lojik tasarım - İtÜ Çözümlü uygulamalar

8/9/2019 Lojik Tasarım - İTÜ Çözümlü Uygulamalar

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!.X.Ù =4.=4.4==" Dfinfsjyjr Kùloghfsif#f DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@C!A HOUTOIOT ! HÕGOK R@G_ RFGJUFJÎFAIJKJIJT3". Rĺgjv Rùrosf ""= hjafajhĺr.4. Nõzotkogioro s`ru s`rkjyĺgĺz.2. A`pyj îoatf#f doifriogogior Ejaùito Hfsfpifg Auruiugj sova ohfioboatfr.

R@T_ "3Du s`rugug $ĺaijr ĺ dfrdfrfghog dj#ĺksĺz ìjrja îõzùiodfifr.j) J$j#ĺhj vorfiog ùî nfrf$if, ùî îĺa ĺ$iĺ nogoi e`gasfy`gu, uynug d`yuttj dfr jhot a`h îõzùbù vo

sjhobo 4 nfrf#if i`cfa dj#ijîijr auiijgjrja tjsjrijyĺgĺz.e :(j,d,b) ; jdb' + d'b

e 4(j,d,b) ; d'b' + dbe 2(j,d,b) ; j'db' + jd'

d) Dfr jhot vorf nfrf$f (\), dfr jhot vorf îĺa ĺ$ĺ (S) ìjg vo 8 jhot H tfpf fzfg nfrf$if tutubu (ijtbl)fîorog dfr hovro tjsjrijgjbjatĺr. Hovrogfg auiijgĺbĺsĺ, \ nfrf$fghog noiog vorfyf du tutubuijr-hjg fstohf#fgo yjzhĺrjdfioboatfr1 S îĺa ĺ$ĺghj hj fstohf#f tutubugug fîorf#fgf nõrodfioboatfr.Hovrogfg noroaif sjyĺhj yjzkj soîko nfrf$f (VR) vo àukj soîko nfrf$f (@R) ìjbjatĺr. VR nfrf$iorf vorfgfg ljgnf tutubuyj yjzĺijbj#ĺgĺ, @R nfrf$iorf fso ljgnf tutubugug fîorf#fgfgîĺa ĺ$j jatjr ĺijbj#ĺgĺ doifrioyoboatfr. Du hovroyf, noroaif d`yutijrhj dfr jhot a`h îõzùbù vo dfrjhot vorf soîfbf auiijgjrja tjsjrijyĺ p îfzfgfz. Xutubuijr ĺg fî yjpĺijr ĺgĺ nõstorkogfzo noroayàtur1 du oiokjgijr ĺ dià hfyjnrjk ìjrja nõstorodfifrsfgfz.

R@T_ 43 {=,",4,2,8} aùkosf fîfgho sjyĺk yjpjbja ìjg dfr sjyĺbĺ tjsjrijgjbjatĺr. Rjyĺbĺgĺg sjjt nfrf$fhĺ$ĺghj dfr ho hogotfk nfrf$f (\) vjrhĺr. \;= ìhu#ughj sjyĺbĺ, sjjt f$jrotfgfg îĺajg aogjrijr ĺghj4 jhĺk norfyo, \;: ìhu#ughj fso " jhĺk fiorfyo sjykjatjhĺr. Rjyĺbĺ fia îjiĺ$kjsĺgj sĺe ĺrsjyĺsĺghjg dj$ijkjatjhĺr.j) Rjyĺbĺyĺ sogar`g jrhĺ$ĺi dfr hovro ìjrja tjsjrijkja ùzoro huruk tjdi`sugu ìu$turuguz.d) Rjyĺbĺyĺ X eifp-ei`pijr ĺ vo noroaif sjyĺhj i`cfa ajpĺ auiijgjrja tjsjrijyĺ p îfzfgfz.b) Jygĺ sjyĺbĺyĺ H eifp-ei`pijr ĺ, uynug d`yuttj vorf soîfbfior auiijgjrja tjsjrijyĺ p îfzfgfz.

Noroakosf hurukughj oa ìjrja tùkioko ajpĺijr ĺ auiijgjdfifrsfgfz.h) Xjsjrijhĺ#ĺgĺz sjyĺbĺ lorljgnf dfr gohogio “?‖ hurukugj noîorso dfr s`grjaf huruku go ìur0

Vuajr ĺ

hjaf d vo b $ĺ

aijr ĺ

ghj yjptĺ

#ĺ

gĺ

z lor faf tjsjr ĺ

k fîfg jyr ĺ

jyr ĺ

bovjpijyĺ

gĺ

z.o) Auiijgĺbĺg fsto#fgo dj#iĺ ìjrja sjyĺbĺgĺg, sjjt f$jrotfgfg p`zftfe yj hj gonjtfe aogjr ĺghjîjiĺ$kjsĺ fstogfy`r vo dugug fîfg hovroyo yogf dfr nfrf$ oaiogfy`r (]G). ]G;= fso sjyĺbĺ p`-zftfe aogjrhj, ]G;: fso gonjtfe aogjrhj îjiĺ$jbjatĺr. Rjyĺbĺgĺg du $oafiho îjiĺ$jdfikosf fîfghovroyo ]G nfrf$fgfg gjsĺi oaiogobo#fgf îfzoroa nõstorfgfz. Xùk syĺbĺyĺ yogfhog îfzkogfzonoroa yàtur, sjhobo ho#f$og a ĺskĺ nõstorodfifrsfgfz.

R@T_ 23j) Dfr XXI tfpf i`cfa ajpĺgĺg sjhobo îĺa ĺ$ ajtĺgĺ îfzoroa îjiĺ$kjsĺgĺ jîĺaijyĺgĺz.d) Dfr XXI ajpĺgĺg îĺa ĺ$ĺgj dj#ijgjbja oiokjg sjyĺsĺ gohog sĺgĺriĺhĺr0 !zfg vorfioghog hjlj

ejzij sjyĺhj oiokjg dj#ijgkjsĺ s`gubughj goior ìjbj#ĺgĺ îĺa ĺ$ ajtĺ ùzorfgho jîĺaijyjrja s`-

ruyu bovjpijghĺr ĺgĺz.



