"!#$&%('*),+,'.-0/213)&!42/6587:9;2/!@-#A5&BC5ED;,#F1A+HGI/"7@? 4BA C/DE...
14 downloads
205 Views
181KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
"!#$&%('*),+,'.-0/213)&!42/6587:9;2/!@-#A5&BC5ED;,#F1A+HGI/
QO PPR2SATUVW ^v_hYIA`0_a~\X8`Q]YAZdZ\p=^I[@ehc@]rgZd^Afg[@_a`:|ehccb@]]@Zd`Qcp,w\`0]m^I YI]@]eY3Zdiu^Ix2c@ehcZdfg^Ic@ehZAc_h]^I`:]sFc2b@ikZdnFjl`F][@mFm_hiy]]Qn@ZdbY3n@ioezc[@c2b@ZIZdpq{^Ap,ehc@cZC]rg`<sFe\_h]M^I|AYA_heI^Is?ion2c2idio_\Yd_\x@p?]ioY3^Ii nv ]^AY3_}iu` t
l
l
0@&& 6&@&& 2¡ g¢6£6¤C¥¤C¦¤o§6¨6¤©QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ 2ª¬«£6¨6®¯¤o£6° QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ 2 ±³²µ´0¶0·6¸u¹0´º¶F¹0§6¶F»¤o§6¨6·EQQQQQQQQQQQQQQQQQQQQQQQQQQ ¼½=¾¿6&@&&À ¼ ±(¡ÁµÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹ÃEQQQQQQQQQQQQQQQQQQQQQQQQQQQQ ± ±(0 g¢6£6¤C¥¤C¦¤o§6¨6¤QQQQQQQQQQQQQQQQQQQQQQQQQQ ± ±(0ª ÄŦƺF¸o¸o°ÈÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨ÉQQQQQQQQQQQQQQQQQQQQ ± ±(0 ± «£6¨6®¯¤o£6°Ê¶F¹0§6¶F»¤o§6¨6·ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨³QQQQQQQQQQQQQ Ë ±(ª¬«¶0£Ì2¥¶0ÂKQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ Í ±( ±«£6¤C¥¢6¶0Î(¹0¤o§6¨6¤QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ Ï ÐÑÒ&6&&¾ ÓÔ=&Õ&Ö&&&×(Ø8¾ Ù ÚÛ Ô=&Õ&Ö&&&×(Ø8Ü8¾:Ý<Þ@&×@ß&@&?à á â¡ãä&å&×(æç&@&?à è é êìë
H
l
í ¶0¸o§6¶0´0¤:¹0¤o¶0£6¨6¨Ê£6¨6¸oÂ2º(îlÂ(ºFÂq¹0¤o¶0£6¨6¨È¢6£6¨6§(Ì(¹0¨(ÌÈ£6¤u»¤o§6¨6·Ê´Kï0¸C¦¶0´0¨(Ì2ðñ´0¤o£6¶IÌ(¹0§6¶0¸u¹0§6¶0· §6¤o¶0¢6£6¤C¥¤C¦¤o§<§6í ¶0¸u¹¨,î&¦¤uò"¨(¹ó¢6¶0§(Ì(¹0¨6¤k¢6£6¤C¥¢6¶0Î(¹0¤d§6¨(ÌK§<º ®¯§6¶Iòô¤o¸u¹0´0¤´F¤o£6¶AÌ(¹0§6¶0¸u¹0§6°µðK£<ºF¸Cõ ¢6£6¤C¥¤C¦¤o§6¨<·, §<ºF¸u¹0¶IÌ(ö¤o·ç¦¤oÂ6÷6¨6¨K´0´0¶d¥¨(¹0¸\Ìç¢6¶0§(Ì(¹0¨6¤¶F¹0§6¶F»¤o§6¨(ÌK¨ç£<ºF¸o¸o®øº3¹0£6¨6´ºFù¹¸hÌ úû3ü}ýCþAýCÿdý Aþ3ü 3þA ý uû 3þ CûAþ "!$#&%%''( %#)* uü}û 3 ü +,#-.+/0 1 +,#2 3 465 7 8:9 5 ;<0= >"7@? 4BA C/DE5FG;,A DGH #dý AJI (KLMNPOKQ (RM ∗
ST ST-UWVXYV[Z\]^VX 6¢ £6¤C¥¢6¶0Î(¹0¤d§6¨6¤¨K¸o´FÌJ_IºF§6§6°l¤¸§6¨6® ¶F¹0§6¶F»¤o§6¨(Ì ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ ¨K¢6¶0£@Ì2¥Â2º(a`B_C¦ÆºcbuºF¤Cõ ®¯°ld¤ _u¥¤o¸uÃ"¸o´0¤C¥¤o§6¨(̨6¸u¢6¶3¦cà _oï2ù¹¸hÌ ´ô¹0¤o¶0£6¨6¨£6¨6¸oÂ2º¢6£6¨¨J_oï2Î6¤o§6¨6¨¸o´FÌJ_d¨¢6£6¤C¥¢6¶0Î(¹0¤o§<¨6· §<º®¯§6¶Iò¤o¸u¹0´0¤ô´0¤o£6¶AÌ(¹0§6¶0¸C¹0§6° ð£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨<· ¸feï2§6Â6÷6¨6¶0§<ºA¦ÆºF®¯¨,î^_IºA¥ÆºF§6§6°l®¯¨§<ºÇu¹0¶0® ®¯§6¶Aòô¤o¸u¹0´0¤F g hji l k
l
m6no
prqtsBu&vdu^wxuzyt{Bu «gï¸u¹0à X | §6¤oÂ@¶F¹0¶0£6¶0¤le ¨6Â@¸o¨6£6¶0´ºF§6§6¶0¤®¯§6¶Aòô¤o¸u¹0´0¶6 ä&å&0Ø8@&8z} ~$fP
-P Q -xP/
c X - a [
c&Jcc1P P6P xP/
c
P
c
