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Конспект лекций по дискретной математике

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I?eeiaeaiea Aoeaaie aeaaa?u e neioaco eiiaeiaoeeiiiuo noai

Aeaie/iay nenoaia eiaeee:

1. Yeaiaiou Aoeaaie aeaaa?u:

a) /enea

b) ia?aiaiiua

n) iia?aoeee

d) au?aaeaiey

e) ooieoeee

f) caeiiu

A) *enea:

Aeaa /enea: eiae/aneee iieue e eiae/aneay aaeeieoea a Aoeaaie aeaaa?a
ioiaeaeanoaeythony n iiiyoeyie “enoeia” e ”eiaeue”.

A) Ia?aiaiiua:

Aoeaau (eiae/aneea, aeaie/iua) ia?aiaiiua iacuaathony ia?aiaiiuie,
i?eieiathueie cia/aiea ec iiiaeanoaa – iieue e aaeeieoea.

N) Iia?aoeee:

1. Io?eoeaiea (eiaa?ney).

2. Eiiuethieoeey (eiae/aneia oiiiaeaiea).

3. Aeecuthieoeey (eiae/aneia neiaeaiea).

Oia?iie yaeyaony iia?aoeey io?eoeaiey.

Iaicia/aiey:

, ( x

2. Eiiuethieoeey a&b, a(b, ab, a(b

3. Aeecuthieoeey a(b

D) Au?aaeaiey:

Ia?aiaiiua, ciaeiiia?aoeee, niaaeeiaiiua aianoa i?e aiciiaeiii iaee/ee
neiaie aeey caaeaiey ii?yaeea auiieiaiey iia?aoeee.

I?ei?eoao caaeaaony ii?yaeeii iia?aoeee.

A) Ooieoeee:

Aoeaaie (eiae/aneie) ooieoeeae iacuaaaony oaeay ooieoeey, a?aoiaioaie
eioi?ie yaeythony aoeaau ia?aiaiiua, e naia ooieoeey i?eieiaao cia/aiea
ec iiiaeanoaa iieue e aaeeieoea.

Iaeanoueth ii?aaeaeaiey Aoeaaie ooieoeee yaeyaony niaieoiiinoue 2n
aeaie/iuo iaai?ia aa a?aoiaioia. Iaai? a?aoiaioia iiaeii ?anniao?eaaoue
eae n-eiiiiiaioiue aeaie/iue aaeoi?.

Oi?iu caaeaiey Aoeaaie ooieoeee:

1. Aiaeeoe/aneay (a aeaea eiae/aneiai au?aaeaiey)

2. Oaaee/iay (a aeaea oaaeeoeu enoeiiinoe)

3. A?aoe/aneay

4. Oaaee/ii-a?aoe/aneay (a aeaea ea?ou Ea?ii)

5. *eneiaay

6. Neiaiee/aneay oi?ia

1) Aiaeeoe/aneay:

_ _

y=(x1 ( x2) x3

_ _ _ _ _ _

y=x1 x2 x3 ( x1 x2 x3 ( x1 x2 x3

2) Oaaee/iay:

x1

x2

x3 _

x1 ( x2

y

0 0 0 1 1

0 0 1 1 0

0 1 0 1 1

0 1 1 1 0

1 0 0 0 0

1 0 1 0 0

1 1 0 1 1

1 1 1 1 0

Ia?aoiae io aiaeeoe/aneie e oaaee/iie iaeiicia/ai! Ia?aoiue ia?aoiae ia
yaeyaony iaeiicia/iui.

Iniiaiua caeiiu (oiaeaeanoaa)

1) ab=ba

a(b=b(a

2) Annioeeaoeaiue:

a(bc)=(ab)c

a((b(c)= (a(b) (c

3) Aeeno?eaooeaiue:

a(b(c)=ab(ac

a((bc)=(a(b)(a(c)

4) Caeii aeaieiiai io?eoeaiey:

=

a=a

5) Oaooieiaee:

aa=a

a(a=a

6) Caeiiu ioeaaiai yeaiaioa:

a0=0

a(0=a

7) Caeiiu aaeeie/iiai yeaiaioa:

a1=a

a(1=1

8) Caeiiu aeiiieieoaeueiiai yeaiaioa:

_

A Aoeaaie aeaaa?a aeiiieieoaeueiui yeaiaioii e a yaeyaony a.