BOUJ] :3

j) e :(j,d,b);jdb' + d'b ; jdb' + jd'b + j'd'be 4(j,d,b);d'b' + db ; jd'b' + j'd'b' + jdb + j'dbe 2(j,d,b);j'db' + jd' ; j'db' + jd'b + jd'b'

23< A`h îõzùbù auiijgĺijbjatĺr. A`h îõzùbùgùg îĺa ĺ!ijr ĺ sĺrjsĺyijr !u kfgtorfkioro ajr !ĺ hù!-

koatohfr3 j'd'b', j'd'b, j'db', j'db, jd'b', jd'b, jdb', jdb.Dugj nõro yuajr ĺhjaf e`gasfy`gijr ĺ norîoaio!tfrkoa fîfg noroaif `ijg îĺa ĺ!ijr UOVJ ajpĺijr ĺ fio

dj"ijgjrja hovro îfzfifr.d) Durjhj a`h îõzùbù ljgnf tutubuyj yjzĺijbj"ĺgĺ (yjgf fzfg vorfiobo"fgf), vorf soîfbf fso ljgnftutubugug fîorf"fgfg îĺa ĺ!j jatjr ĺijbj"ĺgĺ doifrior. Dugj nõro hovro j!j"ĺhjaf nfdf îfzfifr3

BOUJ] 43 j) Z4+Z:

+Z=

+ \

Z4Z:Z= = :

=== =:: ==:

==: :== =:=

=:= === =::

=:: ==: :==:== =:= ===

:=: ÜÜÜ ÜÜÜ

::= ÜÜÜ ÜÜÜ

::: ÜÜÜ ÜÜÜ

d)

Z4+

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Z4Z: == =: :: :=== = = = :=: = = : =

:: Ü Ü Ü Ü

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:= = = Ü Ü

Z4Z: == =: :: :=

== : = : ==: : = : :

:: Ü Ü Ü Ü := : = Ü Ü

Z4Z: == =: :: :=

== : : : :=: = : : =

:: Ü Ü Ü Ü := = = Ü Ü

Z4Z: == =: :: :=

== = = = :=: = = : =

:: Ü Ü Ü Ü := : : Ü Ü

Rjyĺbĺ K``ro k`hoifgo nõro tjsjrijgĺr.

Îĺa ĺ!ijr3S4;Z4S:;Z:S=;Z=

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Vuajr ĺhj duiugjg X fejhoiorfgo nõro hovro îfzfifr.b) H eifp-ei`pijr ĺ vo vorf soîfbfior fio tjsjr ĺk yjpjraog Ajrgjunl tjdiìjr ĺgĺ yogfhog ìu!turkjyjnoroa yàtur, sjhobo huruk tjdi`sugu auiijgkja yotorifhfr. Ùî huruk ho"f!aogf ìhu"ugj nõro<3# vorf soîfbfior auiijgĺiĺr vo vorf soîfbfiorfg soîko uîijr ĺgj huruk ho"f!aogiorf dj"ijgĺr.j) !ĺaa ĺghjaf huruk tjdi`sugug sjtĺrjijr ĺgj dja ĺijrja vorf soîfbfiorfg nfrf!iorfgo noioboa ìjg

ho"orior doifriogfr.

h)

d) !ĺaa ĺ fîfg3 ? huruku Z4Z#Z= ;#=# hurukuhur. Durhjg ljgnf hurukj nfhfiobo"f X ho"oriorf doifriogfraog Ü‖iorfg gjsĺi soîfihf"fgo dj"iĺhĺr. Dugu X fejhoiorfgfg doifrioghf"f tjdiìjrhjg jg-ijyjdfifrfz yj hj X fejhoiorfgo #=# ho"orfgf a`yjrja losjpijyjdfifrfz. Dugj nõro #=# hurukughj3\;= fîfgX4 ; : hokoa af Z4 ho"or ho"f!tfroboa yjgf Z4+ ; = X: ; = hokoa af Z# ho"or ho"f!tfrkoyoboa yjgf Z:+ ; = ==: hurukugj noîfifr. X= ; = hokoa af Z= ho"or ho"f!tfrkoyoboa yjgf Z=+ ; : \;: fîfgX4 ; : hokoa af Z4 ho"or ho"f!tfroboa yjgf Z4+ ; =

X: ; : hokoa af Z# ho"or ho"f!tfroboa yjgf Z:

+

; : =:: hurukugj noîfifr. X= ; = hokoa af Z= ho"or ho"f!tfrkoyoboa yjgf Z=+ ; : b) !ĺaa ĺ fîfg3 ? huruku Z4Z#Z= ;#=# hurukuhur. Durhjg ljgnf hurukj nfhfiobo"f vorfsoîfbfiorfg ? g`.iu nfrf!iorfgo go dj"ijghĺ"ĺgj dj"iĺhĺr. D`! dĺrja ĺijg nfrf!ior i`cfa # ìjrja ho"or-ioghfrfihf"fghog yuajr ĺhjaf tjsjr ĺkhj lok \;= lok ho \;# fîfg ### hurukughj noîfifr.o) ]G;= ìhu"ughj sjjt f!jrotfgfg aoghfsf, ]G;# ìhu"ughj sjjt f!jrotfgfg tùkioyogfgfg hovroyouynuijgkjsĺ fstogfy`r. Du f!iok og djsft ìjrja dfr VJ HJ dj"ijbĺ fio norîoaiogodfifr.

H

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BOUJ] 23

j)

Z# fiotfkho, Z4 aosfkhoyaog îĺa ĺ! Ubb‖yo yja ĺg dfr ho"or, yjgf i`cfa # ìur.Z# aosfkho, Z4 fiotfkhoyaog îĺa ĺ! yjaij!ĺa =.4U, yjgf i`cfa = ìur.Lor faf trjgzfst`r hj aosfkhoyso îĺa ĺ! yùasoa okpohjgs hurukughj ̀ iur.