c&Jcc -Y Q ⊆ X × X ¶0´0¶0£Ì(¹Aî2Î(¹0¶ôÇC¦¤o®¯¤o§(¹0° x, y ∈ X §<ºAð¶I¥<Ì(¹0¸hÌ´M¶F¹0§6¶F»¤o§6¨6¨ Q î2¤o¸C¦¨ (x, y) ∈ Q Çu¹0¶F¹
eQºFÂ(¹
_IºF¢6¨6¸o°l´ºF¤u¹0¸\̹Fº3Â(ò¤
xQy
m6nEm
st{B uzsB¡ ¢ å&"£@åñz} ¤a[dtPYPxP/
c I = {(x, x)| x ∈ X } ⊆ X ×X ¥ P0d¦f § fc, fPxP/
c[
¨fP
-PdX © ª «¬ ® ¯ «/ª J° J±t&²Y³&d-fP WxP
X = {x, y, z} ³
xfP
-P´Jcµ1Y Px-W &¶ §6¦
·PP¶ -,0¸&³0& d-f /²P-¦fPºfP
-P I ¹ /Wd »J-¼P¶[½ x X ×X¹ X x x ◦ y · z ·
y · ◦ ·
z · · ◦
¢ å&"£@åñz}¿¾l± &¶ §6¦
jPP¶ -,ºJcµ, P fP
-Pº-À»Jf^&²Y³0& d-f P0ÁxP/
c X = {x, y, z, t}¹ - Yd§6¦
±PÂ--/P P Ã-c c-Y ¹ P ¹
ÄP dP ¹ PJ-§6¦
ÅfP
-Pºxa&c& &-Y¨Æ ÈÇ x y z t
m6nGÉ
ÊÃËaÌÎÍBÏ&ÐÑËaÒÓÌÎÐ"yBÌaÔÕuzyt{tÍ
x · ◦ ◦ ·
y ◦ · · ◦
z · · · ·
t · · · ·
g ¢6¨(»¤o® §6¤oÂ@¶F¹0¶0£6°l¤¸o´0¶0·6¸u¹0´º(î&Â@¶F¹0¶0£6°l¤×ÖFïI¥<ï@¹:¨6¸o¢6¶3¦Ãc_d¶0´º3¹0Ã0¸hÌ ¥6¦<ÌEºFÂ2¸o¨6¶0®øº3¹0¨6Î6¤o¸oÂ@¶[bo¶ _IºA¥ÆºF§6¨(̶F¹0§6¶F»¤o§6¨6·, «¶3¦§6¶F¹3º( Ź0§6¶F»¤o§6¨6¤ Q §<ºK®¯§6¶Iòô¤o¸u¹0´0¤ X §<º _d°l´ºF¤u¹0¸hÌÙØÚ Û ÜaÝ îl¤o¸C¦¨.¥6¦<Ì.¥´Fïð • ¢6£6¶0J¨ _d´0¶3¦Ã0§6°µð ÇC¦¤o®¯¤o§(¹0¶0´ ¢6¶Â6£<ºF·6§6¤o· ®¯¤o£6¤¶d¥§<º¢<ºF£<º (x, y) ¨2¦¨ (y, x) Ì6´3¦<Ì6¤u¹0¸\ÌÇC¦¤o®¯¤o§(¹0¶0® Q x, y ∈ X
± ²µ¨6®¯®¯¤u¹0£6¨6Î6§<¶0¸u¹Ã6( Ź0§6¶F»¤o§6¨6¤ Q §<º®¯§6¶Iò¤o¸u¹0´0¤ X §<º _d°l´ºF¤u¹0¸hÌ ®äYÝÎÝ.¬¯ © äYåY ÜaÝ î¤o¸Cõ • ¦¨ (x, y) ∈ Q ´3¦¤oÎ6¤u¹ (y, x) ∈ Q £<ºF§J_d¨(¹0¨6´0§6¶0¸u¹0Ã6( Ź0§6¶F»¤o§6¨6¤ Q §<º®¯§6¶Iò¤o¸u¹0´0¤ X §<º _d°l´ºF¤u¹0¸hÌ ¯ © ª ç ä ¯ äY« ÜaÝ î0¤o¸C¦¨ • æ ´3¦¤oÎ6¤u¹ (x, z) ∈ Q (x, y) ∈ Q, (y, z) ∈ Q §(¹0¨6¸o¨6®¯®¯¤u¹£6¨<Î6§6¶0¸u¹Ã6¯ Ź0§6¶F»¤o§6¨6¤ §<º®¯§6¶Iò¤o¸C¹0´0¤ §<º _d°l´ºF¤u¹0¸\Ì ª ¯ ä[®äYÝÎÝ.¬¯/é • è © äYåY ÜaÝ î<¤o¸C¦¨ (x, y) ∈ Q, (y, x) ∈ Q Q´3¦¤oÎ6¤u¹ x = y X ¤eg¦¤oÂ2¸o¨6´0§6¶0¸u¹0Ã6@ Ź0§6¶F»¤o§6¨6¤ Q §<ºÅ®¯§6¶Iòô¤o¸u¹0´0¤ X §<º _d°l´ºF¤C¹0¸\Ì ©¬/ë Û ¬ ì ®äY« ÜaÝ îF¤o¸C¦¨ • ê î<¨2¦¨,î(¥£(ïYbo¨6®¯¨¸C¦¶0´ºF®¯¨,î I ⊆ Q (x, x) ∈ Q, x ∈ X Ź0®¯¤u¹0¨6®lî,Î(¹0¶¢6¶3¦§6¶0¤¶F¹0§6¶F»¤o§6¨6¤¸M§6¤o¶[ÖIð¶d¥¨6®¯¶0¸u¹0Ã0ùìÌ6´3¦<Ì6¤u¹0¸hÌó£6¤ eg¦¤oÂ@¸o¨6´0§6°l® í ¸o®l ï2¢6£<º3ò"§6¤o§6¨6¤Ï(îCî¨ç§6¨6Â(ºFÂ6¨6¤k¥£(ïYbo¨6¤´c_IºF¨6®¯¶0¸o´FJÌ _d¨K®¯¤uò¥<ïEÇu¹0¨6®¯¨*¸o´0¶0·6¸u¹0´ºF®¯¨,0¶0[¶ ÖFö¤ bo¶0´0¶0£@Ì,î6§6¤¢6£6¨6¸uï@¹0¸u¹´Fï2ùŹ¨í ï2¢6£<º3ò"§6¤o§6¨6¤QÏ(ª[îC ñ ï ê "Àð i l k
l í ¥ÆºF§6§6¶0® ¢<º3£
X
É´no ò¨óBËÎ{BËaÒ*wxuzytÐ"yBÌaÏ&ÐÑô ¼z} -} ä&å&0Ø8@&& ä&å&0Ø8@&8¼z} ¤a1õz&,P P ¹ YdtP-»YPÛd tÚ ,cJ -fP PdfP
-P Q - xP/
c X - a [
¨fP
--÷ö ì « äY«/ª ¬ ¯ ® ¯ ä ø ¦<Ì.¶F¹0§6¶F»¤o§6¨(ÌÈÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸C¹¨ÈÎ<ºF¸u¹0¶*¨6¸o¢6¶3¦Ãco_ ï0¤u¹0¸\Ìȸo¢6¤o÷6¨<ºA¦Ã0§6¶0¤ó¶[0Ö ¶cd_ §<ºIÎ6¤o§6¨6¤ eQºFÂ(¹ (x, y) ∈ Q I_ ºF¢6¨6¸o°l´ºF¤u¹0¸\Ì´´0¨2¥¤ x ∼ y =«£6¨6®¯¤o£6¶0®ñ¶F¹0§6¶F»¤o§6¨(Ì ÇoÂ6´0¨6´ºI¦¤o§(¹0§6¶0¸u¹¨ ®¯¶Aòô¤u¹¸C¦<ï@ò"¨(¹0öF¹0§6¶F»¤o§6¨6¤M£<ºF´0¤o§6¸u¹0´º§<º X ù x ∼ y ⇐⇒ x = y.