_ _

a(a=1; aa=0

9) Aeaienoaaiiinoe (aeaIi?aaia):

__ _ _

ab=a(b

___ _ _

a(b=a b

Ceaaenoaey: ab=a(b; a(b=a b

10) Iiaeiuaiey:

a(ab=a

a(a(b)=a

11) Nie?auaiey:

_

a(ab=a(b

_

a(a(b)=ab _ _ _ _

Ceaaenoaey: a(ab=a(b; a(a(b)=ab

12) Neeaeaaiey:

_ _

ab(ab=a; (a(b)(a(b)=a

Eiiiaioa?ee:

1) Aeey aeieacaoaeuenoaa caeiiiaiiaeii eniieueciaaoue:

a) Iaoiae niaa?oaiiie eiaeoeoeee.

a) Eniieueciaaiea iaeieo caeiiia aeey aeieacaoaeuenoaa ae?oaeo caeiiia.

Iaoiae niaa?oaiiie eiaeoeoeee ninoieo a aeieacaoaeuenoaa yeaeaaeaioiinoe
eaaie e i?aaie /anoe ia anai iiiaeanoaa iaai?ia a?aoiaioia. Aeey yoiai
ninoaaeyaony oaaeeoea enoeiiinoe.

2) Aieueoeinoai caeiiia caaeaaony ia?ie niioiioaiee, i?e yoii iaeii
niioiioaiea iiaeii iieo/eoue ec ae?oaiai caiaiea iia?aoeee eiiuthieoeee
ia aeecuthieoeeth eee aeecuthieoeeth ia eiiuthieoeeth (iaoiae ia
i?eiaiei a caeiiao, a eioi?uo o/anoaotho eiinoaiou). N eiinoaioaie aea
eiinoaiou caiaiythony ia i?ioeaiiieiaeiua cia/aiey. (Aeoaeueiinoue
caeiiia Aoeaaie aeaaa?u)

3) Iaeioi?ua caeiiu iiaeii ?ani?ino?aiyoue ia i?iecaieueiia /enei
yeaiaioia.

4) A ethaii caeiia iiaeii caiaieoue ethaoth aoeao ia i?iecaieueiia
eiae/aneia au?aaeaiea.

5) Caeiiu i?eiaiythony aeey oi?iuaiey Aoeaauo ooieoeee.

?aciiia?acea Aoeaauo ooieoeee.

1. Aoeaaay ooieoeey io iaeiie ia?aiaiiie.

Iaicia/aiea a?aoiaioa e ooieoeee Cia/aiey a?aoiaioa e ooieoeee

1 1 Eiae/aneay aaeeieoea

2. Aiciiaeiua ooieoeee io aeaoo ia?aiaiiuo.

Iaicia/aiea a?aoiaioia e ooieoeee Cia/aiea a?aoiaioia e ooieoeee

1 1 1 1 “1” Eiae/aneay aaeeieoea + –

Ii?aaeaeaiea: Aoeaaa ooieoeey io n a?aoiaioia fn(x) iacuaaaony
au?iaeaeaiiie ii a?aoiaioo xi, anee aa cia/aiea ia caaeneo io yoiai
aiaoiaioa, oi anoue aeey anao iaai?ia a?aoiaioia eiaao ianoi ?aaaionoai:

f(x1, x2, … , xi-1, 0, xi+1, … , xn) = f(x1, x2, xi-1, 1, xi+1, …
, xn).

Ooieoeey cai?aoa x1(x2 i?eieiaao cia/aiea, ?aaiia ioeth i?e ?aaainoaa
cai?auathuae ia?aiaiiie (x2) aaeeieoea e iiaoi?yao cia/aiea a?aoiaioa x1
i?e ?aaaionoaa cai?auathuae ia?aiaiiie ioeth.

Iiiyoea eiieeeaoeee a Aoeaaie aeaaa?a ioiaeaeanoaeyaony n au?aaeaieai
neaaeiaaiey (anee … oi … ).