$af trjgzfst`r jygĺ jghj fiotfkho ìjkjz.

d) Dfr i`cfa oiokjgĺg îĺa ĺ!ĺgj îà sjyĺhj oiokjg dj"ijgkjsĺ hurukughj du oijkgĺg îĺa ĺ!ĺghjg

jajg ja ĺk kfatjr ĺ jrtjr. Du hj oiokjgĺg h`"ru !oafiho îjiĺ!kjsĺgĺ ognoiior.Îĺa ĺ! i`cfa =‖hjyaog Z4 fiotfkhohfr. Hĺ!jrhjg noiog ja ĺk du trjgzfst`r ùzorfghog t`prj"j jajr.Ja ĺkĺg jrtkjsĺ hurukughj Z4‖gfg fî hfrogbf gohogfyio durjhjaf norfifk ho jrtjr U;T*F.Dõyiobo îĺa ĺ! norfifkf i`cfa = ho"orfgf j!kjyj dj!ijr.Îĺa ĺ! i`cfa #‖hoyaog Z# fiotfkhohfr. Îĺa ĺ! ja ĺkĺ du trjgzfst`r ùzorfghog hĺ!jr ĺyj h`"ru jajr.Îoafiog ja ĺkĺg jrtkjsĺ hurukughj Z#‖fg fî hfrogbf vo Tb hfrogbf gohogfyio durjhjaf norfifk hojrtjr. Du norfik Ub hog îĺajr ĺijbj"ĺ fîfg îĺa ĺ!tjaf norfifk jzjiĺr vo i`cfa # ho"orfgfg jitĺgj hù!-koyo dj!ijr.

ĺaĺ

Ubb

Tb

Z:

Z4

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!.X.Ù ">."".4=== Dfinfsjyjr Kùloghfsif#f Dõiùkù Vrh.H`î.Hr. Eozj D_SI_BJ

I@C!A HOUTOIOT ! ". VFI!Î! RFGJUF

JÎFAIJKJIJT3

". Rĺgjv Rùrosf <= hjafajhĺr.4. Nõzotkogioro s`ru s`rkjyĺgĺz.2. A`pyj îoatf#f doifriogogior Ejaùito Hfsfpifg Auruiugj sova ohfioboatfr.

R@T_ "3

j) Vuajr ĺhj vorfiog i`cfa hovroyo fif$afg fejhoyf îjrpĺkijr ĺg t`pijkĺ $oaifgho yjzĺgĺz. Xùkio-

yogiorf nõstorkoa fîfg ho#f$aogiorfg ùstùgo îfznf a`yuguz ( j nfdf).d) Duihu#uguz fejhoyf D`ìo bodrfgfg jasfy`k vo to`rokiorfgf auiijgjrja djsftio$tfrfgfz.b) Djsftio$tfrfikf$ fejhoyf sjhobo XUO (GJGH) dj#ijîijr ĺ fio norîoaioyfp îfzfgfz. Dj#ijîijr ĺg

nfrf$ sjyĺijr ĺghj dfr sĺgĺrijkj yàtur.

R@T_ 43e(j,d,b,h) ; :(4,2,5,<,::,:2,:?) + Υ(=,8,>,6,:=)

j) Vuajr ĺhj vorfiog e`gafsy`gug tùk tokoi fîorog tjdjgĺgĺ Ajrgjunl hfyjnrjkĺ fio duiuguz.d) J$j#ĺhj vorfiog kjifyot arftorfgf auiijgjrja soîogoaior tjdi`sugu ìu$turup fghfrnoyfgfz.

Vjptĺ#ĺgĺz fghfrnokogfg j$jkjijr ĺgĺ a ĺsjbj jîĺaijyĺgĺz. !ghfrnoko s`gubu oiho ottf#fgfz ’ogubuz‛ e`gasfy`gug fejhosfgf vo t`pijk kjifyotfgf yjzĺgĺz.Kjifyot Arftorf3 Lor dfr ho#f$aog 4 dfrfk, lor tùkioko f$iokf " dfrfk.

b) E`gafsy`gug ’og ubuz‛ fejhosfgf sjhobo 4 nfrf#if XUO dj#ijîijr ĺ auiijgjrja norîoaioyfpîfzfgfz.

j d b h

x



!.X.Ù ">."".4=== Dfinfsjyjr Kùloghfsif#f Dõiùkù Vrh.H`î.Hr. Eozj D_SI_BJ

I@C!A HOUTOIOT ! ". VFI!Î! RFGJUF BOUJ]IJTF

BOUJ] "3

j) x ; jdb‖h‖ + jdb‖h + jdb + j‖db + jbh + jd‖bh + j‖dbh‖

d) !ghfrnoko R`gubu3 (Jrj f$iokior durjhj nõstorfikokf$tfr)x ; jd + db + jbh

b)

j d b h

x



BOUJ] 43

j) Xùk tokoi fîorog tjdjgĺ3j‖h‖, j‖b, d‖b, bh, jh, jd‖, d‖h‖

J D B H O E N

5 ? ? 8 8 ? 5

d)

4 2 5 < "" "2 "? K

J \ \ 5D \ \ \ ?B \ \ \ ?H \ \ \ 8O \ \ \ 8E \ \ ?N \ \ \ 5

4 2 5 < K

J \ \ 5D \ \ \ ?B \ \ ?H \ 8E \ ?N \ \ \ 5

4 2 5 < K

D \ \ \ ?H \ 8E \ ?N \ \ \ 5

Dugj nõro og ubuz yotorif tjdjg3 e ; O+D+E ; jh + j‖b + jd‖

X`pijk Kjifyot ; "8 dfrfk.

b)

∓∓∓∓

O noroaif fîorog ìhu#u fîfg jiĺghĺ.

D, J vo B‖yf õrtor. Kjifyotf J‖hjg jz, B fio o$ft. Dugj õroJ vo B‖yf sfiorfz.H!AAJX3 D, H‖yf ho õrtor, jkj H hjlj ubuz ìhu#u fîfgsfifgkoz.H!AAJX3 N, E‖yf õrtor, jkj E hjlj ubuz ìhu#u fîfgsfifgkoz.

D noroaif fîorog ljifgo noihf vo jiĺghĺ. < gàtjsĺgĺ E hjlj

ubuzj õrttù#ù fîfg E ho jiĺ

ghĺ

j d b h

∓∓∓∓

∓∓∓∓



!.X.Ù 4"."4.4==" Dfinfsjyjr Kùloghfsif#f DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@C!A HOUTOIOT ! 4. VFI!Î! RFGJUF

JÎFAIJKJIJT3

".

Rĺgjv Rùrosf "== hjafajhĺr.4.

Nõzotkogioro s`ru s`rkjyĺgĺz.2. A`pyj îoatf#f doifriogogior Ejaùito Hfsfpifg Auruiugj sova ohfioboatfr.