í ´0¨2¥<ï£6¤eg¦¤oÂ@¸o¨6´0§6¶0¸u¹¨¦ùtÖ0¶[bo¶¶F¹0§6¶F»¤o§6¨(Ì ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ Q ¨6®¯¤o¤o® I ⊆ Qî<¢6¶0Çu¹0¶3õ ®ï¶F¹0§6¶F»¤o§6¨6¤£<ºF´0¤o§6¸u¹0´ºÌ6´3¦<Ì6¤u¹0¸\Ì §<ºF¨6®¯¤o§6ÃF»¨6®Ùí ¢6¶"´0Â2¦ùÅÎ6¤o§6¨6ùxî¨J_M´0¸o¤Cð ´0¶c_d®¯¶Iò"§6°µð ¶F¹0§6ø¶F»¤o§6¨6· ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ §<º X £(Yï bo¨6®ì¢6£6¨6®¯¤o£6¶0® ¶F¹0§6¶F»¤o§6¨(Ì ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨ ®¯¶Aòô¤u¹ ¸C¦<ï@ò"¨(¹0à ´0¸o¤o[¶ Öcú?¤o®8¦ùö¤o¤ ¶F¹0§6¶F»¤o§6¨6¤ T = X × X îÂ2¶3¹0¶0£6¶0¤Fî,¶0Î6¤o´0¨2¥§6¶6îÌ6´3¦<Ì6¤u¹0¸hÌó§<ºF¨ Ö0¶3¦ÃF»¨6Õ® í ¹3ºFÂ(ò¤¢6¶´0Â2¦ùyõ Î6¤o§6¨6xù îlJ¨ _Q¶F¹0§6¶F»¤o§6¨6·ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨§<º X ù Q ⊆ T z¼ } - } ûØ8-× ü ü0Á ¾ ýÕ&&×(Ø8@&6&& ü06 ø ¦<º Ì e¨6Â@¸o¨6£6¶0´ºF§6§6[¶ bo¶MÇC¦¤o®¯¤o§(¹Fº x ∈ X £<ºF¸o¸o®¯¶F¹0£6¨6®ç¸o¶0´0¶0Â(ï2¢6§6¶0¸u¹0à K(x) ´0¸o¤Cð"ÇC¦¤o®¯¤o§(¹0¶0´ J¨ _ X î<¤o®ïÇoÂ6´0¨6´ºA¦¤o§(¹0§6°µð ù X
X
X
K(x) = {y ∈ X : y ∼ x},
¨Ê§< º _d¶0´0¤o® Çu¹0¶ ®¯§6¶Iò¤o¸u¹0´0¶ ì Û ª ®® Ú Ýjö ì « äY«/ª Û ¬ ¯ Ú ® ¯ ä ÇC¦¤o®¯¤o§(¹3º Å«¶0§(Ì(¹0§6¶6îyÎ(¹0¶ ¥6¦<Ì ¢6£6¶0¨J_d´0¶3¦Ã0§6¶[bo¶ y ∈ K(x) ®¯¶Iò"§6¶6îy´ ¸o´0¶0ù ¶0Î6¤o£6¤C¥Ã6îÅ¢6¶0¸u¹0£6¶0¨(¹0ÃqxÂ2¦ÆºF¸o¸ K(y) î¢6£6¨6Î6¤o®
Ë
ST ST-UWVXYV[Z\]^VX
6î ¹0¶k¤o¸u¹0ÃkÂ2¦ÆºF¸o¸MÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ K(x) ®¯¶Iòô¤u¹ºÖ0° ¹0Ãk¢6¶0£6¶Iò¥¤o§¦ùtÖ0°l® ¨J_ ¸o´0¶0¨2ðÇC¦¤o®¯¤o§(¹0¶0´6 þ¯¸C¦¨òô¤ îA¹0¶6îF¶0Î6¤o´0¨2¥§6¶6î ¨ K(x) §6¤ ®¯¶[buï@¹¨6®¯¤u¹0ö[ÖFö¨2ð ¹0¶0Î6¤oÂÅí ¨6§<ºIÎ6¤BÖ0°q§<ºF£(ï@»ôºA¦ÆºFz¸oÃ6∈¹0K(x) £<ºF§J_d¨(¹¨6´0§6¶0¸u¹Ã¶F¹0§6¶FK(z) »¤o§6¨(ÌÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨&îC<«¶0Çu¹0¶0®ï Â(º3ò¥Æº3̹0¶0Î6Â2º x ∈ X ¢6£6¨6§<ºA¥6¦¤uò"¨(¹ô¶d¥§6¶0®ï¨¹0¶3¦Ã0Â@¶¶d¥§6¶0®ïÂ2¦Æº3¸o¸uï"ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨=î ¶F¹0Â(ïI¥Æº´0° ¹0¤oÂ2ºF¤u¹ ÿ @å&, £×Ez¼ } l¶,fº-xP/
c X ccPffP
-Pt³, P /&-fPf ∼ 0 X , µP [
õ -P-PWPP& c§6¦fY²P xP/
í} î X = K ,
K(y) = K(x)
X
0 × ½
xP/
c
--§ ∼
λ
c& f[JP µ P[
³, P /& -fPf P fP λ∈Λ
Kλ
µ² ¶0´0¶0Â(ï2¢6§6¶0¸u¹0à ´0¸o¤\ðñÂ2¦ÆºF¸o¸o¶0´*ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨Ê§<º _d°l´ºF¤u¹0¸hÌ ëxª ì¯ Ú ©/éÝÎ Ú ¬/® ¯ « Ú Ý X ¢6¶ ¶F¹0§6¶F»¤o§6¨6ù ÇoÂ6´F¨6´ºA¦¤o§(¹0§6¶0¸u¹¨ î&¨E¶[Ö0¶c_d§<ºIÎ<ºF¤u¹0¸hÌ °ÖFïI¥¤o® {K , λ ∈ Λ} ¨6¸o¢6¶3¦Ãc_d¶0´º3¹0ÃE¹3ºFÂ(ò¤ Ö0¶3¦¤o¤¦ÆºFÂ@¶0§6∼¨6Î6§6¶0¤ ¶[Ö0¶c_d§<ºIÎ6¤o§6¨6¤ XX/ ∼= Å 0 ¹ [ ¶ 0 Ö < £ 3 º ò¤o§6¨6¤ K : = X/ ∼ îI¸u¹3ºF´FÌ(ö¤o¤µ´¸o¶0¶F¹0´0¤u¹0¸u¹´0¨6¤µÂ(º3ò¥¶F®¯ïgÇC¦¤o®¯¤o§(¹0ï x ∈ X ¤bo¶ÅÂ2¦ÆºF¸o¸HÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ X →X î ÖFïI¥¤o® §< º _d°l´º3¹0Ã ìª Ú äYåc¬/®ì äYÝ K(x) ø ¦<Ìk¶F¹0§6¶F»¤o§6¨(Ì£<º3´0¤o§6¸u¹0´º I Â2¦ÆºF¸o¸o°qÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨Ì6´3¦<Ì6ù¹0¸\̶d¥§6¶F¹0¶0Î6¤oÎ6§6°l®¯¨ ù î?