I?eia?: Eiatho ianoi aeaa i?inouo auneacuaaiey.

A. Ia iaaa oo/e.

A. Eaeao aeiaeaeue. A(A

A A A(A

f f t

f t f

t f t

t t t

Ec enoeiu ia iiaeao neaaeiaaoue eiaeue!

Iaeioi?ua ooieoeee io o?ao ia?aiaiiuo.

Cia/aiea a?aoiaioia Cia/aiea ooieoeee

Noiia ii iiaeoeth 2 Eneeth/athuaa EEE Ooieoeey iaaei?eoa?iinoe

x1 x2 x3 x1(x2(x3 XOR (x1,x2,x3) x1#x2#x3

0 0 0 0 0 0

0 0 1 1 1 0

0 1 0 1 1 0

0 1 1 0 0 1

1 0 0 1 1 0

1 0 1 0 0 1

1 1 0 0 0 1

1 1 1 1 0 1

Ooieoeey – noiia ii iiaeoeth 2 e eneeth/athuaa EEE yaeythony
yeaeaaeaioiuie oieueei aeey aeaoo a?aoiaioia.

n

Iauaa ?aciiia?acea ooeioeee io n a?aoiaioia ?aaii 22

, aaea n-eiee/anoai a?aoiaioia, a N-aeanyoe/iue yeaeaaeaio aeaie/iiai
iaai?a cia/aiee ooieoeee ia oii?yaei/aiiii iiiaeanoaa a?aoiaioia.

I?eia?:

Iaau?iaeaeaiiua ooieoeee io aeaoo ia?aiaiiuo n aeiaaaeaieai ooieoeee
io?eoeaiey i?eiyoi iacuaaoue ooieoeeyie Aoeaaie aeaaa?u. N o/aoii
ia?auaaiinoe iaeioi?uo aaciauo ooieoeee e iaeioi?ui a?aoiaioai, eo iauaa
eiee/anoai ?aaii aeaayoe.

Ii?iaeueiua oi?iu Aoeaauo ooieoeee

Ii?iaeueiua oi?iu – yoi iniaue eeann aiaeeoe/aneeo au?aaeaiee,
eniieuecoaiuo i?e ?aoaiee caaea/e ieieiecaoeee Aoeaauo ooieoeee e aeey
ia?aoiaea io oaaee/iie oi?iu caaeaiey e aiaeeoe/aneie. Ii?iaeueiua oi?iu
no?iyoueny ia iniiaaiee iia?aoeee eiiuthieoeee, aeecuthieoeee e
io?eoeaiey, i?e/ai io?eoeaiea oieueei aaeeinoaaiiie ia?aiaiiie.

Ii?aaeaeaiea: Yeaiaioa?iie eiiuthieoeeae (aeecuthieoeeae) iacuaaaony
eiiuthieoeey (aeecuthieoeey) eiia/iiai /enea iiia?ii ?acee/eiuo
ia?aiaiiuo eee eo io?eoeaiee.

Yeaiaioa?ioth eiiuthieoeeth (aeecuthieoeeth) i?eiyoi iacuaaoue
eiiuthieoeaiui (aeecuthieoeaiui) oa?iii.

A /anoiii neo/aa oa?i, eae eiiuthieoeaiue oae e aeecuthieoeaiue iiaeao
ninoiyoue ec aaeeinoaaiiie aoeau (eeoa?aea). Iiae aoeaie aoaeai
iiieiaoue a?aoiaio Aoeaaie ooieoeee e aai io?eoeaiey.

I?eia?u eiiuthieoeaiuo oa?iia:

_ _

x1, x2, x1x3, x2x4x5 (oa?i)

___ _

x1x2, x1x2x3 (ia oa?i)

?aiaii oa?ia iacuaaaony eiee/anoai aoea aoiaeyueo a iaai.

Aeecuthieoeaiie (eiiuthieoeaiie) ii?iaeueiie oi?iie Aoeaaie ooieoeee
iacuaaaony aeecuthieoeey (eiiuthieoeey) eiia/iiai /enea iiia?ii
?acee/eiuo eiiuthieoeaiuo (aeecuthieoeaiuo) oa?iia.

