2 jezn 1 cenr zihxwqic dwihnzna ziteq dpiga xeqpn wite`z :dvxnd f"qyz '` :xhqnq 290107 :jix`z zery 2 1 2 :dpigad jyn ` :cren :mipgapl ze`xed ly dpr,zel`y yely likn 'a wlg zecewp 30 ly daeg zg` dl`y likn '` wlg :miwlg dyelyn zakxen dpigad 1 zecewp 15 dl`y lk,weica zel`y izy lr dpr zel`y yely likn 'b wlg zecewp 20 dl`y lk,weica zel`y izy oeaygna `l mb,edylk xfr xnega ynzydl oi` 3 cala legk e` xegy hra aezkl `p 4 zecewp 30 -a dkfny daeg zl`y likn df wlg ` wlg jxtd e` gked dcewp s`a dkfn `l xaqd `ll daeyz zecewp 3 lra sirq lk,mitirq 10 yi dl`ya :1 dl`y :ze`ad zeprhd z` q p -e p q :zibel milewy md mi`ad miweqtd ipy (1) P (A B) = P (A) P (B) f`,odylk zeveaw izy B -e A m` (2) lr g f`,lr f g -y jk odylk zeivwpet izy g : A C -e f : B C m` (3) iaiqwltx `ed R S qgid f`,miiaiqwltx S -e R m` A dveaw dze`a miqgi ipy S -e R eidi (4) A\B A f`,diipn-za dpi`y A dveaw ly diipn-za dveaw-zz B m` (5) n 2 n 1 `ed miyp` n za dveawn y`x ayei mr dcre xegal mipeyd mikxcd xtqn (6) 81 `ed 2, 3, 5 mixtqdn cg` s`a miwlgzn `l 1, 2,, 300 oian mixtqnd xtqn (7) n xcql mipeyd mikxcd xtqnl zxveid divwpetd mirav r -a zeiperav zerahn ly laben izla xb`n oezp (8) 1 (1 rx)(1 x) r `ed dnixra mixeft mwlge dxeya mixceqn mwlgy zerahn a n = α 4 n + β 2 n `ed a n = 6a n 1 8a n 2 dbiqpd llk ly illkd oexztd (9) ( 7 3) 8 5 `ed 4 dbxcn 9 cewcewdy jk 1, 2,, 9 micewcewd lr mixtqennd mivrd xtqn (10) weica zel`y izy xga zecewp 25 dkfn dl`y lk,zel`y yely likn df wlg a wlg :(x 1, y 1 ), (x 2, y 2 ) E xear :`ad ote`a X = R\{(0, 0)} dveawa xcben E qgid :1 dl`y (x 1, y 1 )E(x 2, y 2 ) c : (x 2, y 2 ) = c(x 1, y 1 ) (zecewp 9) zeliwy qgi `ed E qgidy d`xd (`) (zecewp 9) mznver z`e zeliwyd zewlgn z` `vn (a) (zecewp 7) X/E dpnd zveaw idn (b) xezt minrt ly ibef-i` xtqn ritez D ze`dy jk A, B, C, D zeize`dn xvil xyt` n jxe`a miln dnk :2 dl`y :ze`ad mikxca ef dl`y z` (zecewp 6) mepiad zgqep zxfra (`) (zecewp 6) zexvei zeivwpet zxfra (a) (zecewp 7) dbiqp illk zxfra (b) (zecewp 6) zixehpianew dgked (c) :d`ad dxeva mixcben -e mixywd :3 dl`y p q p q T F T F T T p q p q T F F F T F (zecewp 7) q -e p miweqtde -e mixywd zxfra ( p) q weqtd z` azek (`) 9) xywd zxfra dlewy dxeva "`l" -e,"mb","e`" mixywd zxfra aezky ibel weqt lk `hal ozip ik gked (a) (zecewp (zecewp 9) q -e p miweqtde xywd zxfra wx p q weqtd z` `hal xyt` m`d (b)