¹3ºFÂEÎ(¹0À¶ eQºFÂ(¹0¶0£2õ ®¯§6¶Aòô¤o¸u¹0´0¶ X ¢6¶¸uï@ö¤o¸u¹´Fï ¸o¶0´0¢<ºA¥ÆºF¤u¹:¸¸dºF®¯¨6® K(x) = {x}, x ∈ X ®¯§6¶Aòô¤o¸u¹0´0¶0® £< º _/Ö0¨6¤o§6¨6¤"§<ºÂ2¦ÆºF¸o¸o° ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ç¢6¶3¦<ï2Î<ºF¤u¹ ¢6£6¨EÇu¹0¶0®Ê§<ºF¨ Ö0¶3¦¤o¤ ¢6¶I¥£6[¶ Ö0§6°l®l ø X¦< Ì:§<ºF¨ Ö0¶3¦ÃF»¤ bo¶¶F¹0§6¶F»¤o§6¨(ÌÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨ T ¨6®¯¤o¤o®È¤C¥¨6§6¸u¹0´0¤o§6§6°y· Â2¦ÆºF¸o¸ K(x)í = X , x ∈ X îH¹3ºFÂ.Î(¹0¶ç¸o¶0¶F¹0´0¤u¹0¸u¹0´Fï(ùö¤d¤:£< º _/Ö0¨6¤o§6¨6¤:¶0Â2 º _d°l´ºF¤u¹0¸hÌq¸dºF®¯°l® bo£(Yï Ö0°l®l ¸o¤¸o¶I¥¤o£òº3¹0¤C¦Ã0§6°l¤g¶F¹0§6¶F»¤o§6¨(ÌkÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0× ¨ _IºF§6¨6®øºFù¹¢6£6¶0®¯¤uòï@¹0¶0Î6§6¶0¤ ¢6¶3¦¶Aòô¤o§6¨6¤®¯¤uò¥<ï I ¨ T Ź0®¯¤u¹0¨6¿® _u¥¤u¸oÃ6î8Î(¹0¶:¶F¹0§6¶F»¤o§6¨6¤ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨K§<º®¯§6¶Aòô¤o¸u¹0´0¤ ®¯¶Iò¤u ¹ _IºA¥ÆºF§6¶ ¢6¶0¸o£6¤C¥¸u¹0´0¶0®K¤ bo¶£< º _/Ö0¨6¤o§6¨(Ì,(«gï0¸u¹0à A , λ ∈ Λ | ¢6£6¶0J¨ _d´0¶3¦Ã0§6¶0¤£< º _/Ö0¨6¤o§6X¨6¤g®¯§6¶Iò¤o¸C¹0´º X §<º§6¤o¢(ï0¸u¹0°y¤M¢6¶d¥®¯§6¶Iò¤o¸u¹0´0 º í A 6= ∅, λ ∈ Λî ù λ
f
f
X
f
X
X
X
λ
λ
X =
X
Aλ .
<¦ Ì x, y ∈ X ¸oÂ2º3ò¤o®lî<Î(¹0¶ x ∼ y î¤o¸C¦¨§<ºF·2¥¤u¹0¸\Ì λ ∈ Λ ¹3ºFÂ2¶0¤QÎ(¹0¶ x, y ∈ A ¸o§6¶6îÎ(¹0¶ ¹FºFÂ@¶0¤y¶F¹0§6¶F»¤o§6¨6¤Å´¥¤o·6¸u¹0´0¨(¹0¤u¦Ã0§6¶0¸u¹¨"Ì6´3¦<Ì6¤u¹0¸h̶F¹0§6¶F»¤o§6¨6¤d®EÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨,î2¢6£6¨6Î6¤o® ¢6¶I¥®¯§6¶Iòô¤o¸u¹0´º A , λ ∈ Λ Ì6´3¦<Ì6ù¹0¸hÌ ¤bo¶ Â2¦ÆºF¸o¸dºF®¯¨ ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨¼í ¸o®l?ï2¢6£<ºAò"§6¤o§6¨6¤ Ï(ª[Í îC z¼ } - }¼ ¢ å&" £@å&¾¿6&@&8· À ýÕ&&×(Ø8@&6&& ü06 «gï¸u¹0à X | ®¯§6¶Aòô¤o¸u¹0´0¶÷6¤C¦°µð Î6¨6¸o¤C¦? º _d¶0´0¤o®ñ¥´ºÎ6¨6¸C¦Æº m, n ∈ X ÇoÂ6´0¨6´ºA¦¤o§(¹0§6°l®¯¨,î ¤o¸C¦¨ m − n ¥¤C¦¨(¹0¸\Ì §<º"@ æ [¶ bh¥Æº X £< º _/Ö0¨6´ºF¤u¹0¸ḩ<º¥´ºÂ2¦ÆºF¸o¸dºÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨ ù ®¯§6¶3õ ò¤o¸u¹0´ºÎ6¤C¹0§6°µðç¨E§6¤oÎ6¤u¹0§6° ðçÎ6¨6¸o¤C¦? µ¶3¦¤o¤"[¶ ÖFö¨6·E¢6£6¨6®¯¤o£ ù ¥6¦<ÌE¢6£6¶0J¨ _d´F¶3¦Ã0§6[¶ bo¶ ÷6¤C¦[¶ bo¶ §< º _d¶0´0¤o® m, n ∈ X ÇoÂ6´F¨6´ºA¦¤o§(¹0§6°y®¯¨,î=¤o¸C¦¨ m − n ¥¤C¦¨(¹0¸ḩ<º k ?