h''qyz ` xhqnq,zihxwqic dwihnznl `ean '` cren - ogand onqheb hxaex lk zveaw xnelk,b zegt A :dveawd ef A B f`,zeveaw B e A m` :dxrd A\B i''r zpneqn `id k''xca.b-l mikiiy `le A-l mikiiyy mixai`d :zniiwnd a 0, a 1, a 2,... dxcqd ly i-n-d xai`l yxetn iehia e`vin (`).1.a n = 4a n 1 + 5a n 2,n 2 xeare,a 1 = 5,a 0 = 3 `ed dly ipiite`d mepiletd okl,a n + 4a n 1 5a n 2 = 0 `id d`eeynd :eiyxey z` `vnp,p(x) = x 2 + 4x 5 x 2 + 4x 5 = 0 (x + 5)(x 1) = 0 x 1 = 1, x 2 = 5. :A, B mireawd z` `vnp,a n = A 1 n + B ( 5) n = A + B ( 5) n okl 3 = a 0 = A + B ( 5) 0, 5 = a 1 = A + B ( 5) 1, A + B = 3 mbe A 5B = 5, okl okl,a = 3 + 1 3 e B = 1 3 okl,3 B 5B = 5 `''f a n = 3 + 1 3 (1 ( 5)n ). itle,a 2 = 4 5 + 5 3 = 5 diqxewxd zgqep itl lynl :wecal xyt`.a 2 = 3 + 1 3 (1 ( 5)2 ) = 3 8 = 5 ep`vny dgqepd.300 20 lye 30 30 ly mitzeynd miwlgnd xtqn edn (a) 30 30 = (2 3 5) 30 = 2 30 3 30 5 30, :miipey`x minxebl mixtqnd z` wxtp 300 20 = (3 100) 20 = (3 4 25) 20 = (3 2 2 5 2 ) 20 = 2 40 3 20 5 40. 1
jk d`xp 300 20 e 30 30 ly szeyn wlgn 2 a 3 b 5 c, 0 a min(30, 40) = 30, 0 b min(30, 20) = 20, 0 c min(30, 40) = 30, e minly mixtqn a, b, c xy`k (a, b, c) zexecqd zeylyd xtqnl deey mitzeynd miwlgnd xtqn okl 0 a 30, 0 b 20, 0 c 30, y jk minly mixtqn ly.mitzeyn miwlgn 31 21 31 yi xnelk `z lkay jk mipey mi`z 5 oia (midf) mixeck 20 wlgl mikxcd xtqn edn.2? mixeck 6 xzeid lkl eidi oey`xd `zae cg` xeck zegtl didi mipey mi`z 5 oia midf mixeck n wlgl zeiexyt`d xtqn z` a n a onqp.mixeck 6 xzeid lkl eidi oey`xd `zae cg` xeck zegtl didi `z lkay jk mixtqna zepexztd xtqnl deey a n f`,i xtqn `za mixeck u i epnyy gipp u 1 + u 2 + u 3 + u 4 + u 5 = n, d`eeynd ly minly dxcqd ly zxveid divwpetd.1 u 2, u 3, u 4, u 5,1 u 1 6 miveli`d mr F (x) = a n x n n=0 = (x + x 2 + x 3 + x 4 + x 5 + x 6 )(x + x 2 + x 3 +...) 4 `id {a n } n=0 = x(1 + x + x 2 + x 3 + x 4 + x 5 )x 4 (1 + x + x 2 + x 3 +...) 4 = x 5( 1 x 6 )( 1 ) 4 1 x 1 x ( 1 ) 5. = (x 5 x 11 ) 1 x y mircei ep`.f (x) a x 20 ly mcwna xnelk,a 20 a mipiipern ep` ( 1 ) 5 (n + 5 1)! ( ) n + 4 ( ) = x n = x n. 1 x (5 1)!n! 4 n=0 n=0 okl,( ) iehiaa x 9 e x 15 ly mincwna mipiiperne ( ) ( ) ( ) ( ) 15 + 4 9 + 4 19 13 a 20 = =. 4 4 4 4 2
R {0} dveawd lr xcben S.iwlg xcq qgi `ed S qgid m`d ewcia (`).3 x 1 m` wxe m` (x, y) S. y `ad ote`a (0 ila miiynnd). x x = 1 okl,x 0 f`,x R {0} idi :zeiaiqwltx e,1s( 1) okl, 1 1 = 1 1 ik,ixhniq ihp` `l S :zeixhniqihp`.iwlg xcq qgi `l S `''f.1 1 la`.( 1)S1 okl, 1 1 = 1 1 f`,ysz e xsy gipp :zeiaihifpxh z`f lka wecap x y 1 mbe y z 1, x z = x y y z = x y y z 1. okl.iaihifpxh S okl,xsz `''f zeveaw izy lkl :d`ad dprhd z` (zicbp `nbec i''r) ekixtd e` egiked (a).p (A B) = P (A) P (B) :miiw B e A f`,b = {2} e A = {1} dpiidz,zicbp `nbec `iap,zixwiy dprhd P (A B) = P ({1, 2}) = {, {1}, {2}, {1, 2}}, P (A) P (B) = P ({1}) P ({2}) = {, {1}} {, {2}} = {, {1}, {2}}..{1, 2} P (A) P (B) la`,{1, 2} P (A B) ik mi`ex divwpet xicbp.y = {x Z : ibef i` x},x = {x Z : ibef x} :onqp.4 :`ad ote`a f : P (Z) P (Z) { X A ziteq dveaw A m` f(a) = A X ziteqpi` dveaw A m` f({1}) = X {1} = X,.f(X),f(Y ),f({1}) z` eayg (`) okl,ziteq dveaw {1}.X Y = ik,f(y ) = Y X = Y okl,ziteqpi` dveaw Y.1 X ik.f(x) = X X = okl,ziteqpi` dveaw X 3