2 jezn 2 cenr weica zg` dl`y xga zecewp 20 dkfn dl`y lk,zel`y izy likn df wlg b wlg dln-zz lka migeztd mixbeqd xtqn -y jk [ mixbeq n -e ] mixbeq n ly w 1 w 2 w 2n dxcq dpezp :1 dl`y dln-zz dze`a mixebqd mixbeqd xtqnn deey e` lecb `ed w 1 w 2 w j (zecewp 8) n = 1, 2, 3 xear dl`k milin ly zeiexyt`d lk z` `vn (`) xear zyxetn `gqep jknn byde,2n jxe`a Dyck ilelqn oial elld milnd zeveaw oia lre r"gg dn`zd `vn (a) (zecewp 12) dl`ya excbedy miln xtqn ziteq dveaw ARB (A B) (B A) `ad ote`a P (Z) dveawa xcben R qgid :2 dl`y (zecewp 6) zeliwy qgi `ed R qgidy d`xd (`) (zecewp 7) diipn za dveaw `id zg` lky gkede zeliwyd zewlgn z` `vn (a) (zecewp 5) P (Z)/R dpnd zveaw znver idn (b) (zecewp 7) (b) -e,(a),(`) ly zeaeyzd eid dn f`,p (R) dveawa qgid mixicbn epiid m` (c) dglvda
2 jezn 1 cenr zihxwqic dwihnzna ziteq dpiga xeqpn wite`z :dvxnd f"qyz '` :xhqnq 150207 :jix`z zery 2 1 2 :dpigad jyn a :cren :mipgapl ze`xed ly dpr,zel`y yely likn 'a wlg zecewp 30 ly daeg zg` dl`y likn '` wlg :miwlg dyelyn zakxen dpigad 1 zecewp 15 dl`y lk,weica zel`y izy lr dpr zel`y yely likn 'b wlg zecewp 20 dl`y lk,weica zel`y izy oeaygna `l mb,edylk xfr xnega ynzydl oi` 3 cala legk e` xegy hra aezkl `p 4 zecewp 30 -a dkfny daeg zl`y likn df wlg ` wlg jxtd e` gked dcewp s`a dkfn `l xaqd `ll daeyz zecewp 3 lra sirq lk,mitirq 10 yi dl`ya :1 dl`y ( 302 2 ) :ze`ad zeprhd z` q p -e (p q) :zibel milewy md mi`ad miweqtd ipy (1) P (A B) = P (A) P (B) f`,odylk zeveaw izy B -e A m` (2) r"gg g f`,r"gg f g -y jk odylk zeivwpet izy g : A C -e f : B C m` (3) iaiqwltx `ed R S qgid f`,miiaiqwltx S -e R m` A dveaw dze`a miqgi ipy S -e R eidi (4) ℵ ℵ 0 = 2 ℵ0 (5) `ed miirav dyelya mixeck ly miteqpi` dveawn mixeck 300 ly dveaw xegal mipeyd mikxcd xtqn (6) 81 `ed 3, 4, 5 mixtqdn cg` s`a miwlgzn `l 1, 2,, 300 oian mixtqnd xtqn (7) n xcql mipeyd mikxcd xtqnl zxveid divwpetd mirav 5 -a zeiperav zerahn ly laben izla xb`n oezp (8) 1 (1 4x)(1 x) 5 `ed dnixra mixeft mwlge dxeya mixceqn mwlgy zerahn a n = 3 n (α + βn + γn 2 ) `ed a n = 6a n 1 9a n 2 dbiqpd llk ly illkd oexztd (9) ( 7 3)( 4 2) 7 3 `ed 3 dbxcn 8 -e 4 dbxcn 9 cewcewdy jk 1, 2,, 9 micewcewd lr mixtqennd mivrd xtqn (10) weica zel`y izy xga zecewp 25 dkfn dl`y lk,zel`y yely likn df wlg a wlg :(x 1, y 1 ), (x 2, y 2 ) E xear :`ad ote`a X = R\{(0, 0)} dveawa xcben E qgid :1 dl`y (x 1, y 1 )E(x 2, y 2 ) c : (x 2, y 2 ) = c(x 1, y 1 ) (zecewp 9) zeliwy qgi `ed E qgidy d`xd (`) (zecewp 9) mznver z`e zeliwyd zewlgn z` `vn (a) (zecewp 7) X/E dpnd zveaw idn (b) z` xezt minrt ly ibef xtqn ritez A ze`y jk A, B, C, D zeize`dn xvil xyt` n jxe`a miln dnk :2 dl`y :ze`ad mikxca ef dl`y (zecewp 6) mepiad zgqep zxfra (`) (zecewp 7) zexvei zeivwpet zxfra (a) (zecewp 6) dbiqp illk zxfra (b) (zecewp 6) zixehpianew dged (c) :d`ad dxeva mixcben -e mixywd :3 dl`y p q p q T F T F T T p q p q T F F F T F (zecewp 7) q -e p miweqtde -e mixywd zxfra ( p) q weqtd z` azek (`) 9) xywd zxfra dlewy dxeva "`l" -e,"mb","e`" mixywd zxfra aezky ibel weqt lk `hal ozip ik gked (a) (zecewp (zecewp 9) q -e p miweqtde xywd zxfra wx p q weqtd z` `hal xyt` m`d (b)
2 jezn 2 cenr weica zg` dl`y xga zecewp 20 dkfn dl`y lk,zel`y yely likn df wlg b wlg dln-zz lka migeztd mixbeqd xtqn -y jk ( mixbeq n -e ) mixbeq n ly w 1 w 2 w 2n dxcq dpezp :1 dl`y dln-zz dze`a mixebqd mixbeqd xtqnn deey e` lecb `ed w 1 w 2 w j (zecewp 8) n = 1, 2, 3 xear dl`k milin ly zeiexyt`d lk z` `vn (`) xear zyxetn `gqep jknn byde,2n jxe`a Dyck ilelqn oial elld milnd zeveaw oia lre r"gg dn`zd `vn (a) (zecewp 12) dl`ya excbedy miln xtqn frg x R : f(x) g(x) Q `ad ote`a R R dveawa xcben R qgid :2 dl`y (zecewp 6) zeliwy qgi `ed R qgidy d`xd (`) (zecewp 6) 2 ℵ dnver zlra dveaw `id zg` lky gkede zeliwyd zewlgn z` `vn (a) (zecewp 7) R R /R dpnd zveaw znver idn (b) (zecewp 6) (b) -e,(a),(`) ly zeaeyzd eid dn f`,q Q dveawa qgid mixicbn epiid m` (c) dglvda
2 jezn 1 cenr zihxwqic dwihnzna ziteq dpiga xeqpn wite`z :dvxnd f"qyz '` :xhqnq zery 2 1 2 :dpigad jyn b :cren :mipgapl ze`xed ly dpr,zel`y yely likn 'a wlg zecewp 30 ly daeg zg` dl`y likn '` wlg :miwlg dyelyn zakxen dpigad 1 zecewp 20 dl`y lk,weica zel`yd zg` lr dpr zel`y izy likn 'b wlg zecewp 25 dl`y lk,weica zel`y izy oeaygna `l mb,edylk xfr xnega ynzydl oi` 3 cala legk e` xegy hra aezkl `p 4 zecewp 30 -a dkfny daeg zl`y likn df wlg ` wlg jxtd e` gked dcewp s`a dkfn `l xaqd `ll daeyz zecewp 3 lra sirq lk,mitirq 10 yi dl`ya :1 dl`y ( 403 3 ) :ze`ad zeprhd z` q p -e (p q) :zibel milewy md mi`ad miweqtd ipy (1) P (A B) = P (A) P (B) f`,odylk zeveaw izy B -e A m` (2) r"gg f f`,r"gg f g -y jk odylk zeivwpet izy g : A C -e f : B C m` (3) iaihifpxh `ed R S qgid