ÁH¹0¶¶F¹0§6¶F»¤o§6¨6¤ k≥2 ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨K£< º _/Ö0¨6´ºF¤u¹ X §<º k Â2¦ÆºF¸o¸o¶0´ ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ K , i = 0, 1, . . . , k − 1î ¢6£6¨6Î6¤o® ´óÂ2¦ÆºF¸o¸ ¢6¶0¢<ºI¥ÆºFù¹Î6¨6¸C¦Æºó´0¨2¥Æº jk + i, j ∈ X î&¹0¶¤o¸u¹0Ã6î¯÷6¤C¦°l¤Î6¨6¸C¦Æº(6£6¨ ¥¤C¦¤o§6¨6¨§<º k ¥ÆºFùKö¨6¤Q¶0¸u¹3º3¹0¶0 i «gï0¸u¹0à Â@¶0§6¤oÎ6§6¶0¤®¯§6¶Aòô¤o¸u¹0´0¶6 ®¯§6¶Aòô¤o¸u¹0´0¶ó´0¸o¤Cðç¤ bo¶ó¢6¶I¥®¯§6¶Iòô¤o¸u¹0´6 Ź0§6¶F»¤o§6¨6X¤ÇoÂ6| ´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ §<º X ®¯¶Iò"X§6¶k=´0´02¤o¸u¹0¨| ¸C¦¤C¥6ï2ùöô¨6®È[¶ Ö0£< º _d¶0® ù ¥6¦<Ì x, y ∈ X λ∈Λ
ø
λ
λ
i
i
X
Í
Þtß/à Vá
\ àâJã
o¸ Î6¨(¹Fí ºF¤o® x ∼ y ´ ¹0¶0®¨ç¹0¶3¦Ã0Â@¶ó´ ¹0¶0®¸C¦<ï2Î<ºF¤Fî¯Â@¶[bz¥Æº:¨2ð箯¶Fö§6¶0¸u¹0¨*¸o¶0´0¢<ºA¥ÆºFù¹ ù |x| = Çu¹0¶0® ¸\¦<ï2Î<ºF¤kÂ(º3ò¥°l· Â2¦ÆºF¸o¸kÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨ C , k = 0, 1, . . . , |X| ¸o¶0¸u¹0¶0¨(¹:¨J_ |y| ®¯§6¶Aòô¤o¸u¹0´e¨6Â@¸o¨6£6¶0´ºF§6§6¶0·®¯¶Fö§6¶0¸u¹0¨ k k X
½ÌÎs vdÌÎó ä&å&0Ø8@&8¼z}¿¤a1õz&,P PBt,cJ -fP Ptc-fPYdtP-»Y-xfP
-Pt- xP/
c X - a [
fP
-P »Y[f-»YP P1J- ¹ rµ -c»Y[
≤ PÀ/
-
-PBP1J- PJP
¹ &0dx
¹ PP-" ¹
WP1J- º- a c§6 Û äé ¬ Y ÜaÝ
ºF£Ì2¥<ï¸l¶F¹0§6¶F»¤u§6¨6¤d® ≤ ¢6¶3¦¤d_ §6¶£<ºF¸o¸o®øº3¹0£6¨6´º3¹0ùFºFÂ(òô¤l¢6£6¶0¨Jd_ ´0¶d¥§6¶0¤l¶F¹0§6¶F»¤o§6¨6¤ < î É´nEm
¢6¶0£6¶Aò¥ÆºF¤o®¯¶0¤¢6¶¢6£<ºF´0¨2¦<ï «£6¨6®¯¤o£6¶0®¿¶F¹0§6¶F»¤o§6¨(Ì í ¦¨6x§6¤o<·6§6y¶[bo⇐⇒ ¶,î ¢6¶0x£Ì(≤¥Â2y,º.y®¯6≤¶Aòôx.¤u¹ñ¸C¦<ï@ò"¨(¹0Ãñ¶[Ö0°lÎ6§6°l·¢6¶0£@Ì2¥¶0 §<º®¯§6¶Aòô¤o¸u¹0´0¤´0¤uö¤o¸u¹0´0¤o§6§<°µð Î6¨6¸o¤C¦? ¤o§6¤o¤¹0£6¨6´0¨<ºA¦Ã0§6°l® ¢6£6¨6®¯¤o£6¶0® ¸C¦<ï@ò"¨(¹E¢6¶0Â@¶0®8õ ¢6¶0§6¤o§(¹0§6°l·*¢6¶0£@Ì2¥¶0Âç´ R î8Â@¶F¹0¶0£6°l·çÌ6´3¦<Ì6¤u¹0¸hÌçï@òô¤kÎ<ºF¸u¹0¨6Î6§6°l® ù ¥6¦<Ì x = (x , . . . , x ) î n
y = (y1 , . . . , yn ) ∈ R
1
n
n
º®¯§6¶Aòô¤o¸u¹0´0¤k´0¸o¤CðE¢6¶d¥x®¯≤§6¶Iòôy ¤o⇐⇒ ¸u¹0´ Xx =≤ 2y , ei =¨6Â21,¸o¨6.£6.¶0.´, ºFn.§6§6¶[bu¶:®¯§6¶Iò¤o¸u¹0´º i
i
´0´0¤o¸u¹0¨Î<ºF¸u¹0¨6Î6§6°l· ¢6¶0£Ì2¥¶0¢6¶´0Â2¦ùÅÎ6¤o§6¨6ù ù ¥6¦<Ì X
X
®¯¶Iò"§6¶
x, y ⊆ X
í ¹0¤o¶0£6¨6¨M£6¨6¸oÂ(ºy´º3ò"§(ï2ùK£6¶3¦Ã¨bo£<ºFù¹Å¢6¶0£@Ì2¥Â6¨Q§<ºy®¯§6¶Aòô¤o¸u¹0´0¤H£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨6·,F«gï¸u¹0à ¸o¶0´F¶0Â(ï2¢6§6¶0¸u¹0ô0¸o¤o´0¶c_d®¯¶Iò"§6°µðM´0¤uö¤o¸u¹0´0¤o§6§6° ðeï2§6Â6÷6¨6·£<ºF¸o¢6£6¤C¥¤C¦¤o§<¨(Ì, ê ºF¸o¸o®¯¶F¹0£6¨6® F| _u¥¤o¸oÃ"§6¤oÂ@¶F¹0¶0£6°l¤M¶F¹0§6¶F»¤o§6¨(Ì¢6¶0£Ì2¥Â2º§<º F ä&å&0Ø8@&8z¼ }¼ ^¯ Ú ª ® ¯ äYåc¬/®ì Ú ¬ Ú ÝÎäY äY© Ú «/ª äc¬ c1J- ¹ »Y
õζ - Â-Y,[c& x ≤ y ⇐⇒ x ⊆ y.
&-Y
G∈F
²-[f-»J ± 0x--cP¶,
!
F ∈ F F ≤1 G ¹
P
B Ö0c¶ _d§<ºIÎ6¨6® eï2§6Â6÷6¨6ùÒ£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨6̸C¦<ï2Î<ºF·6§6¶0·´0¤C¦¨6Î6¨6§6° ¶[bz¥Æº¢6£6¨ £<ºF¸o¢6£6¤C¥¤C¦¤o§<¨6¤ FX + a ¸u¹0¶dð2ºF¸u¹0¨6Î6¤d¸oÂ6¨¥¶0®¯¨6§6¨6£(ï0¤u¹k£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨6¤ XXù æ F (x) ≥ G(x), x ∈ R.