R@T_ "3

j)

Dfr jhot p`zftfe aogjr totfaiokoif H eifp-ei`pu vo noroaif sjy ĺhj i`cfa dj#ijî auiijgjrja, p`zftfe aogjrtotfaiokoif dfr RT eifp-ei`pu tjsjrijyjrja îfzfgfz.

d)

Jgj-uyhu (kjstor-sijvo) tfpf dfr RT eifp-ei`pu fio, aogjr totfaiokoif dfr RT eifp-ei`pugug îji ĺ$kjsĺ jrjsĺghj go ejra vjrhĺr0 Du ejra ĺ nõstorog dfr zjkjgijkj hfyjnrjkĺgĺ îfzfgfz.

R@T_ 43

j)

Vuajr ĺhj îfzfkf vorfikf$ ìjg sogar`g jrhĺ$ĺi hovroyf îõzùkioyoroa huruk noîf$/îĺa ĺ$ tjdi`sugu vohuruk noîf$ hfyjnrjkĺgĺ îfzfgfz.d)

Jygĺ hovroyf vorf soîfbf auiijgkjhjg, i`cfa ajpĺijr vo H tfpf eifp-ei`pijr fio tjsjrijyĺ p îfzfgfz.

R@T_ 23

Dfr ijkdjyĺ a`gtrì otkoa ùzoro1 dfr sjjt nfrf$f, faf jhot vorf nfrf$f (\,V) vo dfr jhot îĺa ĺ$ĺ (S) ìjg dfrsogar`g jrhĺ$ĺi hovro tjsjrijgjbjtĺr. Rjjt f$jrotfgfg porfy`hu " sjgfyo ìjg hovroho, \ nfrf$f og jz 4 sjgfyo

d`yugbj '"' ìursj S ; " ìjbja vo ijkdj yjgjbjatĺr. Ijkdjgĺg yjgkjhĺ#ĺ hurukijrhj (S ; =) V nfrf$fg-hog noiog ho#oriorfg dfr otafsf yàtur. Ijkdj yjgĺy`raog p`zftfe dfr sjjt aogjr ĺghj V;" ìursj ijkdjsõgor vo ijkdjgĺg toarjr yjgkjsĺ fîfg \ nfrf$fgfg og jz 4 sjgfyo d`yugbj '"' ìkjsĺ doaiogfr. Ijkdjgĺgyjghĺ#ĺ hurukijrhj \ nfrf$fghog noiog ho#oriorfg dfr otafsf yàtur.Hfaajt3 \ nfrf$fgfg og jz 4 sjgfyo d`yugbj '"' ìkjsĺ fîfg, sjjt f$jrotfgfg 2 aogjr ĺghj \ ;" ìkjiĺhĺr. Hov-rogfg zjkjgijkj hfyjnrjkĺ j$j#ĺhj nõstorfikf$tfr3

j) Hovrogfg huruk noîf$ hfyjnrjkĺgĺ îfzfgfz, huruk/îĺa ĺ$ tjdi`sugu ìu$turuguz.d)

Hovroyf X eifp-ei`pijr ĺ vo noroaif sjyĺhj i`cfa dj#ijî auiijgjrja tjsjrijyĺ p îfzfgfz.

H ZBIA

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!.X.Ù 4"."4.4==" Dfinfsjyjr Kùloghfsif#f DõiùkùVrh.H`î. Hr. Eozj D_SI_BJ

I@C!A HOUTOIOT ! 4. VFI!Î! RFGJUF BOUJ]IJTF

BOUJ] "3j)

RT Eifp-ei`p3 Z(t+:); R + Z(t)T', RT;= H Eifp-ei`p3 Z(t+:); H

Dugj nõro H tfpf eifp-ei`pug nfrf$fgo R + Z(t)T' fejhosf uynuijgjbjatĺr.

d) Jgj-uyhu (kjstor-sijvo) tfpf eifp-ei`pijrhj, sjjt f$jrotf '"'ho ìhu#u sùrobo nfrf$o noiog ho#orior, îĺa ĺ$ĺ doifriokoa ùzoro ho#orioghfrfifr. Doifriogog ho#or sjjt f$jrotfgfg fgog aogjr ĺghj ajyĺt ohfifr vo îĺa ĺ$jyjgsĺtĺiĺr.Aogjr totfaiokoif eifp-ei`pijrhj nfrf$ior sjhobo sjjt f$jrotfgfg otafg aogjr ĺghj ho#orioghfrfifr vo îĺa ĺ$jyjgsĺtĺiĺr.

Vuajr ĺhjaf hfyjnrjkhj sjjt f$jrotf "'hoyaog R;" T;= ìuy`r. Jgj-uyhu eifp-ei`p du nfrf$fho#orioghfrfy`r vo fîorf#fgf " yjpkjyj ljzĺrijgĺy`r. Hjlj s`grj yfgo sjjt f$jrotf "'hoyaog R;= T;=ìuy`r. Du nfrf$ fîorfa ho#f$koyoboa (yjgf jgj-uyhu ee "'ho ajijbja) hokoatfr. Rjjt f$jrotfgfg fgogaogjr ĺghj fso ljzĺrijgjg ho#or (yjgf ") jgj uyhu eifp-ei`pug îĺa ĺ$ĺgj yjgsĺy`r. Ajr $ĺij$tĺrkjyj jihĺ#ĺkĺzgonjtfe aogjr totfaiokoif eifp-ei`p fso sjhobo fgog aogjrhjaf nfrf$iorf ho#orioghfrfy`r. Durjhj R;=,T;=ìhu#ugj nõro efip-ei`pug fîorf#f ho#f$kfy`r vo ='hj ajiĺy`r.

BOUJ] 43

j) H tfpf ee'ijrhj Z+;H ìhu#ugj nõro Z=

+ vo Z:

+ doifriokoa fîfg H= vo H: doifriogkoif.Hovrohog H= ; J

⋏D vo îĺa ĺ$ĺg S ; J

⋏D

⋏Z:' + Z='ìhu#u nõrùiùy`r.