f`,miiaihifpxh S -e R m` A dveaw dze`a miqgi ipy S -e R eidi (4) ℵ ℵ 0 = ℵ (5) `ed miirav drax`a mixeck ly miteqpi` dveawn mixeck 400 ly dveaw xegal mipeyd mikxcd xtqn (6) 81 `ed 3, 5, 7 mixtqdn cg` s`a miwlgzn `l 1, 2,, 300 oian mixtqnd xtqn (7) n xcql mipeyd mikxcd xtqnl zxveid divwpetd mirav r -a zeiperav zerahn ly laben izla xb`n oezp (8) 1 (1 x r )(1 x) r `ed dnixra mixeft mwlge dxeya mixceqn mwlgy zerahn a n = α4 n + (β + nγ)2 n `ed a n = 6a n 1 8a n 2 2 n dbiqpd llk ly illkd oexztd (9) ( 9 5) 8 4 `ed 5 dbxcn 9 cewcewdy jk 1, 2,, 9 micewcewd lr mixtqennd mivrd xtqn (10) weica zel`y izy xga zecewp 25 dkfn dl`y lk,zel`y yely likn df wlg a wlg :(x 1, y 1 ), (x 2, y 2 ) E xear :`ad ote`a X = R\{(0, 0)} dveawa xcben E qgid :1 dl`y (x 1, y 1 )E(x 2, y 2 ) (x 1 = x 2 = 0) (y 1 = y 2 = 0) c : (x 2, y 2 ) = c(x 1, y 1 ) (zecewp 9) zeliwy qgi `ed E qgidy d`xd (`) (zecewp 9) mznver z`e zeliwyd zewlgn z` `vn (a) (zecewp 7) X/E dpnd zveaw idn (b) B ze`de minrt ly ibef xtqn ritez A ze`y jk A, B, C, D zeize`dn xvil xyt` n jxe`a miln dnk :2 dl`y :ze`ad mikxca ef dl`y z` xezt minrt ly ibef-i` xtqn ritez (zecewp 8) mepiad zgqep zxfra (`) (zecewp 8) zexvei zeivwpet zxfra (a) (zecewp 9) dbiqp illk zxfra (b) :d`ad dxeva mixcben -e mixywd :3 dl`y p q p q T F T F T T p q p q T F F F T F (zecewp 7) q -e p miweqtde -e mixywd zxfra ( p) q weqtd z` azek (`) 9) xywd zxfra dlewy dxeva "`l" -e,"mb","e`" mixywd zxfra aezky ibel weqt lk `hal ozip ik gked (a) (zecewp (zecewp 9) q -e p miweqtde xywd zxfra wx p q weqtd z` `hal xyt` m`d (b)
2 jezn 2 cenr weica zg` dl`y xga zecewp 20 dkfn dl`y lk,zel`y yely likn df wlg b wlg dln-zz lka micg` xtqn -y jk 1 zeize` n -e 0 zeize` n ly w 1 w 2 w 2n zix`pia dxcq dpezp :1 dl`y dln-zz dze`a miqt`d xtqnn deey e` lecb `ed w 1 w 2 w j (zecewp 8) n = 1, 2, 3 xear dl`k milin ly zeiexyt`d lk z` `vn (`) xear zyxetn `gqep jknn byde,2n jxe`a Dyck ilelqn oial elld milnd zeveaw oia lre r"gg dn`zd `vn (a) (zecewp 12) dl`ya excbedy miln xtqn ziteq dveaw ARB (A B) (B A) `ad ote`a P (Z) dveawa xcben R qgid :2 dl`y (zecewp 6) zeliwy qgi `ed R qgidy d`xd (`) (zecewp 6) diipn za dveaw `id zg` lky gkede zeliwyd zewlgn z` `vn (a) (zecewp 6) P (Z)/R dpnd zveaw znver idn (b) (zecewp 7) (b) -e,(a),(`) ly zeaeyzd eid dn f`,p (R) dveawa qgid mixicbn epiid m` (c) dglvda