X
ø
¦<Ì
a∈R P{Wa = a} = 1
a>0
¶[Ö0¶c_d§<ºIÎ6¨6® W ¸C¦<ï2Î<ºF·6§(ï2ù ´0¤C¦¨6Î6¨6§(ï ¸ô´0°l£6¶Iò¥¤o§6§6°l®ñ£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨<¤o® Æ Å¹0§6¶F»¤o§6¨6¤M¸u¹0¶dð@ºF¸u¹¨6Î6¤o¸dÂ@¶[bo¶¥¶0®¯¨6§6¨6£6¶0´ºF§6¨(̧<º®¯§6¶Iòô¤o¸u¹0´0¤ FX ≤1 FX+a , a > 0.
a
´0¸o¤Cð ´0°l£6¶Iò¥¤o§6§6°µðÀeï2§6Â6÷6¨6·:£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨6Ì:¸o¶0¶F¹0´0¤u¹0¸u¹´Fï0¤o¹¶[Ö0°lÎ6§6¶0®ï¦¨6§6¤o·6§6¶0®ï¢6¶3õ £@Ì2¥Â(ï§<º Rù F ≤ F ⇐⇒ a ≤ b. ø ¦<Ìó´F¤uö¤o¸u¹´0¤o§6§6° ðÎ6¨6¸o¤C¦ a < b ¨ p ∈ [0, 1] [¶ Ö0c¶ _d§<ºIÎ6¨6® B(a, b, p) ¸C¦<ï2Î<ºF·6§(ï2ù ´0¤C¦¨2õ Î6¨6§(︣<ºF¸o¢6£6¤C¥¤C¦¤o§6¨6¤d"® µ¤o£6§(ïI¦6¦¨ ù {FWa , a ∈ R}
Wa
1
Wb
P{B(a, b, p) = a} = 1 − p, P{B(a, b, p) = b} = p,
Ï º
ST ST-UWVXYV[Z\]^VX
¸u¶0¶F¹0´0¤u¹0¸u¹´Fï2ùÅöMï(ùÙeï2§6Â6÷6¨6ù£<ºF¸o¢6£6¤C¥¤C¦¤o§6¨(Ì, æ ¶[bz¥Æº í o¸ ®l<£6¨6¸uï2§6¶0 î |
F (a, b, p)
F (a1 , b1 , p1 ) ≤1 F (a2 , b2 , p2 ), a1 ≤ a2 , b1 ≤ b2 , p1 ≤ p2 .
Ź0®¯¤u¹0¨6®lî0Î(¹0¶g¤o¸C¦¨ 6
b1 ≤ a 2
îA¹0¶g¶F¹0§6¶F»¤o§6¨6¤ F (a , b , p ) ≤ F (a , b , p ) ¢6£6¨M¦ùtÖ0°µð 1
1
1
1
2
2
2
p1 , p2
0 # F (a1 , b1 , p1 ) @
1 − p2
R @ I @
1 − p1
@
F (a2 , b2 , p2 ) a1
a2
b1
-
b2
$=]@_I %I'&&^AZ)(Fid_h^I]{@e\_hm0ZIel[2ZIp,]c]n@ZdY3ioc]2eyn2id_hbn@ez[2ezx2ehc]+*-,ehnc|uxx@] É´nGÉ sBu&vxqBÌ/t. ÐÑuzyt{Bu ä&å&0Ø8@&8¼z} Ð0W1PPt,cJ -fP PfP
-P - xP/
c
- a c X
¨fP
-PÓc&P»Y
-Y
º¸C¦¤C¥<ï2ùö¤o®ç£6¨6¸uï2§6Â@¤Å¨Jd_ ¶[0Ö £<º3òô¤o§6¶Q¶3¹0§6¶F»¤o§6¨6¤Å¢6£6¤C¥¢6¶0Î(¹0¤o§<¨(Ì"§<ºÎ6¤u¹0°l£6¤Cð6ÇC¦¤d®¯¤o§(¹§6¶0®
®¯§6¶Aòô¤o¸u¹0´0¤
{x, y, z, t}
x y z t
x ◦ ◦ ◦ ◦
y · ◦ ◦ ◦
z · ◦ ◦ ◦
t · · · ◦
í [î
«£6¤C¥¸u¹3ºF´3¦<Ì6¤u¹k¨6§(¹0¤o£6¤o¸Q¸o´FÌJ_duò¥<ï¢6¶0£Ì2¥Â@¶0® ¨¢6£6¤C¥¢6¶0Î(¹0¤o§<¨6¤o®lîP_IºA¥Æº3§6§6°l® §<ºô¶d¥6õ §6¶0® ¨¹0¶0®*òô¤®¯§6¶Iò¤o¸u¹0´0¤ X ä&å&0Ø8@&8z¼ }Ú l¶,ff- X cc- fPÚ3
2 Û -YÚ 1 »Y[f-»YP P1J- ≤ c& P»Y
-Y
6&P»Y
-P
- a [
®
ª ® «/ª ÜaݼlP1J-,0
x ≤ y =⇒ x y.
≤¹
P
樺F¦Â,ùtîFÖ0¶F°l¹0§6® ¶F»¤bo¤o¶§6¨6¢6¤y¶d¥¢6¢6£6¶0¤C£@¥Ì2¢6¥¶0Â@Î(¶0¹0®l¤d§6 ¨(̨J_Wí [î&¸o¶[bz¦ÆºF¸o¶0´ºF§6¶¸µ¦¨6§6¤o·6§6°l® ¢6¶0£Ì2¥Â@¶0® x ≤ y ≤ z ≤ t í ¸hÌ6Â@¶0¤"¶F¹0§6¶F»¤o§6¨6¤k¢6£6¤C¥¢6¶0Î(¹0¤o§<¨(Ìç¢6¶0£6¶Iò¥ÆºF¤u¹Aî?Â(ºF ¢6¶0Â2º d_ °l´ºF¤u¹:¸\¦¤C¥<ï2ùöº3̦¤o®¯®øº(î
¶F¹0§6¶F»¤o§6¨6¤MÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹¨§<ºô¹0¶0®*òô¤®¯§6¶Iò¤o¸u¹0´0¤F
4
Þtß/à Vá