Eaiiie/aneay ii?iaeueiay oi?ia.

Eiinoeooaioie aaeeieoeu (ioey) iacuaaaony eiiuthieoeaiue
(aeecuthioeaiue) oa?i iaeneiaeueiiai ?aiaa. O.a. aeey Aoeaaie ooieoeee
io n ia?aiaiiuo eiinoeooaioa aeeth/aao a naay n aoea.

Naienoai eiinoeooaiou: Eiinoeooaioa aaeeieoeu (ioey) i?eieiaao cia/aiea
aaeeieoeu (ioey) ia iaeiii e oieueei iaeiii iaai?a a?aoiaioia.

I?eia?: _ _

n=4 x1x2x3x4 (1010)=1

_ _ _

x1(x2(x3(x4=0

Ii?aaeaeaiea: Aeecuthieoeaiay (eiiuthieoeaiay) ii?iaeueiay oi?ia
iacuaaaony eaiiie/aneie, anee ana aa aeecuthieoeaiua (eiiuthieoeaiua)
oa?iu i?aaenoaaeytho niaie eiinoeooaiou aaeeieoeu (ioey). Eiiaaea
eaiiie/aneea oi?iu iacuaatho niaa?oaiiuie.

I?eia? iieo/aiey eaiiie/aneeo oi?i:

y=x1(x2

2

EAeIO – eaiiie/aneay aeecuthieoeaiay ii?iaeueiay oi?ia:

_ _

y=x1x2(x1x2

EEIO – eaiiie/aneay eiiuthieoeaiay ii?iaeueiay oi?ia:

_ _

y=(x1(x2)(x1(x2)

1) N iiiiuueth eaiiie/aneeo oi?i iaeaieaa i?inoi inouanoaeyaony ia?aoiae
io oaaee/iie oi?iu caaeaiey Aoeaaie ooieoeee e aiaeeoe/aneie.

2) N iiiiuueth eaiiie/aneeo oi?i iiaeii inouanoaeoue i?aia?aciaaiea
ethaie ooieoeee a Aoeaa aacen.

3) Ethaay Aoeaaa ooieoeey ca eneeth/aieai eiae/aneiai ioey e eiae/aneie
aaeeieoeu eiaao aaeeinoaaiiua EAeIO e EEIO. Eiae/aneoth aaeeieoeo iiaeii
i?aaenoaaeoue a aeaea EAeIO e eiae/aneee iieue a aeaea EEIO.

4) I?aaeei ia?aoiaea io oaaee/iie oi?iu caaeaiey Aoeaaie ooieoeee e
aiaeeoe/aneie:

a) a oaaeeoea enoeiiinoe auaeaeythoueny ana iaai?u a?aoiaioia, i?e
eioi?uo ooieoeey ?aaia aaeeieoea (ioeth).

a) aeey eaaeaeiai ec yoeo iaai?ia ninoaaeytho eiinoeooaiou aaeeieoeu
(ioey).

a) iauaaeeiaieai eiinoeooaiou aaeeieoeu (ioey) ciaeaie aeecuthieoeee
(eiiuthieoeee) iieo/aaony aiaeeoe/aneay oi?ia a aeaea EAeIO (EEIO).

Iiyniaiea: i?e ninoaaeaiee eiinoeooaio aaeeieoeu (ioey) eniieuecotho
neaaeothuaa i?aaeei:

Anee iaeioi?ue a?aoiaio i?eieiaao ia iaai?a cia/aiea ?aaiia ioeth, oi a
eiinoeooaioo aaeeieoeu ii aoiaeeo n io?eoeaieai, a a eiinoeooaioo ioey
aac iaai.

5) EAeIO e EEIO i?aaenoaaeytho niaie aeaa ?acee/iua, ii yeaeaaeaioiua
aiaeeoe/aneea oi?iu aoeaaie ooieoeee. Yoi icia/aao, /oi ec iaeiie oi?iu
iiaeii iieo/eoue ae?oaoth, eniieuecoy caeiiu Aoeaaie aeaaa?u.

