Basic Numeracy Set Theory Venn Diagrams Functions Relations

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Tfpg` Pot Thof Thofry ry,, Qoee Qoee Dglmr Dglmrlks lks Aue`tgfes & Rocltgfes @cgì


Pot Thofry, Qoee Dglmrlks Aue`tgfes & Rocltgfes Pots L sot gs l `fcco`tgfe fa wocc doageod fbjo`ts. Tho fbjo`ts fa tho sots lro `lccod ocokoets. (g) Pots lro usulccy doeftod by `lpgtlc cottors L cottors L,, B, @ ,..., X ,..., X , W , Z . (gg) Tho ocokoets fa tho sots lro doeftod by sklcc cottors cgio l, b, ` ,..., x ,..., x , y , z ot`. z ot`. Roprosoetltgfe fa Pots Pots lro usulccy dos`rgbod getf twf wlys. (g) Tlbuclr lbuclr afrk afrk fr rfstor rfstor afrk, afrk, ge thgs thgs afrk, afrk, lcc tho ocokoets ocokoets fa tho tho sot lro lro soplrltod by `fkkls led oe`cfsod botwooe tho brlìot { }. Afr oxlkpco (l) Tho sot fa vfwocs v fwocs fa Oemcgsh Lcphlbot ls L ; {l {l, o, g , f, u) (b) Tho sot fa eukbors fe l `cfì alò gs wrgttoe ls B ; (<, 2, 0, 4, ?, 1, =, 8, 7, <>, <<, <2}


(gg) Pot bugcdor arfk: ]o doageo l sot by stltgem prfportgos whg`h gts ocokoets kust sltgsay. Afr oxlkpco tho sot fa lcc ovoe getomors. Thoe, wo uso tho cottors usulccy x, led wo wrgto L ; {x | x gs gs le ovoe getomor} Thgs gs tf bo rold ls L gs l sot fa eukbors x su`h thlt x gs le ovoe getomor. Tho vortg`lc cgeo cgeo – | ‛ tf bo rold ls –su`h thlt‛ sfko tgkos

wo uso x ge pclò fa vortg`lc cgeo. L ; {x : x gs le ovoe getomor} getomor} om, @ ; {<,w, w2} ; {x | x0 ‐ < ; >} Ga le fbjo`t x gs le ocokoet fa l sot L, wo wrgto x  L whg`h gs rold rold

ls –x bocfem tf L‛ led ga le fbjo`t x gs eft l kokbor fa L wo wrgto x  L led rold ls –x dfos eft bocfem tf L‛.

Pfko Gkpfrtlet Torks

(g)

Okpty fr Eucc sot Tho sot whg`h `fetlges ef ocokoets gs `lccod tho okpty sot fr tho eucc sot. Tho okpty sot gs wrgttoe ls a. Thus, a ; { } ls thoro gs ef ocokoet ge tho okpty sot. Afr oxlkpco9 tho sot fa fdd eukbors dgvgsgbco by 2 gs tho eucc sot.


(gg) Pgemcotfe sot L sot `fetlgegem fecy feo ocokoet gs `lccod l sgemcotfe afr oxlkpco, {<}, {4} lro sgemcotfe sots. (ggg) Oqulcgty fa sots. Tho Tho sots L led B lro oqulc ga thoy hlvo slko kokbors thlt gs ga ovory ocokoets fa L gs le ocokoet fa B led ovory ocokoet fa B gs le ocokoet fa L, thoe L ; B om, ga L ; { c, 0,?,=} led B ; {=, 0, <, ?}, thoe L ; B Ga tho twf sots lro eft oqulc wo wrgto L  B Gkpfrtlet Afrkuclo

<. L sot dfos eft `hlemo `hlemo ga gts gts ocokoets, ocokoets, lro lro ropoltod. ropoltod. 2. L sot dfos dfos eft `hlemo ovoe ga tho frdor fa gts ocokoets ocokoets gs dgaaoroet. dgaaoroet. (gv) Agegto led Geagegto sot. Tho sot whg`h `fetlges l doagegto eukbor fa ocokoets gs `lccod l agegto sot. Tho sot whg`h `fetlges le geagegto eukbor fa ocokoets gs `lccod le geagegto sot. om, (G) Tho sot fa dlys ge l wooi. om, (GG) Tho sot fa elturlc eukbors. (v) Dgsjfget sot. Twf sots L led B lro slgd tf bo dgsjfget, ga thoy df eft hlvo ley ocokoet ge `fkkfe. om, L ; { <, 2, 0}, B ; { 4, ?, 1} lro dgsjfget sots.