\ àâJã
5 ,£x£× ¼z}l¶,f 6 [dW Y[ &0
fP
-Plc&P»Y
-Y¨-
fP
-P ∼ ¹ cc í± î x ∼ y ⇐⇒ x y, y x, PJP
fP
-P Õ³, P /&-fPf¨- X ¹
ºf ¹ YdtP-»Y-" ¹ B,cJ fP -"Óº&1õz&,P -"jfP
-P 8 Õ,×(Þ@×66Ø8Ü&ü06 ¦¤o®¯®¯°È¢6£6¤C¥¶0¸u¹FºF´3¦<Ì6¤u¹0¸\ÌÎ6¨(¹3º3¹0¤C¦Æù í ï2¢6£<º3ò"§6¤o§6¨6¤MÏ( ± Cî í ¢6£6¨6®¯¤o£6¤ºí [îH¨6®¯¤o¤o® í ¨,î6Â@¶0§6¤oÎ6§6¶òô¤Fî6Â2º3ò¥°l·¨J_ ÇC¦¤o®¯¤o§(¹0¶0´ x, y, z, t ÇoÂ6´0¨6´ºAõ ¦¤o§(¹0¤o§ ¸dºF®¯¶0®ï¸o¤0Ö ¤Cî y ∼ z þ¯¸u¹0¤o¸u¹´0¤o§6§6¶g¹3ºFÂ(òô¤µ£<ºF¸o¸o®øº3¹0£6¨6´º3¹0Ãg¢6£6¶0¨Jd_ ´0¶I¥§6¶F¤µ¶F¹0§6¶F»¤o§6¨6¤ ¸u¹0£6¶[ob ¶[ob ¶¢6£6¤C¥¢6¶0Î(¹0¤o§6¨(Ì §≺ <º X ù í ¢6£6¨6®¯¤o£6¤×í [µî ¨6®¯¤o¤o® x ≺ y ∼ xz ≺≺ ty ⇐⇒ x y, y 6 x. «gï0¸u¹0à Xf | Qe ºFÂ(¹0¶0£2õ ¢6£6¶0¸u¹0£<ºF§6¸u¹´0í ¶ X ¢6¶E¶F¹0§6¶F»¤o§6¨6ù ÇoÂ6´0¨6´ºA¦¤o§(¹0§6¶0¸u¹0¨Õí ±[uî îHº K | Â(ºF§6¶0§6¨6Î6¤o¸oÂ@¶0¤Q¶F¹0¶[Ö0£<º3òô¤o§6¨6¤ ´ ¢6£6¨6®¯¤o£6¤×í [6®¯¤o¤o® X
f X
¤u¹0£(ïd¥§6¶À_IºF®¯¤u¹0¨(¹0Ã6îÎ(¹0¶ ¶3¹0§6¶F»¤o§6¨6¤¢6£6¤C¥¢6¶0Î(¹0¤o§6¨(Ì
6¢ ¶0£@Ì2¥¶0¢6¶¢6£<ºF´0¨2¦<ï ¢6£6¨6Î6¤o® í ¢6£6¨6®¯¤o£6פ í [ î
7 0
X
¢6¶0£6¶Iò¥ÆºF¤u¹ §<º X ¦¨6§6¤o·6§6°l·
K(x) = {x}, K(y) = K(z) = {y, z}, K(t) = {t}.
f
K(x) ≤ K(y) ⇐⇒ x y, K(x) < K(y) ⇐⇒ x ≺ y. K(x) < K(y) = K(z) < K(t).
9
: Q i Q ð
<;>="@?@AQCB ð ä&å&0Ø8@&8:Ð.} l¶,f-
xP/
c X cc¨»Y[f-»Y-/¨P1J- ≤ 7D -/ - a [
ÀP¶ µa c§6¦fY P µ:X →R
¹
¶ - Â--
è§<ºA§<¦ºA° ¦¶[bo´0¨6¸uÎ6¤§6¶0¶§6¨¶0¢6§<£6º ¤C_d¥°l¤C´¦<ºFÌ6ùù¹0¹0¸h¸hÌÌó¹F§6ºF¤oÂ(´0òô¶c_d¤£<®¯ºF¸u¶0¹3§6ºF¶Fù¹0ö¶0§6¨<§6·,°lî,®¯´0¨ ¶c_dí £<¢6º3¶¸u¹F¢6ºF¶0ù£öÌ2¥¨6Â(·:ï ¨ïYîCÖ0 °l´ºFùö¨6·¨eï2§6Â6÷6¨6¶3õ ≤ ä&å&0Ø8@&8:Ð.} l¶,f -ºxP/
c X ccP fP
-Pfc&P»Y
-Y D ¶ - Â-P-/ µ : X → R - a [
ÀP¶ µa c§6¦fY ¹ P x ≤ y =⇒ µ(x) ≤ µ(y).
è§<ºA§<¦ºA° ¦¶[bo´0¨6¸uÎ6¤§6¶0¶§6¨¶0¢6§<£6º ¤C_d¥°l¤C´¦<ºFÌ6ùù¹0¹0¸h¸hÌÌó¹F§6ºF¤oÂ(´0òô¶c_d¤£<®¯ºF¸u¶0¹3§6ºF¶Fù¹0ö¶0§6¨<§6·,°lî,®¯´0¨ ¶c_dí £<¢6º3¶¸u¹F¢6ºF£6ù¤C¥ö¢6¨6¶0·:Î(¹0¨¤o§6ïY¨6Ö0ù°l´ºFùîC ö¨6·¨eï2§6Â6÷6¨6¶3õ ²µ´F ÌJd_ uò¥<ï´0´0¤C¥¤o§6§6°l®¯¨¢6¶0§(Ì(¹0¨(Ì6®¯¨®¯¶0§6¶3¹0¶0§6§6¶0¸u¹0¨ï0¸u¹3ºF§<ºF´3¦¨6´ºF¤u¹ ¢ å&08Ø82æ@&& Ð.} l¶,f -ºxP/
c X cc »Y[f-»Y-/ ÅP1J- ≤ Pc 0 c- E ¨õζ - Â-P-/ c-PW-YjfP
-PW &P»Y
-Y PJ-§6¦fEP¨dP
--" × c,x P-µ ×xa7 & fPc& P»Y
--§ ¹ PµJP:
X → ±dR¹ P
--"j
ÓWxa&WPfP1J- ¶ ≤ 8 Õ,×(Þ@×66Ø8Ü&0ü 6 ¢6£6¤C¥¶0¸u¹3ºF´3¦<Ì6¤u¹¸hÌÎ6¨(¹Fº3¹0¤C¦ù í ï2¢6£<º3ò"§6¤o§6¨6¤QÏ( Ë,Cî x y =⇒ µ(x) ≤ µ(y).