_ _ _ _ _ _ _
_ _ _

y=(x1(x2)(x1(x2)=x1x1(x1x2(x2x1(x2x2=x1x2(x2x1=x1x2(x1x2 (EAeIO)

6) I?eioeeieaeueii nouanoaoao ae?oaie niinia iieo/aiey EEIO:

a) ninoaaeyaony EAeIO, ii ia aeey naiie, a aeey aa io?eoeaiey.

a) aa?aony io?eoeaiea iaae iieo/aiie EAeIO, eioi?ia nieiaaony n
i?eiaiaieai caeiia aeaienoaaiiinoe.

_ _ _ = ———— —– —-
_ _

y=x1x2(x1x2, y=y=x1x2(x1x2=x1x2 x1x2=( x1(x2)(x1(x2)

?aciiia?acea aeaie/iuo aeaaa?

A nayce n oai, /oi ethaoth neieue oaiaeii neiaeioth Aoeaaoth ooieoeeth
iiaeii i?aaenoaaeoue a eaiiie/aneeo oi?iao, oi anoue caienaoue aa n
iiiiuueth iia?aoeee io?eoeaiey, eiiuthieoeee e aeecuthieoeee yoa nenoaia
Aoeaauo iia?aoeee iaeaaeaao naienoaii ooieoeeiiaeueiie iieiiou, o.a.
ia?acoao oae iacuaaaiue aacen. Anoanoaaiii i?aaeiieiaeeoue, /oi nenoaia
Aoeaauo iia?aoeee yaeyaony ia aaeeinoaaiiie, n iiiiuueth eioi?ie iiaeii
ia?aciaaoue iaeioi?ue aacen.

A i?eioeeia ethaoth ec aaciauo ooieoeee iiaeii ioiaeaeanoaeoue
niioaaonoaothuae iia?aoeeae e ia iniiaa niaieoiiinoe yoeo iia?aoeee
iino?ieoue aeaie/iua aeaaa?u, ioee/iua io Aoeaaie. E iaeaieaa
?ani?ino?aiaiiui aeaie/iui aeaaa?ai ioiinyony: aeaaa?a AEeaaeeeia ((,
&); aeaaa?a Aaaaa (Ie?na) ((); aeaaa?a Oaooa?a ( | ). A eaaeaeie ec yoeo
aeaaa? aeaenoaotho nianoaaiiua caeiiu. Anoanoaaiii nouanoaotho acaeiii
iaeiicia/iua ia?aoiaeu io iia?aoeee iaeiiai aacena e iia?aoeeyi
ae?oaiai.

*eneiaia i?aaenoaaeaiea Aoeaauo ooieoeee

Aeey ethaie Aoeaaie ooieoeee iiaeii i?aaeeiaeeoue aeaa /eneiaua oi?iu,
iniiaaiiua ia ia?a/eneaiee aeanyoe/iuo yeaeaaeaioia iaai?ia a?aoiaioia
ia eioi?uo ooieoeey i?eieiaao cia/aiea aaeeieoeu (ioey).

(0,2,6,7) – io yoie /eneiaie oi?iu eaaei ia?aeoe e EAeIO iooai caiaiu
eaaeaeiai ec iaai?ia a ia?a/eneaiee eiinoeooaiou aaeeieoeu.

_ _ _ _ _ _ _ _
_ _ _

y=x1x2x3(x1x2x3(x1x2x3(x1x2x3=x1x3(x2(x2)(x1x2(x3(x3)=x1x3(x1x2
(AeIO)

&(1,3,4,5)

_ _ _ _ _ _

y=(x1(x2(x3) (x1(x2(x3) (x1(x2(x3) (x1(x2(x3) (*)

I?aia?aciaaiea i?iecaieueiie aiaeeoe/aneie oi?iu Aoeaaie ooieoeee a
ii?iaeueioth

A Aoeaaie aeaaa?a a aeaea oai?aiu aeieacuaaaony neaaeothuaa
ooaa?aeaeaiea: nouanoaoao aaeeiue eiino?oeoeaiue iiaeoiae, iicaieythuee
i?aia?aciaaoue aiaeeoe/aneia au?aaeaiea Aoeaaie aeaaa?u a i?iecaieueiie
oi?ia e ii?iaeueiie oi?ia.