Uorf soîfbf fso hovrogfg lor huruku fîfg vorf nfrf$iorfghog dfrfgf îĺa ĺ$ĺgj jatjrjbjatĺr. Dugj nõro3Z: Z= H:

== J⋏D

=: (J⋏D)':= J+D:: D'

H ZBIA

R

T

Z

Z'

Rjjt

Rjjt

R

T

Z (J-_)

Z (AX)

Z: Z= == =: := ::

== ==,: ==,: ==,: ::,:

=: :=,= :=,= :=,= =:,:

:= ==,: :=,: :=,: ::,:

:: :=,= ==,= :=,= =:,=

Z+Z

+S

J D Xjdi`yu ìu$turkja fîfgvorf soîfbfgfg fejhosfgfyjzkjyj noroa yàtur.Vjghjaf tjdi` Ajrgjunlhfyjnrjkĺ ho#fihfr.



b) Vuajr ĺhj duiugjg tjdi` Ajrgjunl hfyjnrjkĺ ljifgonotfrfifp Z:

+ fîfg yogfhog yjzĺiĺr3

H: ; D'⋏

Z= + J'⋏

Z:'⋏

Z= + J⋏

D⋏

Z=' + D⋏

Z:⋏

Z=' + J⋏

Z:⋏

Z='Rjhobo H" nfrf$fgo noiog hovro ho#f$oboatfr. Vuajr ĺhjaf fejhoyo nõro hovro îfzfifr.

R@T_ 23

Sjkjgijkj hfyjnrjkĺghjg hj nõrùihù#ù nfdf nfrf$iorfg îĺa ĺ$ ùzorfghoaf otafsf sjjt f$jrotfghog s`grjnorîoaio$koatohfr. Du gohogio hovro K``ro k`hoifgo nõro tjsjrijgjbjatĺr.

(Nfrf!ior \V !oaifgho nõstorfikf!tfr.)

X:; \⋏Z"'⋏Z= + \'⋏Z"⋏Z= + V⋏Z"⋏Z=' X=; \⋏Z"'⋏Z=' + \'⋏Z= + ⋏Z"⋏Z= S; Z"⋏Z='

Vuajr ĺhjaf fejhoioro nõro hovro îfzfifr.

R == =: := ::

H: H:,: H:,: H:,: H8,:

H4 H2,= H2,= H2,= H4,:

H2 H:,: H2,: H2,: H8,:

H8 H2,= H:,= H2,= H4,=

R+

S J D

H"

H4

H2

H8

==,:

=:,:

:=,:

::,:

::,:::,:

==,=

=:,=

:=,=

=:,:

:=,:

==,:

::,:

==,=

:=,=

::,=

=:,=

Z: Z= == =: :: :=

== = = : =

=: : : = :

:: : = = :

:= = : : :

Z:+

J D

J,=

D,=

B,=

H,"

==

=:

==

=:

:=

::

:=

::

==

=:

:=

::==

:=

=:

::

Dj!ijgnĺî

Huruku R == =: :: := S

J J J D D =

D J J B B =

B J J H H =H H J J H :

R+

\V

Z: Z= == =: :: := S

== == == =: =: =

=: == == :: :: =

:: == == := := =

:= := == == := :

Z:+ Z=

+ \V

Z: Z

= == =: :: :=

== = = = =

=: = = : :

:: : : = =

:= = : : =

X: \V

Z: Z= == =: :: :=== = = : :

=: : : = =

:: : : : :

:= = = = =

X= \V



Ľ.X.Ù =6.=:.4==2Dfinfsjyjr KùloghfsifĐf DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@CĽA HOUTOIOT Ľ HÕGOK R@G_ RFGJUFJÎFAIJKJIJT3:. Rĺgjv Rùrosf :=? hjafajhĺr.4. Nõzotkogioro s`ru s`rkjyĺgĺz.

2. Xùkioyogiorf nõstorkoa fîfg hoĐfŐaogiorfg ùstùgo îfznf a`yuguz ( j nfdf).

R@T_ :3j) JŐjĐĺhj vorfiog i`cfa e`gasfy`gug dfrfgbf ajg`gfa jîĺiĺkĺgĺ (îjrpĺkijr ĺg t`pijkĺ) yjzĺgĺz.

DuihuĐuguz fejhoyf D`ìo bodrfgfg jasfy`k vo to`rokiorfgf auiijgjrja sjhoioŐtfrfgfz.e(j,d,b,h) ; :(=,:,2,8,?,:2)

d) JŐjĐĺhj vorfiog 2 nfrfŐif, faf îĺa ĺŐiĺ e`gasfy`gu sjhobo uynug d`yuttj dfr jhot a`h îõzùbù vonoroaif i`cfa djĐijîijr auiijgjrja tjsjrijyĺ p îfzfgfz.

e :(j,d,b) ; jb' + dbe 4(j,d,b) ; (j'+d)(d'+b)(j+b)

R@T_ 43j) JŐjĐĺhj vorfiog e`gasfy`gug tùk tokoi fîorog tjdjgĺgĺ Zufgo-KbBiusaoy yõgtokf fio duiu-guz. e(j,d,b,h) ; :(?, >, :=, ::, :4, :8, :?) + Υ(2, 8, <)

d) JŐjĐĺhj vorfiog kjifyot arftorfgf auiijgjrja soîogoaior tjdi`sugu ìuŐturup fghfrnoyfgfz. Vjp-tĺĐĺgĺz fghfrnokogfg jŐjkjijr ĺgĺ a ĺsjbj jîĺaijyĺgĺz. Ľghfrnoko s`gubu oiho ottfĐfgfz ’og ubuz‛

e`gasfy`gug fejhosfgf vo t`pijk kjifyotfgf yjzĺgĺz.Kjifyot Arftorf3 Lor dfr hoĐfŐaog 4 dfrfk, lor tùkioko fŐiokf : dfrfk.b) E`gasfy`gug ’og ubuz‛ fejhosfgf sjhobo 4 nfrfŐif XUO djĐijîijr ĺ auiijgjrja norîoaioyfp îfzf-gfz.