(vg) Pubsot. Ga ovory ocokoet ge sot L gs lcsf le ocokoet fa lefthor sot B. Thoe L gs `lccod l subsot fa B. Lcsf B gs slgd tf bo supor sot fa L. Pykbfcg`lccy, Pykbfcg`lccy, wo wrgto L  B (go, L `fetlgeod ge B) B  L (go, B `fetlges L) Kfro spo`gag`lccy L  B ga x L  x  B om, (G) Cot L ; { 2, 4, =}, B ; { <, 2, 0, 4, =} Thoe, L Å B sgeò ovory ocokoet fa L gs ge B. om, (GG) L ; {x | x l rolc eukbor} led B ; {x | x gs le getomor} Thoe, L  B <. Ga thoro gs lt colst feo ocokoet fa L whg`h gs eft ge B, thoe L gs eft l subsot fa B wrgttoe ls L  B. 2. Ovory sot gs l subsot fa gtsoca gtsoca go, L  L. 0. Ga L  B led B `  L, thoe L ; B. (vgg) Tho Eucc sot a gs l subsot fa ovory sot L. (vggg) ^rfpor Pubsot: L gs l prfpor subsot fa B. ga L  B led L  B led gs wrgttoe ls L  B go, ga B `fetlges lt colst feo ocokoet kfro thle L, thoe L gs l prfpor subsot fa B


^fwor sot: Pot fa lcc tho subsots fa l sot gs `lccod (gx) ` lccod tho pfwor sot om, L ; {l, b, `} subsots fa L lro a, {l}, {b}, {`}, {l, b}, {b, `}, {`,l}, {l, b, `} L) ; Ya, {l}, {b}, {`}, {l, b}, {b, `}, {`, l}, {l, b, `}[ Hoeò, ^ ( L Ga e gs tho eukbor fa ocokoets fa l sot L, thoe tho eukbor fa subsot fa L go, tho eukbor fa ocokoets fa ^ (L (L)) ; 2e. _egvorslc sot: Ga lcc tho sots uedor `fesgdorltgfe lro tho subsots (x) fa l agxod sot _ , thoe _ gs `lccod tho _egvorslc sot. _egfe fa sots

_egfe fa twf sots L led B gs tho sot fa lcc ocokoets whg`h bocfems tf L fr B (fr tf bfth) led gs wrgttoe ls L  B (go, L uegfe B) Tho slko gs doageod ge sot bugcdor afrk ls L  B ; {x |x  L fr x  B} Ga L ; {<, 0, ?, =, 7} led B ; {2, 4, ?, 1, 7} Thoe, L  B ; { <, 2, 0, 4, ?, 1, =, 7}


<. Arfk tho doagegtgfe doagegtgfe fa _egfe fa sots L u B ; B u L (@fkkutltgvo (@fkkutltgvo Clw) Ga L gs ley sot, thoe L  L ; L led L  a ; L L  B) led B  ( L L  B) 2. Ga L led B lro ley twf sots, thoe L  ( L Ga x Å L  B, thoe x  L fr x  B led ga x  L  B, thoe x  L led x  B. B  @ ) ; ( L L  B)  @ 0. Ga L, B, @ lro throo sots, thoe L  ( B Getorso`tgfe fa Pots

Ga L led B lro ley twf sots, thoe getorso`tgfe fa L led B gs tho sot fa lcc ocokoets whg`h lro ge L led lcsf ge B. Gt gs wrgttoe ls L Í B led gs rold ls – L getorso`tgfe B‗ B‗ Ga L ; {2, 4, 1, 8} led B ; {4, ?, 1, 7} Thoe L  B ; {4, 1} <. Arfk tho doagegtgfe fa tho getorso`tgfe, gt afccfws afccfws L  B ; B  L (@fkkutltgvo (@fkkutltgvo Clw) 2. Ga L gs ley sot, thoe L  L ; L led L  a) ; a 0. Afr ley twf sots L led B. L  B ; L led L  B  B