I?eia?:

_ _ _ _
_ _

y=f4(x)=(x1x2(x2x3)(x1|x4)=(x1x2(x2x3)(x1x4)=(x1x2(x2x3)(x1(x4)=

_ _ _ _ _ _ _
_ _ _

=x1x2(x1x2x4(x1x2x3(x2x3x4=x1x2(x1x2x3(x2x3x4=x1(x2(x2x3)(x2x3x4=

_ _ _ _

=x1(x2(x3) (x2x3x4=x1x2(x1x3(x2x3x4
(EAeIO)

Caia/aiey:

1) A iauai neo/aa ethaay Aoeaaa ooieoeey iiaeao eiaoue ianeieueei
EAeIO, ioee/athueony eeai eiee/anoaii oa?iia, eeai eiee/anoaii aoea a
yoeo oa?iao.

2) I?e iino?iaiee eiiaeiaoeeiiiie noaiu, ?aaeecothuae aeaiioth
ooieoeeth ii aa ii?iaeueiie oi?ia i?aaeii/oeoaeueiae oa, eioi?ay
iaeaaeaao iaeiaiueoei /eneii oa?iia e iaeiaiueoei eiee/anoaii aoea a
yoeo oa?iao.

3) Ii n?aaiaieth ni noaiie, iino?iaiiie ii AeIO, noaia, iino?iaiiay ii
neiai/iie oi?ia (*), yaeyaony aieaa i?aaeii/oeoaeueiie o.e. i?e iaeiii e
oii aea /enea eiae/aneeo yeaiaioia (E, EEE) niaea?aeao iaiueoaa /enei
aoiaeia (9 aianoi 10).

Caaea/a i?aia?aciaaiey ii?iaeueiie oi?iu Aoeaaie ooieoeee a
neiai/iie oi?ia iacuaatho caaea/ae oaeoa?ecaoeee.

4) Nouiinoue eiino?oeoeaiiai iiaeoiaea i?e iieo/aiee AeIO ninoieo a
neaaeothoai:

a) i?aia?aciaaiea iia?aoeee ia-Aoeaaiai aacena e iia?aoeeyi Aoeaaiai
aacena (ni. iineaaeiea no?iee oaaeeoeu)

a) niyoea io?eoeaiee iaae au?aaeaieyie n i?eiaiaieai caeiiia
aeaienoaaiiinoe

a) ?ane?uoea neiaie n i?eiaiaieai aeeno?eaooeaiiai caeiia

a) oi?iuaiey au?aaeaiey n i?eiaiaieai caeiia iiaeiuaiey

I?eaaaeaiea i?iecaieueiuo ii?iaeueiuo oi?i Aoeaaie ooieoeee e
eaiiie/aneei

Aeey i?eaaaeaiey i?iecaieueiie AeIO e EIO iaiaoiaeeii eniieueciaaoue
i?aaeei aeecuthieoeaiiai ?acaa?ouaaiey i?eiaieoaeueii e eaaeaeiio ec
iaiieiuo eiiuthieoeaiuo oa?iia.

_ _

P=P(xi(xi)=Pxi(Pxi, aaea P-iaiieiue eiiuthieoeaiue oa?i (?aia yoiai
oa?ia

iaiueoa n), a xi – iaaeinoathuee a
oa?ia a?aoiaio.

I?eia?:

_ _ _ _ _ _
_ _ _ _

y=x1(x2x3(AeIO)=x1(x2(x2)(x3(x3) (x2x3(x1(x1)=x1x2x3( x1x2x3(

_ _ _ _ _ _ _ _ _

(x1x2x3(x1x2x3( x1x2x3( x1x2x3 (EAeIO)

Caia/aiea:

Iinea ?ane?uoey neiaie iiaoo iieo/eoueny iaeeiaeiaua oa?iu, ec eioi?uo
ioaeii inoaaeoue oieueei iaeei.

(0,1,2,3,5)=f3

I?aia?aciaaiea EIO e EEIO ?aaeecoaony iooai i?eiaiaiey i?aaeea
eiiuthieoeaiiai ?acaa?ouaaiey e eaaeaeiio iaiieiiio aeecuthieoeaiiio
oa?io.