R@T_ 23Dfr sjjt nfrfŐf, faf jhot vorf nfrfŐf (\,V) vo dfr jhot îĺa ĺŐĺ (S) ìjg jrhĺŐĺi dfr sogar`g hovro tjsjr-ijgjbjatĺr. \ nfrfŐf dfrdfrfgf fzioyog og jz 4 p`zftfe sjjt aogjr ĺghj i`cfa : ìhuatjg s`grj, S;: ìur. Îĺa ĺŐĺg toarjr sĺe ĺr ìkjsĺ fîfg V nfrfŐf dfrdfrfgf fzioyog og jz 4 p`zftfe sjjt aogjr ĺghj i`cfa :ìkjiĺhĺr. Îĺa ĺŐtj '=' vjraog (S;=) V nfrfŐfghog noiog hoĐorior îĺa ĺŐĺ otafiokoz. Dogzor Őoafihoîĺa ĺŐtj ':' vjraog (S;:) \ nfrfŐfghog noiog hoĐorior îĺa ĺŐĺ otafiokoz. Hovrogfg zjkjgijkj hfyjn-

rjkĺ jŐjĐĺhj nõstorfikfŐtfr3

j) Hovrogfg huruk noîfŐ hfyjnrjkĺgĺ îfzfgfz, huruk/îĺa ĺŐ tjdi`sugu ìuŐturuguz.

d)

Hovroyf X eifp-ei`pijr ĺ vo noroaif sjyĺhj i`cfa djĐijî auiijgjrja tjsjrijyĺ p îfzfgfz.

RJJX

\

V

S



Ľ.X.Ù =6.=:.4==2Dfinfsjyjr KùloghfsifĐf DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@CĽA HOUTOIOT Ľ HÕGOK R@G_ RFGJUF ÎÕSÙKIOT Ľ

BOUJ] :3j) Dfrfgbf ajg`gfa jîĺiĺk3

e(j,d,b,h) ; j'd'b'h' + j'd'b'h + j'd'bh + j'db'h' + j'db'h + jdb'hA`gsjgsùsior auiijgĺijrja fghfrnoko yjpĺijdfifr. Ľghfrnoko s`gubu3

e(j,d,b,h) ; j'b' + db'h + j'd'hd) Dfr jhot 23< a`h îõzùbù vo UOVJ djĐijîijr ĺ auiijgĺijbjatĺr. Xjsjr ĺk fîfg fejhoiorfg :.ajg`gfa jîĺiĺkijr ĺgĺ yjzkja (yj hj õrttùaiorf h`Đru g`atjijr ĺ duikja) noroafr.

e :(j,d,b) ; jdb' + jd'b' + jdb + j'db (5, 8, >, 2 gukjrjijr)e 4(j,d,b) ; (j'+d)(d'+b)(j+b) (îjrpĺk yjpĺiĺp sjhoioŐtfrfifr)

; j'b + db; j'db + j'd'b + jdb (ajg`gfa jîĺiĺk) 2, :, > gukjrjijr

Jrj djĐijgtĺijr uynug Őoafiho yjpĺiĺr.

BOUJ] 434) e(j,d,b,h); (?,>,:=,::,:4,:8,:?) + υ(2,8,<)

A.G` j d b h8 = : = =< : = = =

2 = = : :? = : = ::= : = : =:4 : : = =

> = : : ::: : = : ::8 : : : =:? : : : :

A.G` j d b h8,? = : = -8,:4 - : = =<,:= : = - =<,:4 : - = =2,> = - : :2,:: - = : :?,> = : - ::=,:: : = : -

:=,:8 : - : =:4,:8 : : - =>,:? - : : :::,:? : - : ::8,:? : : : -A.G` j d b h

2,>,::,:? - - : :<,:=,:4,:8 : - - =:=,::,:8,:? : - : -

23< AÎ

@=@:

@4

@2

@8

@?

@5

@>s4 s: s=

j d b

@2

@8

@5

@>

e :

@:

@2

@>

e 4

ĽŐjrotsfz (d`yjgkjkĺŐ) `ijg a`kdfgoz`gijr tokoifîorogioro ajr Őĺ hùŐorior.



Xùk tokoi fîorogior 3 j‖db‖ , db‖h‖ , j‖dh , bh , jh‖ , jb Kjifyot 3 (<) (<) (>) (8) (?) (8)Õrtùiogior ? :4 ?,> >,::,:? :=,:4,:8 :=,::,:8,:?

:. tjdi`

? > := :: :4 :8 :? Kjifyotj‖db‖ \ <

db‖h‖ \ <j‖dh \ \ >jb \ \ \ \ 8jh‖ \ \ \ ?bh \ \ \ 8

j‖dh, j‖db‖ dfioŐogfgf õrtor.jh‖, db‖h‖ dfioŐogfgf õrtor.

4. tjdi`

? > := :: :4 :8 :? Kjifyotj‖dh \ \ >jb \ \ \ \ 8jh‖ \ \ \ ?bh \ \ \ 8

DjŐiĺbj Gàtjijr 3 (?) , (:4) Ľinfif sjtĺr vo aì`gijr sfifgfr.Roîfiog DfioŐogior 3 j‖dh , jh‖

2. tjdi`

:: :? Kjifyotjb \ \ 8bh \ \ 8

DfioŐogiorfg kjifyotiorf oŐft ìhuĐughjg ’og ubuz‛ e`gasfy`g fejhosf

f) e; j‖dh + jh‖ + jb (Kjifyot ; :5 dfrfk)

voyj

ff) e; j‖dh + jh‖ + bh (Kjifyot ; :5 dfrfk) ìur.



BOUJ] 23NfrfŐiorfg îĺa ĺŐ ùzorfghoaf otafsf sjjt fŐjrotfghog s`grj norîoaioŐkoatohfr. Du gohogio hovroK``ro k`hoifgo nõro tjsjrijgjbjatĺr.

(NfrfŐior \V Őoaifgho nõstorfikfŐtfr.)

X:; \⋏Z:'⋏Z= + V⋏Z:⋏Z=' X=; \⋏Z:'⋏Z=' + \'⋏Z:'⋏Z= + V⋏Z:⋏Z='+ V'⋏Z:⋏Z='S; Z:

Vuajr ĺhjaf fejhoioro nõro hovro îfzfifr.