L led B hlvo ef ocokoets ge `fkkfe go, L led B lro dgsjfget, 4 Ga L thoe L  B ; a Ga x  L  B ; x  L led x  B om, (G) Ga L ; { 2, 0, 1, 8, 7} led B ; (<, 0, ?, 1, =, 7}, thoe L  B ; {0, 1, 7} om,(GG) Ga L ; {x <|3 <|3 x 3 4 } led B ; {x |2 |2 3 x 3 ?}, thoe L  B ; {x |2 |2 3 x 3 4} Ga L, B, @ lro throo sots, thoe L  B)  @ ; L  ( B B  @ ) Lssf`gltgvo Clw (g) ( L B  @ ) ; ( L L  B)  ( L L  @ ) Dgstrgbutgvo Clw (gg) L  ( B Dgaaoroeò fa Pots Tho dgaaoroeò fa twf sots L led B gs sot fa ocokoets whg`h bocfems L ‐ B B tf L but df eft bocfem tf B. Thgs gs wrgttoe ls L ‐ L ‐ L ‐ B B ; {x| x  L led x  B} L ‐ B B subsot fa L go, L ‐ B B  L <. Pot L ‐ L ‐ B B) led B lro dgsjfget go, ( L ‐ L ‐ B B)  B ; a 2. Pot ( L ‐ L ‐ B B ; ( L L  B) ‐ ( L L  B) 0. L ‐


Pykkotrg` Dgaaoroeò fa Pots Tho sykkotrg` dgaaoroeò fa twf sots L led B gs (L ‐ B)  (B ‐ L) L) led gs

wrgttoe ls L D B Thus, L D B ; (L ‐ B)  (B ‐ L) L) Ge tho sot bugcdor afrk L D B ; {x | x  L fr x  B, but x  L  B} Dokfrmle Clws Ga L, B, @ lro throo sots, thoe (g) L ‐ (B  @) ; (L ‐ B)  (L ‐ @) (gg) L ‐ (B  @) ; (L ‐ B)  (L ‐ @)

@fkpcokoet fa l Pot Cot L bo l subsot fa uegvorslc sot _ , thoe tho `fkpcokoet fa L gs doeftod by L@ gs doageod by L@ ; {x  _ , x  L} S x  L@  x  L om, (G) Ga _ ; {<, 2, 0, 4, ?, 1} led L ; { <, 0, ?}, thoe L@ ; {2, 4, 1} om, (GG) _ bo tho sot fa lcc cottors ge Oemcgsh lcphlbot led L gs l sot fa lcc vfwocs, thoe L@ gs tho sot fa lcc `fesfelets. L  B)@ ; L@  B@ L  B)@ ; L@  B@ <. ( L 2. ( L


Qoee Dglmrlks

L sgkpco wly fa oxpclgegem oxpclgegem tho rocltgfe rocltgfe botwooe sots gs by l dglmrlk dglmrlk whg`h gs `lccod Qoee dglmrlk. Ge thgs l sot gs moeorlccy roprosoetod by l `gr`co led gts ocokoets by pfgets ge tho `gr`co. @lso G: L G: L  _ led B  _ led L  B  a

Horo L Horo L led B lro roprosoetod by l `gr`co. L ‐ B gs tho cgeod romgfe B ‐ L gs dfttod romgfe led L led L  B gs pcleo romgfe.


@lso GG: L ; {l, `, o}, B ; {b, d}

L  B ; a led L led L ‐ B ; L led B ‐ L ; B


@lso GGG: ]hoe L  B  _ Ge ldjfgegem agmuro, ge Qoee dglmrlk

L  B ; B, L  B ; L led L led L ‐ B ; a Pfko rosucts arfk tho t ho Qoee Qoee dglmrlk (g) e(L  B) ; e (l) + e(B) ‐ e(L  B) (gg) e(L  B) ; e (l) + e(B), whoe L  B ; a (ggg) e(L ‐ B) + e (L  B) ; e(L) (gv) e(B ‐ L) + e (L  B) ; e(L) (v) e(L ‐ B) + e (L  B) + e (B ‐ L) ; e (L  B)


Oxlkpco <: Ga ge l al`tfry fa 0> wfriors, <> tlio tol but eft `faaoo led <4 tlio tol. Thoe hfw kley tlio fecy `faaoo 6 Pfcutgfe. Tftlc eukbor fa wfriors ; e (T  @) ; 0> Eukbor fa wfriors whf tlio tol e (T) ; <4

]hf tlio tol but eft `faaoo ; e (T ‐ @) ; <> ]hf drgeis bfth `faaoo led tol ; e (T) ‐ e (T ‐ @) ; <4 ‐ <> ; 4 ]hf tlios fecy `faaoo ; e (@ ‐ T) ; x Arfk tho agmuro ; x + 4 + <> ; 0> ; x ; 0> ‐ <4 ; <1 Tho wfrior whf drgeis fecy `faaoo ; <1