_ _

P=P(xixi=(P(xi)(P(xi)

_ _ _ _ _ _ _ _
_ _

y=x1(x2x3(AeIO)=(x1(x2)(x1(x3)(EIO)=(x1(x2(x3x3)(x1(x3(x2x2)=

_ _ _ _ _ _ _ _

=(x1(x2(x3)(x1(x2(x3)(x1(x2(x3)(x1(x2(x3)(EEIO)

(4,6,7)

Ieieiecaoeey aoeaauo ooieoeee ia ea?oao Ea?ii(ni. I?aeoeeo).

Iaoiae Eaaeia-IaeEeannee aace?oaony ia eoae/aneii i?aaenoaaeaiee
aoeaauo ooieoeee.

Eoae/aneia i?aaenoaaeaiea aoeaauo ooieoeee.

A eoae/aneii i?aaenoaaeaiee aoeaaie ooieoeee io n ia?aiaiiuo ana
iiiaeanoai ec 2n iaai?ia aa a?aoiaioia ?anniao?eaaaony eae iiiaeanoai
eii?aeeiao aa?oei n-ia?iiai eoaa n aeeeiiie ?aa?a ?aaiie 1. A
niioaaonoaee n yoei iaai?u a?aoiaioia, ia eioi?uo aoeaaa ooieoeey
i?eieiaao cia/aiea ?aaiia 1 i?eiyoi iacuaaoue nouanoaaiiuie aa?oeiaie.

Nouanoaaiiua aa?oeiu ia?acotho oae iacuaaaiua iieue-eoau (0-eoau).
Iaaeaeo 0-eoaaie nouanoaoao ioiioaiea ninaaenoaa e ii?aaeaeaia iia?aoeey
neeaeaaiey. Aeaa 0-eoaa iacuaathony ninaaeieie anee iie ioee/athony
oieueei ii iaeiie eii?aeeiaoa.

I?eia? : n=4 0101

0001 – aeaa ninaaeieo 0-eoaa

?acoeueoao neeaeaaiey : 0x01 (*)

Neeaeaaiea 2-o ninaaeieo 0-eoaia aeaao a ?acoeueoaoa 1-eoa.
Eii?aeeiaoa, ioia/aaiay neiaieii o, iacuaaaony naiaiaeiie (iacaaeneiie,
ianaycaiiie), a inoaeueiua (/eneiaua) eii?aeeiaou iacuaathony
caaeneiuie (naycaiiuie). Aiaeiae/iia ioiioaiea ninaaenoaa nouanoaoao
iaaeaeo 1-eoaaie, a ?acoeueoaoa neeaeaaiey eioi?uo iieo/aaony 2-eoa.

0o01

0o11 – 0oo1 (**)

A i?iaeieaeaiee aiaeiaee aeaa r-eoaa iacuaathony ninaaeieie anee iie
ioee/athony oieueei ii iaeiie (anoanoaaiii caaeneiie) eii?aeeiaoa. r-eoa
niaea?aeeo r iacaaeneiuo e n-r caaeneiuo eii?aeeiao. A ?acoeueoaoa
neeaeaaiey 2-o ninaaeieo r-eoaia ia?acoaony (r+1)-eoa niaea?aeauee r+1
iacaaeneioth eii?aeeiaoo.

Iia?aoeey neeaeaaiey iaae eoaaie niioaaonoaoao i?eiaiaieth caeiia
neeaeaaiey e eiiuthieoeaiui oa?iai, ioiaeaeanoaeyaiui n yoeie eoaaie.

I?eia? : aeey neeaeaaiey (*)

_ _ _ _ _ _ _

o1o2o3o4( o1o2o3o4= o1o3o4

(0101) (0001) (0o01)

_ _ _ _

aeey (**) o1o3o4( o1o3o4= o1o4

(0o01) (0o11) (0oo1)

Ii?aaeaeaiey.