J,=

D,=

B,:

H,:

==

=:

==

=:

:=

::

:=

::

==

:=

=:

::

==

:=

=:

::

DjŐijgnĺ

îHurukuR == =: :: := S

J J J D D =

D J J B B =

B B H H B :

H B J J B :

R+

\V

Z: Z= == =: :: := S

== == == =: =: =

=: == == :: :: =

:: :: := := :: =:= :: == == :: :

Z:+ Z=

+ \V

Z: Z= == =: :: :=

== = = : :

=: : : = =

:: = : : =

:= : = = :

Z: Z= == =: :: :=

== = = = =

=: = = : :

:: = = = =

:= = : : =

X: \V

X= \V

X ZZ=

Z'=

X=

BIA

X ZZ:

Z':

X:

BIA

Eifp-ei`pRùrko

Hovrosf

E S

Rjjt

\

V



8 Ľ.X.Ù =:.::.4==4Dfinfsjyjr KùloghfsifĐf DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@CĽA HOUTOIOT Ľ :. VFIĽÎĽ RFGJUF

JÎFAIJKJIJT3

:. Rĺgjv Rùrosf 6= hjafajhĺr.4. Nõzotkogioro s`ru s`rkjyĺgĺz.2. A`pyj îoatfĐf doifriogogior Ejaùito Hfsfpifg Auruiugj sova ohfioboatfr.

8. Xùkioyogiorf nõstorkoa fîfg hoĐfŐaogiorfg ùstùgo îfznf a`yuguz ( j nfdf).

R@T_IJT3

:) j. JŐjĐĺhj nõstorfiog to`rokfg ("Dfe`rk ajroior a`gsjgsùsiorfgf yutjr") h`Đru ìhuĐugu,D`ìo bodrfgfg jasfy`k vo to`rokiorfghog yjrjrijgjrja ajgĺtijyĺgĺz. Ajgĺt, h`Đruiua tjdi`suyj hj Ajrgjunl hfyjnrjkĺ auiijgĺikjhjg, bodfrsoi ìjrja yjpĺijbjatĺr.

jd + j'b + db ; jd + j'b

d. Vuajr ĺhjaf to`rfkfg hujifgf yjzĺ p jygĺ Őoafiho h`Đru ìhuĐugu nõstorfgfz.

4) JŐjĐĺhj vorfiog i`cfa e`gasfy`gug dfrfgbf ajg`gfa jîĺiĺkĺgĺ (kfgtorfkiorfg t`pijkĺ) yjzĺgĺz.DuihuĐuguz fejhoyf bodfrsoi ìjrja djsftioŐtfrfgfz vo sjhobo 4 nfrfŐif XUO djĐijîijr ĺ auiijgjrja tjsjrijyĺ p îfzfgfz.

e(j,d,b,h) ; (j+d') ¾ (d+h) ¾ (j'+b)

2) JŐjĐĺhj vorfiog i`cfa e`gasfy`gug tùkioyogfgf Ho K`rnjg to`rokfgf auiijgjrja duiuguz.DuihuĐuguz fejhogfg h`Đru ìhuĐugu, D`ìo bodrfgfg jasfy`k vo to`rokiorfghog yjrjrijgj-rja bodfrsoi ìjrja nõstorfgfz. Ľejhogfg h`ĐruiuĐugu nõstorfraog toarjr Ho K`rnjg to`rokfgfauiijgjkjzsĺgĺz.

e(j,d,b,h) ; jh' + db' + j'd'

8) j. JŐjĐĺhj vorfiog e`gasfy`gug tùk tokoi fîorog tjdjgĺgĺ Ajrgjunl hfyjnrjkĺghjgyjrjrijgjrja duiuguz.

e(j,d,b,h) ; :(=,2,8,>,<,:=,::,:4,:8,:?)

d. JŐjĐĺhj vorfiog kjifyot arftorfgf auiijgjrja soîogoaior tjdi`sugu ìuŐturup fghfrnoyfgfz.VjptĺĐĺgĺz fghfrnokogfg jŐjkjijr ĺgĺ a ĺsjbj jîĺaijyĺgĺz. Ľghfrnoko s`gubu oiho ottfĐfgfz ’ogubuz‛ e`gasfy`gug fejhosfgf vo t`pijk kjifyotfgf yjzĺgĺz.Kjifyot Arftorf3 Lor dfr hoĐfŐaog 4 dfrfk, lor tùkioko fŐiokf : dfrfk.b. E`gasfy`gug ’og ubuz‛ fejhosfgf sjhobo 4 nfrfŐif XUO djĐijîijr ĺ auiijgjrja norîoaioyfpîfzfgfz.



8 Ľ.X.Ù =:.::.4==4Dfinfsjyjr KùloghfsifĐf DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@CĽA HOUTOIOT Ľ :. VFIĽÎĽ RFGJUF ÎÕSÙKIOT Ľ

:) j) jd + j‖b + db ;(0) jd + j‖bjd + j‖b + db ; jd + j‖b + db(j+j‖)

; jd + j‖b + dbj + dbj‖; j(d+db) + j‖(b+db); jd + j‖b

d) (j+d)(j‖+b)(d+b) ; (0) (j+d)(j‖+b) H_JI(j+d)(j‖+b)((d+b)+j⋏j‖) ; (j+d)(j‖+b)(d+b+j)(d+b+j‖)

; ( j + d⋏(d+b) )( j‖ + b⋏(d+b) )

; (j+d)(j‖+b)

4) H`Đruiua tjdi`su auiijgjrja, Ajrgjunl hfyjnrjkĺ yjrhĺkĺyij yj hj bodfrsoi ìjrja duiugjdfifr.Ljtĺrijtkj3 :. ajg`gfa jîĺiĺk kfgtorfkiorfg t`pijkijr ĺghjg ìuŐur. Lor kfgtorfk sjhobo dfrh`Đru gàtjyj (a`kdfgoz`g) ajr Őĺ noifr vo kfgtorfkiorho tùk hoĐfŐaogior duiugur.

Ajrgjunl Hfyjnrjkĺ yjrhĺkĺyij3

j‖d‖b‖h + j‖d‖bh + jd‖bh + jdbh‖ + jdbh: 2 :: :8 :?