Oxlkpco 2: Le ocfùtgfe `fkpotgtgfe `fkpotgtgfe wls hocd ge Oemcgsh Oemcgsh led Hgedg. Fut

fa 8> studoets, 4? tffi plrt ge Oemcgsh, 0? ge Hgedg,
(b) ]hf tffi plrt ge Hgedg but eft ge Oemcgsh. (`) ]hf tffi plrt ge ogthor Oemcgsh fr Hgedg. (d) ]hf tffi plrt ge eogthor.


Puppfso O gs tho sot fa studoets whf tffi plrt ge Oemcgsh, Hgs tho sot fa studoets whf tffi plrt ge Hgedg, thoe O e H mgvos tho sot fa studoets whf tffi plrt ge bfth Oemcgsh led Hgedg. (l) (l) Tho euk eukbor fa stud studooets ets whf whf tff tffi plrt lrt ge Oemcg mcgsh but eft eft ge Hgedg gedg ; e(O) ; e(O  H ) ; 4? ‐ (b) (b) Tho euk eukbor fa stud studooets ets whf whf tff tffi plrt lrt ge Hged Hgedgg but but eft eft ge Oemcg emcgsh sh ‐ e(O  H ) ; 0? ‐ ; e(H ) ‐ e (`)Tho (`) Tho eukbor fa studoets whf tffi plrt ogthor ge Oemcgsh fr ge Hgedg gs e(O  H ) ; e(O ) + e(H ) ‐ e ‐ e(O  H ) ; 4? + 0? ‐ ‐ 1? ;
Pfcutgfe.

Frdorod ^lgr

Ga l, b bo ley twf fbjo`ts, thoe tho plgr (l, b) gs `lccod tho frdorod plgr. Tho fbjo`t l gs `lccod tho agrst `ffrdgelto (fr agrst eukbor) led b gs `lccod tho so`fed `ffrdgelto (fr so`fed eukbor) fa tho frdorod plgr (l, b). <. Tho frdoro frdorodd plgr plgr (l, b)  (b, l) Twf frdorod plgrs (l, b) led (`, d) lro slgd tf bo oqulc, ga led fecy ga l ; ` led b ; d.


@lrtosgle ^rfdu`t fa Pots L led B gs tho sot fa Ga L led B bo ley twf sots, thoe `lrtosgle prfdu`t fa L lcc frdorod plgr (l, b), whoro l  L led b  B @lrtosgle prfdu`t fa L led B gs wrgttoe ls L Ý B (go, L `rfss B) go L Ý B ; {(l, b) | l Å L led b Å B} om, Ga L ; {l, b, `} led B ; {<, 2}, thoe L Ý B ; {(l, <), (l, 2),(b, <),(b, 2),(`, <),(`, 2)} B Ý L ; {(<, l),(<, b),(<, `),(2, l),(2, b),(2, `)} Thus, L Ý B · B Ý L L Ý L ; {(l, l), (l, b), (l, `),(b, l),(b, b),(b, `) (`, l),(`, b),(`, `)} B Ý B ; {(<, <), (<, 2),(2, <),(2, 2)) B  @ ) ; ( L L Ý B)  ( L L Ý @ ) <. L Ý ( B B  @ ) ; ( L L Ý B)  ( L L Ý @ ) 2. L Ý( B Rocltgfes L rocltgfe gs l sot fa frdorod frdorod plgrs. Ga (x , y) gs l kokbor fa l rocltgfe R, wo wrgto gt ls x R y (go rocltgfe R tf y). om, Ga R gs tho frdorod plgrs fa pfsgtgvo getomors whoro R ; {(x , y)| x 2 ; y} Tho rocltgfe gs y gs l squlro fa x led tho sot gs {(<, <), (2, 4),(0, 7),(4,

<1),...}


Typos fa Rocltgfes (g) Roacoxgvo: L rocltgfe R fe l sot L gs slgd tf bo L

roacoxgvo afr ovory x 

(x , x )  R Rocltgfe: L rocltgfe R fe l sot L gs slgd tf bo sykkotrg` (gg) Pykkotrg` Rocltgfe: L ga x R y  y R x y,x )  R (x, y)  R ; ( y,x

om, Cot L ; {<, 2, 0} led R ; {(<, <), (2, 2),(<, 0),(0,<)} @colrcy, R gs l sykkotrg` rocltgfe. (ggg) Trlesgtgvo Rocltgfe: L rocltgfe R ge l sot L gs `lccod trlesgtgvo ga x R y led y R z  x R z om, Cot R bo l rocltgfe ge tho rolc eukbor doageod by –x coss thle y‛ thoe x 3 y led y

3z;x3z

(gv) Oqugvlcoeò Rocltgfe L rocltgfe whg`h gs roacoxgvo, roacoxgvo, sykkotrg` sykkotrg` led trlesgtgvo gs l oqugvlcoeò rocltgfe.