F
ST ST-UWVXYV[Z\]^VX
H ="@?@AQCBJIQð
LKQNMl7O
lQP
G
í ´0¤C¥¤o®¨_u¥¤o¸oç6¤oÂ@¶F¹0¶0£6°l¤¶[Ö0¶c_o§<ºAÎ6¤o§6¨(Ì,îFÂ2¶F¹0¶0£6°l¤l¢6¶c_d´0¶3¦<Ì(¹´Â6£<º3¹0Â@¶0®Ãe¶0£6®øºA¦Ã0§6¶0®:´0¨2¥¤ _IºF¢6¨6¸dº3¹0ÃM¸o´0¶0·6¸u¹0´ºQ¶F¹0§6¶F»¤o§6¨6·,î2£<ºF¸o¸o®¯¶F¹0£6¤o§6§6°y¤Å´¢<ºF£<ºcbo£<ºce¤g2 ±(3ºFÂ(ò¤y¢6£6¨bo¶d¥<Ì(¹0¸\Ì"´ ¥Æí ºA¦Ã0§6¤o·(»¤o®Ê¢6£6¨ ºF§<ºA¦¨J_d¤¶F¹0§6¶F»¤o§6¨6·E§<º®¯§6¶Iò¤o¸u¹0´ºAðó´0¤o£6¶IÌ(¹0§6¶0¸u¹0§6°µð £<ºF¸o¢6£6¤C¥¤C¦¤o§<¨6·, ¢<º3£
º _d¶0´0¤o® ®¯§6¶Iòô¤o¸u¹0´0¶ I -[ -/P § X × X ø ºA¦¤o¤Fî(¶0¢6£6¤C¥¤C¦¨6® ¥´Fïð¶F¹0§6¶3»¤o§6¨6· §<º î2Â(ºF®¯§6¶Iòô¤Cõ ¸u¹0´0¶ô´0¸o¤Cð"¹0¶0Î6¤o (x, y) ∈,0XxP× X-Â-¹3-ºF§ Â6¨2Sðî6=Î(¹0Q¶ô◦§
X
Q x y x ◦ ◦ y ◦ ◦ z · ·
R x y x · · y · ◦ z · ◦
z · · ·
S z · ◦ ◦
x x · y · z ·
y ◦ ◦ ·
z ◦ ◦ ·
Ź0®¯¤u¹0¨6®lîÆÎ(¹0¶Â@¶0®¯¢6¶c_d¨6÷6¨6ù®¯¶Iò"§6¶f_IºF¢6¨6¸dº3¹0ôô¦¶[bo¨6Î6¤u¸oÂ2¶0® ´0¨2¥¤Fî<Â2ºF ((x, y) ∈ S) =
[
([(x, z) ∈ Q] ∩ [(z, y) ∈ R]) ,
í Ë,î
bh¥¤ ∪ ¶[Ö0¶c_d§<ºIÎ<ºF¤u¹¦¶cbo¨6Î6¤o¸oÂ2¶0¤ ,P îº ∩ | ¦¶[bo¨6Î6¤o¸oÂ@¶0¤ ¢6¶e¶0£6®¯¤ºí Ë î §<ºF¢6¶0®¯¨6§<ºF¤u¹"¢6£6¶3õ ¨Jd_ ´0¤C¥¤o§6¨6¤M®øº3¹0£6¨6÷, ø ¦<Ìñ¶F¹0§6¶F»¤o§6¨(Ì Q §<º X ¶0¢6£6¤C¥¤C¦¨6® t, -- Qc ¸\¦¤C¥<ï2ùöô¨6® ¶[Ö0£<º _d¶0® Qc = 3 §6¶Iò¤o¸u¹0´0¶ Qc ®¯¶Iò"§6¶Q¢6£6¤C¥¸u¹FºF´3¦<Ì(¹0Ãô¸o¤0Ö ¤Fî2Â2ºFÂk¶F¹0£<º3òô¤o§6¨6¤ {(x, y) ∈ X × X | (y, x) ∈ Q} ¶F¹0§6¶0¸o¨(¹0¤u¦Ã0§6¶ô¥¨<ºcob ¶0§<ºA¦¨ I <«g£6¨6®ø¤u£¶F¹0£<º3òô¤o§6¨(Ì¢6£6¨6´0¤C¥¤o§§<º¸C¦¤C¥<ï2ùöô¤o® £6¨6¸uï2§6Â@¤F Q z∈X
X
Q x y x ◦ · y ◦ · z · ·
z · ◦ ◦
Qc x y x ◦ ◦ y · · z · ◦
z · · ◦
`¸o¢6¶3¦Ãc_oïFÌ´0´0¤\¥¤o§6§6°l¤M¢6¶0§(Ì(¹0¨(̨¶[Ö0¶c_d§<ºIÎ6¤o§6¨(Ì,î<¸o´0¶0·6¸u¹0´º¶F¹0§6¶F»¤o§6¨6· ¨J_Q¢<ºF£<ºcbo£<ºceQº 2 ±ô®ø¶Iò"§6¶ô¶0¢6¨6¸dº3¹0Ã"¸C¦¤C¥<ï2ùö¨6® ¶[0Ö £<º d_ ¶0® ù ¶F¹0§6¶F»¤o§6¨6¤ Q §<º X Ì6´3¦<Ì6¤u¹0¸hÌ &1õz&,P -" î<¤o¸C¦¨ I ⊆ Q •
î<¤o¸C¦¨
•
YdtP-»Y-"
•
c-fPYdtP-»Y-"
•
t,cJ -fP -"
•
PP-"
î6¤o¸C¦¨
î(¤o¸C¦¨
X
Q = Qc
î6¤o¸C¦¨
Q ∩ Q c ⊆ IX
Q ◦ Q ⊆ Q
Q ∪ Qc = X × X
R
Þtß/à Vá
\ àâJã
T>U
lQP
S
ãä&å&×(æç&@&&âz} ° cf ¹ »Y
Ãc-, P-P±fP
-P×PJP
r&1õ^&,P -" ãä&å&×(æç&@&&zâ } Wt&-f×cPYdP¼fP
-E !
PJ-§6¦fY²PPP-"x ¹ PBP PJ-§6¦fY²P-×YdtP-»Y-"x ¹ -×t,cJ -fP -"x ¹ ±c-fPYdtP-»Y-"xJ¸ PJ-§6¦fY²P YdtP-»Y-"x ¹ PBP PJ-§6¦fY²P-×P1P-"x ¹ -×t,cJ -fP • -"x ¹ ±c-fPYdtP-»Y-"x ¹ -&1õz&,P -"xJ¸ , P§´tPcP¶ §6¦fP
¶
Pºc² P¶,0Y² /JcV Åd ½¶Åc P
x fP •
-E ¹ -Pc--"²¨ Y[E,[õalÆ X W, ãä&å&×(æç&@&&âz} ¼ ° cff& x¶ W, Æ ãä&å&×(æç&@&&âz} Ð ° cfrc&&/-P- Y Èǹ c& P-
P Pö,
»,-P & [ Z
0ÓW x &\Z º õa1 x¶Pc& , ãä&å&×(æç&@&&âz} ÚW cf ¹ »Y
×fP
-P ¹ cc -P
×,-ÂPt, P W,ÈÇ Æ ¹ P JP
r/
-
-PÓ³, P /&-fPf ¹ P1&/½c§6¦fP fxP/
A , λ ∈ Λ λ PJ-§6
À³&d-fx¨
c
¶ §6¦
õc
1PGxP/
ãä&å&×(æç&@&&âz} â[]^c-Pcf c& &-PÀ-,cÂ-- ,0xP -Â--·J^ XY_¨Å - S = PPPc ¶ ÀP,c
1& f
P ,
1&f-,GxP/
c--aºP,cÂ-- ¹ f (Q, R) ¹ P,cÂ-- /J c--aBlPÂ-PõÎ-»J, tP¶ f¶& P
c&Jcc -Y -ccP- d ¹ P,¹ cÂ--¨»J-Y Q = {y ∈ X | (x, y) ∈ Q} Q = {x ∈ X | (x, y) ∈ Q} ¹ •
x·
·y