:. Ajg`gfa jîĺiĺk

Jygĺ fejhogfg bodfrsoi ìjrja duiugkjsĺ3 Õgbo îjrpĺk yjpĺiĺr, jrhĺghjg lor k`g`khjaf oasfahoĐfŐaogf tjkjkijkja fîfg fejhoyo D`ì bodrfgo uynug oaiokoior yjpĺiĺr.ÕrgoĐfg jdb k`g`kughj h hoĐfŐaogf oasfatfr. Du gohogio jdb yorfgo `gj oŐft ìjg jdbh + jdbh' yjzĺiĺr.

e; (j+d‖)(j‖d+db+j‖h+bh) ; jdb+ jbh + j‖d‖h + d‖bhe; jdbh + jdbh‖ + jd‖bh + j‖d‖bh + j‖d‖b‖h :. Ajg`gfa jîĺiĺk

RjhoioŐko s`gubu 3

e; j‖d‖h + jdb + jbh (:)e; j‖d‖h + jdb + d‖bh (4)

jd

bh

== =: :: :=

==

=:

::

:=

:8

2

:?

::

:



8) j) Xùk tokoi fîorog tjdjgĺ3Ľejhoior 3 bh, b‖h‖, jb, jh‖Rfknoior3 J D B H

Kjifyotior3 8 5 8 ?

d)

= 2 8 > < := :: :4 :8 :? KJ \ \ \ \ 8D \ \ \ \ 5B \ \ \ \ 8H \ \ \ \ ?

:= :8 KB \ \ 8H \ \ ?

Dugj nõro og ubuz yotorif tjdjg3 e ; J+D+B ; bh + b‖h‖ + jb

X`pijk Kjifyot ; :8 dfrfk.

b)

J vo D noroaif tokoi fîorogiorhfr.B H‖yf õrtor vo kjifyotf hjlj hùŐùatùr. Du hurukhj Hsifgfr.

j d b h

e



Ľ.X.Ù :2.:4.4==4Dfinfsjyjr KùloghfsifĐf DõiùkùVrh.H`î.Hr. Eozj D_SI_BJ

I@CĽA HOUTOIOT Ľ 4. VFIĽÎĽ RFGJUF

JÎFAIJKJIJT3

:. Rĺgjv Rùrosf :== hjafajhĺr.4. Nõzotkogioro s`ru s`rkjyĺgĺz.2.

A`pyj îoatfĐf doifriogogior Ejaùito Hfsfpifg Auruiugj sova ohfioboatfr.

R@T_ :3

Vjghj dià hfyjnrjkĺ nõstorfiog, 2 jhot vorf nfrfŐf (j,d,b), dfr jhot soîko nfrfŐf (s)ìjg dfr i`cfa hovro tjsjrijgjbjatĺr. Hovrogfg s nfrfŐfgo i`cfa = uynuijgĺrsj z îĺa ĺŐĺghje(j,d,b) i`cfa e`gasfy`gugug hoĐorf, s nfrfŐfgo i`cfa : uynuijgĺrsj jygĺ e e`gasfy`gugtùkioyogf (e') oiho ohfikoatohfr.

s;= vorfifrso z ; e(j,d,b) ; j'd +db + d'b'

s;: vorfifrso z ; e'(j,d,b)Xjrfe ohfiog hovroyf, og ejzij faf jhot tùkioko ajpĺsĺ vo og dùyùĐù 83: ìjg noroaif sjyĺhj vorf soîfbfiorauiijgjrja tjsjrijyĺ p îfzfgfz. Dugijr ĺg hĺŐĺghj lorljgnf dfr i`cfa djĐijî auiijgĺikjyjbjatĺr.

R@T_ 43

j) Lok Z lok ho Z' îĺa ĺŐĺ ìjg dfr jhot p`zftfe aogjr totfaiokoif H eifp-ei`pu vo sjhobo dfr jhot uynug d`yuttj vorf soîfbf auiijgjrja, p`zftfe aogjr totfaiokoif dfr CA "eifp-ei`p"u tjsjrijyjrja îfzfgfz.

d) Vjghjaf Őoafiho nõstorfihfĐf nfdf dfr X eifp-ei`pugug nfrfŐfgo dfr VJ HJ (O\@T) djĐijbĺ djĐijgjrja dfr \ "eifp-ei`p"uoiho ohfikfŐtfr. Du \ "eifp-ei`p"ugug fŐiov tjdi`sugu (\ nfrf-Őfgo nõro "eifp-ei`p" fîorfĐfgfg hoĐfŐfkf) îfzoroa îjiĺŐkjsĺgĺ

jîĺaijyĺgĺz. \ "eifp-ei`p"ugug zjkjg hfyjnrjkĺgĺ îfzfgfz.

R@T_ 23

Vuajr ĺhj îfzfkf vorfikfŐ ìjg sogar`g jrhĺŐĺi hovroyf îõzùkioyoroa huruk noîfŐ/îĺa ĺŐ tjdi`sugu vo hu-ruk noîfŐ hfyjnrjkĺgĺ îfzfgfz.

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I@CĽA HOUTOIOT Ľ 4. VFIĽÎĽ RFGJUF BOUJ]IJTF

BOUJ] 43

R;= fso z ; e(j,d,b) ; j‖d + db + d‖b‖R;: fso z ; e ‖(j,d,b)

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BOUJ] 43

j) CA Eifp-ei`p fŐiov tjdi`su3C A Z(t+:)= = Z(t)= : =: = :: : Z(t)'

d) X tfpf eifp-ei`p3 Z(t+:) ; X⊟ Z(t)Uorfiog eifp-ei`p3 X ; \ ⊟ Z(t) ↔ Z(t+:) ; ( \ ⊟ Z(t) ) ⊟ Z(t)

Z(t+:) ; \ H tfpf eifp-ei`p nfdf hjvrjgĺr.ĽŐiov tjdi`su3\ Z(t+:) = = (sjjt fŐjrotfgfg îĺajg aogjrijr ĺghj): : (sjjt fŐjrotfgfg îĺajg aogjrijr ĺghj)υ Z(t) (sjjt fŐjrotfgfg hfĐor hurukijr ĺghj)

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BOUJ] 23

C;(J¾D)' + Z= Z=+ ; C¾Z=' + A'⋏Z=

A;(J¾D)' + Z: Z=+ ; ((J¾D)' + Z= )⋏Z=' + ((J¾D)' + Z:)'⋏Z=

Z=+ ; J'¾Z=' + D'¾Z=' + J¾D¾Z=¾Z:'

X; (J+D)¾Z:' Z:+ ; X ⊟ Z

Z:+ ; J + D + Z:

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==,:Huruk-noîfŐ hfyjnrjkĺghjnfrfŐior vo îĺa ĺŐ JD,S ŐoaifghonõstorfkfŐtfr.

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