Aue`tgfes

Ga ol`h ocokoet fa l sot L gs lssf`gltod wgth oxl`tcy feo ocokoet ge tho sot B, thoe thgs lssf`gltgfe gs `lccod l aue`tgfe arfk L tf B. Tho sot L gs `lccod tho dfklge led tho sot B gs `lccod tho `f-dfklge fa tho aue`tgfe. @fesgdor : L ; {<, 2}, led B ; (0, 4, ?, 1), thoe {(<, 4),(2, ?)) gs l aue`tgfe {(<, 4),(2, ?),(2, 1)} gs eft l aue`tgfe sgeò ocokoet 2 ge tho sot L hlvo twf gklmos ? led 1 ge tho sot B <. Ol`h Ol`h ocok ocokoet oet fa L kust bo lssf`gltod wgth oxl`tcy feo ocokoet ge tho sot B. 2. Lcc tho ocoko ocokoets ets fa tho sot sot B eood eft hlvo tho lssf`gltgfe. 0. Tho sot fa fa ocoko ocokoets ets fa fa B whg`h lro lssf`gltod wgth tho ocokoets fa tho sot L gs `lccod tho ‘rlemo‗ fa tho aue`tgfe. 4. Tho rlemo rlemo gs tho tho subsot subsot fa tho `f-dfk `f-dfklge. lge. Typos fa Aue`tgfes

(g) Feo-feo Feo-feo Aue`tgfe Aue`tgfe (gejo`tg (gejo`tgfe): fe): L aue`tgfe aue`tgfe a : L  B gs slgd tf bo l feo-feo aue`tgfe ocokoets fa L hlvo dgaaoroet gklmos ge B go,


; z x, x  {<, 2, 0} a ; {(<, 2),(2, 4),(0, 1)} Kley-feo Aue`tgfe: L aue`tgfe a : L  B gs slgd tf bo l kley(gg) feo aue`tgfe ga twf lro e fa L hlvo tho slko gklmos ge B. Fetf Aue`tgfe: L Aue`tgfe: L aue`tgfe a : L  B gs `lccod le fetf aue`tgfe (ggg) ga ovory ocokoet fa B gs le gklmo fa sfko ocokoets fa L go, ga `f-dfklge ; rlemo. om, Cot L ; {l, b, `, d} led B ; {<, 2, 0} a ; {(l, 0),(b, 2),(`, 2),(d, <)) Getf Aue`tgfe: L aue`tgfe a: L ‐ L ‐ B B gs `lccod le getf aue`tgfe ga (gv) `f-dfklge  rlemo. Oxlkpco 0: L gs sot fa prgko eukbors coss thle 2>, 2>, wrgto L ge Rfstor afrk. Pfcutgfe. ^rgko eukbors Coss thle 2> lro 2, 0, ?, =, <<9 <0, <=, <7 sot L ge Rfstor afrk. L ; {2, 0, ?, =, <<, <0, <=, <7} a (x )


Cot L ; {4, ?, 1, =} led B ; {1, 4, =, ?}, thoe Pfcutgfe. {4, ?, 1, =} ; {1, 4, =, ?}, sgeò ol`h fa tho ocokoets 4 , ?, 1, = bocfems tf Bled ol`h fa tho ocokoets 1, 4, =, ? bocfems tf L, thoe L ; B. Tho sot dfos eft `hlemo ga gts ocokoets lro rolrrlemod. Oxlkpco ?: L ?: L ; {x 2 ; <1, x gs fdd}, thoe Pfcutgfe. L gs l okpty sot. x 2 ; <1 Õ x ; + 4 fr x ; ‐ 4, 4, but x gs eft fdd S L dfos eft `fetlge ley ocokoet, L ; a Oxlkpco 1: Rowrgto tho afccfwgem stltokoets usgem sot eftltgfes. x dfos eft bocfems tf L L gs eft l subsot fa B (l) (b) H dfos eft ge`cudo D d gs l kokbor fa O . (`) (d) Oxlkpco 4:

Pfcutgfe.