.zikxr-cg-cg dpi` f -y egiked (a) y jk,a, B P (Z) zepey zeveaw izy `evnl witqn z`f gikedl ick okl zeiteq odizy,b = {3},A = {1} gwp.f(a) = f(b) f(a) = X {1} = X, f(b) = X {3} = X,.r''gg `l f okl.f(a) = f(b) `''f.f ly dpenza dpi` {1, 2} dveawdy e`xd (b) dlilya gipp.f(a) = {1, 2} y jk A P (Z) zniiw `ly gikedl jixv yi f(a) a `''f,f(a) = X A X f` ziteq A m`.z`fk A zniiwy.1 f(a) ik!dxizq - miibef mixtqn wx - miibef mixtqn oi` f(a) a `''f,f(a) = A X f` ziteqpi` A m`.f ly dpenza dpi` {1, 2} okl.2 f(a) ik!dxizq R inewn-ec qgi oniq dlikny dtya mihwicxtd aiygza weqt enyix (`).5 zeliwy zewlgn yely zegtl yiye zeliwy qgi `ed R qgidy orehy.zepey qgi.oeieeyd oniq -e inewn-ec qgi oniq R,L = {R, } `id dtyd.iaihifpxhe,ixhniq,(dpand ly mlerd lr) iaiqwltx qgi `ed zeliwy :R lr z`f extqiy miweqt 3 meyxp α = xr(x, x), β = x y(r(x, y) R(y, x)), γ = x y z((r(x, y) R(y, z)) R(x, z)). -i` 3 yiy cibdl witqn zeliwy zewlgn 3 zegtl yi R-ly cibdl ick weqt meyxp,(r qgia micner `l) milewy `l mdn 2 lky mipey mixa δ = x y z[ (x y x z y z) (R(x, y) R(x, z) R(y, z))]. :z`f xn`iy.α β γ δ didi eply weqtd weqtd z` δ mewna zgwl xyt`,α z` llek eply weqtdy llba :dxrd δ = x y z (R(x, y) R(x, z) R(y, z)), `ad.dl`yd lr dper α β γ δ weqtd mb f`e :dibelehe`h `ed `ad weqtd m`d ewcia (a) [( p q) (r q)] (p r). 4
dnyd zniiw f`,dibelehe`h `l α m`.α l''pd weqtl `xwp g : {p, q, r} {T, F }, e g(p) = F `''f,f jxr p r weqtl zpzep g okl.f jxr α l zpzepy p q okl,t jxr ( p q) (r q) weqtl zpzep g.g(r) = T (r q) mb.g(q) = T lawp,g(p) = F y oeekne,g dnyda T jxr lawn eplaiw `''f.g(q) = F lawp,g(r) = T y oeekne,g dnyda T jxr lawn jxr α l zpzepy dnyd zniiw `l okl!dxizq - g(q) = F e g(q) = T.dibelehe`h α `''f,f.zn` zlah zxfra mb xeztl xyt`y oaenk :dxrd ( xp (x)) ( xq(x)) x(p (x) Q(x)). :d`ad zibeld dxixbd z` egiked (`).6 ipniq Q e P xy`k {P, Q, } dtya dpan M = ( M, P M, Q M ) idi, M mlerd ly dveaw zz `ed M a mdly yexitd `''f,miinewn-1 qgi y jk a M miiw'' aezkl mewna.q M M e P M M xnelk gipp.''a P M miiw'' xeviwa aezkp ''a P M ( xp (x)) ( xq(x)) xp (x) m`.m a izin` xq(x) e` M a izin` xp (x) f`,m a izin`,a dveaw lkl a P M A miiw okl,a P M miiw f`,m a izin` x(p (x) Q(x)) `''f,a P M Q M miiw hxta.m a izin` miiw okl,a Q M miiw dnec ote`a f`,m a izin` xq(x) m` `''f,a P M Q M miiw hxta,a dveaw lkl a A Q M x(p (x) Q(x)).dxixbd z` epgked okl.m a izin`,`l m`e,dze` egiked,ok m`? dpekp mb dketdd zibeld dxixbd m`d (a).milewy mpi` miweqtd eay mi`zn dpan z`ivn ici-lr ekixtd dtya dpan M = ( M, P M, Q M ) idi :dgked,dpekp dketdd dxixbd mb gipp.miinewn-1 qgi ipniq Q e P xy`k {P, Q, } x(p (x) Q(x)) 5
m`.a Q M e` a P M okl,a P M Q M miiw f`,m a izin` f`,a P M xp (x),{p, Q, } dtya α weqt lkl,m a izin` ( xp (x)) α okl,m a izin` ( xp (x)) ( xq(x)) hxta xq(x).m a izin` f`,a Q M m`,{p, Q, } dtya α weqt lkl,m a izin` α ( xq(x)) okl,m a izin` ( xp (x)) ( xq(x)) hxta.dketdd dxixbd z` epgked.m a izin` -petd mr zilnxet xzei dxeva mitirqd ipy z` gikedl xyt`y oaenk :dxrd.jixvy enk df z` miyery i`pza,g dnyde val divw 6