMOI\OJ SE Wi %urqJosqe Icq *Q uo ,s uog!lred-luaw%as E auyap pue (A) W S!ql I[%?3 aM l "a > "ti %&S!,l?S N )SWl aq, ayisl aM *Q 3 it baAa '04 Uiq& 'pgtw ST '{q ' l l ' 'I} 3 N '"a > "k su0ye~a.I y aqj JO au0 lsea[ $I? *Q 3 A yxa "Od :aAlq QM (% * . l l "a) = : (a)s pug (% 'l * "A) = : (h)s lnd aM .+Q 3 Ii 'OJ JI :!iuea~u lwp ?brtquo:, aql aurnssE a~ OS aho.xd 01 SU~XII~J %u~y)ou ~ULJ I+ u1 yl%. : aql 01 papualxa si 3 upzqa aql A >i b icql qms [( I+ u1 'y %)J '( UI 'y 'D)J) u (I = : *a 3 i lUNK3~~ U?? SlSiXa aJay) Ji 'MON '(W '3 'V)js a aAl'?y aM OS '3 JO juxUa[a lsalea&I ayl ay a ia? l tu ylZ?ua~ 30 3 u!eq3-,s ue sisixa araqi spaqlocihy uoilmpu! mo 01 %urplomv '[(W 'y '?I)J '") u Q 01 S JO uojbs!3$sal ayl alouap ,s la? * I + lu q@uaI 30 [( I+ l.0 'y "D)J 'u) u Q 5 u!sq~-s UB slsixa aJay l~ql ahold 01 aiwq aM 'Q uo uogiErcd-luaw9as aaa!d-y i! aq s la1 put? '( I+ IU 'y %)I pw? v uaaM)aq swap-v aq N 3 Q I;VI '(I+ u1 'y 'v) 1~ OS osr,~! saop I! leql aAoJd 01 aney am uayl pue (?!A 'y 91) le luawalels aql says~lm / leql qms Jasalu! ah&sod ua@ IT ST tb ~eyl auxnsse aM 'tu uo uoilmpu! asn aM asodmd s!qi OJ~ 'N 3 w qlfM (U.A 'y l v) saId!Jl III? 303 luat.ualt?ls aql ahold 01 amy ahi uau 'v 30 1uawaIa uaA!% B aq v ial MON '"~~---~'~3tu 'V3V l#M (l4-4'~--y'V) saldlq 1111 303 scloy luatualels aql lay1 pm 2" Ja%alur paxy t! s! y leql s!say,odhq uoympu! aql ayeur ah MON 'f+.j 3 LU '#f 3 v ql!M (UA ' 1 'v) saldg II?? 304 an.31 si )uatuaiels slf '(z) 6q '1s~1d 'w pm y uo uo!impui Aq ??wrual mo aAord aM '(2) JO (I) sasm aql 01 %u$iboIaq (144 'y %) sqd!q II?? 103 ura3oayl aw JO ~uawal~ls aql says!les 'sE~nullo3 aawl %oqe aql ftq ua@ !J uogmn3 aql ieql pm luals!suo=, am (2) pm (I) wql I+gl s! 11 .M
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