(l) (`)

L  B (b) d  O (d) Oxlkpco =: Cot L ; {l, b, `}9 go, L `fetlges ` fetlges tho ocokoets l, b, `, stlto whothor ol`h fa tho afur stltokoets gs `frro`t fr ge`frro`t tocc why. x  L H  D


(l) (`)

l Å L {l} Å L

(b) (d)

l Ë L {l} Ë L

Pfcutgfe.

(l) l  L, `frro`t. (b) Ge`frro`t. Tho sykbfc  kust `feeo`t twf sot gt gedg`ltos thlt feo sot l subsot fa fthor. Thoroafro, l  L gs ge`frro`t sgeò l gs l kokbor fa L, eft l subsot. (`)Ge`frro`t. Tho sykbfc l `feeo`ts le fbjo`ts tf l sot. Gt gedg`ltos thlt fbjo`t gs l kokbor fa tho sot. Thoroafro, {l}  L gs ge`frro`t sgeò {l} gs l subsot fa L. (d) @frro`t. Oxlkpco 8: Ga P bo tho uegvorslc sot fa Oemcgsh lcphlbot led cot L ; {l, b, `}, thoe `fkpcokoet fa L gs Pfcutgfe. L@ ; {d, o, a … x, y, z} Oxlkpco 7: Ga L ; {<, 2, 0, 4} led B ; {2, 4, 1, 8}, aged L ‐ L ‐ B B, B ‐ B ‐ L L led L D B. Pfcutgfe. L ‐ L ‐ B B ; {<, 0},


L ‐ B B) `fetlges tho ocokoets fa L but eft tho ocokoets fa B. Pgkgclrcy, B ( L ‐ ‐ L L ; <1,8< B ‐ L L) `fetlges tho ocokoets fa B but eft tho ocokoets fa L. ( B ‐ L D B ; ( L ‐ L ‐ B B)  ( B ‐ B ‐ L L) ; {<, 0, 1, 8} Oxlkpco <>: Ga P ; {<, 2, 0, 4, ?}, L; {<, 2, 4}, B; {2, 4, ?} B ‐ L L (o) L@  B (a) L  B@ (m) Aged (l) L  B (b) L  B (`) B@ (d) B ‐ L@  B@ (h) B@ ‐ L@ L@ (g) ( L L  B)@ L  B)@ (j) ( L Pfcutgfe.

(l) L Í B ; {<, 2, 4, ?} (b) L Í B ; {2, 4} (`) Tho `fkpco `fkpcoko koet et fa B `fesgsts fa cottors whg`h lro ge P but eft ge B, thoroafro B@ ; {<, 0} B ‐ L L `fesgstod fa ocokoets ge B whg`h lro eft ge L go, B ‐ B ‐ L L ; {?} (d) B ‐ (o) L@ ; {0, ?} led B ; {2, 4, ?}, thoroafro, L@  B ; {2, 0, 4, ?} (a) L ; {<, 2,4) led B@ ; {<, 0}, thoroafro, L  B@ ; {<, 2,0,4} (m) L@ ; {0, ?}, led B@ ; {<, 0} 9 thoroafro, L@  B@ ; {0} L@ ; {<} (h) B@ ; {<, 0}, led L@ ; (0,?)9 thoroafro, B@ ‐ L L  B)@ ; {<, 0,?} (g) L  B ; {2, 4}, thoroafro, ( L L  B)@ ; {0} (j) L  B ; {<, 2, 4, ?}9 thoroafro, ( L


Oxlkpco <<: L <<: L ; { <, 2, 0} led B ; {l, b}, thoe aged L Ý B led B Ý L

Ý B ; {{<, l}, {<, b}, {2, l}, {2, b), {0, l}, {0, b}} led B Ý L ; {{l,<}, {l, 2}, {l, 0}, {b, c}, {b, 2}, {b, 0}} S L Ý B  B Ý L sgeò tho frdorod plgr (<, l)  (l, <) Oxlkpco <2: Ga tho sot L `fetlges 4 ocokoets led sot B `fetlges 0 ocokoets, thoe L Ý B `fetlges Pfcutgfe. Tho sot L Ý B `fetlges <2 ocokoets. Pfcutgfe. L


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