Eoae/aneei eiiieaenii K0(f) aoeaaie ooieoeee f iacuaaaony iiiaeanoai
0-eoaia yoie ooieoeee. A iauai neo/aa eoae/aneei eiiieaenii Kr(f)
aoeaaie ooieoeee f iacuaaaony iauaaeaiaiea iiiaeanoa eoaia anao
?acia?iinoae yoie ooieoeee

m

k(f)=UKr(f) m-iaeneiaeueiay ?acia?iinoue eoaia ooieoeee f.

r=0

I?eia? iieo/aiey eoae/aneeo eiiieaenia

f3(x)=V(1,2,3,6,7) |001 (1) |0x1 (1-3) (1)

(f=1) |010 (2) |01x (2-3) (2)

K0(f)=|011 (3) K1(f)=|x10 (2-4) (3)
K2(f)=|x1x (2-5)

|110 (4)
|x11 (3-5) (4)

|111 (5) |11x (4-5)
(5)

K3(f)=ionoiio iiiaeanoao

K(f)=K0(f)UK1(f)UK2(f)

Aeey iieo/aiey eoae/aneiai eiiieaena K(f) iaiaoiaeeii i?iaanoe
anaaiciiaeiua iia?aoeee neeaeaaiey iaae 0-eoaaie, 1-eoaaie e o.ae. aei
oao ii? iiea ia i/a?aaeiii oaaa ia iieo/eony Kr+1(f)=ionoiio iiiaeanoao.
I?e neeaeaaiee 1-eoaia 2-eoau i?aaenoaaeaiu a 2-o yecaiiey?ao eae
?acoeueoaou neeaeaaiey 2-o ?acee/iuo ia? 1-eoaia.

?ani?ino?aiyy yoio i?eioeei iiaeii ooaa?aeaeaoue, /oi r-eoau eae
?acoeueoao neeaeaaiey (r-1)-eoaia iieo/athony a r-e?aoiii eiee/anoaa
yecaiiey?ia.

Eoa, aoiaeyuee a ninoaa eoae/aneiai eiiieaena K(f) iacuaaaony
iaeneiaeueiui, anee ii ia anooiaao ie a iaeio iia?aoeeth neeaeaaiey.

A i?eaaaeaiiii i?eia?a iaeneiaeueiuie eoaaie yaeythony o1o e 0o17.

Aaiiao?e/aneay eioa?i?aoaoeey eoaia iaeie ?acia?iinoe. A?aoe/aneia
i?aaenoaaeaiea aoeaauo ooieoeee.

Iiaeiaiue iiaeoiae iineo ia?aie/aiiue oa?aeoa? e eae i?aaeei yaeyaony
iaaeyaeiui aeey aoeaauo ooieoeee io 2-o e 3-o ia?aiaiiuo.

F3(x)=V(1,2,3,6,7)

(f=1)

Aaiiao?e/aneei ianoii 0-eoaa yaeyaony oi/ea, i?aaenoaaeythuay
nouanoaaiioth aa?oeio.

Aeaa ninaaeieo 0-eoaa yaeythony eiioeaie eaeiai-eeai ?aa?a.

Aaiiao?e/aneei ianoii 1-eoaa yaeyaony ?aa?i, caiueaaiia neeaeaathueieny
0-eoaaie, ia?acothueie aeaiiue 1-eoa.

Aeaa ia?aeeaeueiuo ?aa?a, ia?acothueo a?aiue, yaeythony ia?acaie
neeaeaathueony 1-eoaia. A niioaaonoaee n yoei aaiiao?e/aneie
eioa?i?aoaoeeae 2-eoaa yaeyaony a?aiue, ia?acoaiay ia?ie ia?aeeaeueiuo
?aaa?. Oae eae ethaoth a?aiue iiaeii ii?aaeaeeoue iaeiie ec ia?
ia?aeeaeueiuo ?aaa?, 2-eoa iiaeao auoue iieo/ai eae ?acoeueoao
neeaeaaiey aeaoo ?acee/iuo ia? 1-eoaia, oi anoue i?aaenoaaeyaony a aeaoo
yecaiiey?ao.

Aaiiao?e/aneei ia?acii 3-eoaa iiaeii n/eoaoue 3-o ia?iue eoa. Oae eae
ii iiaeao auoue ia?aciaai 3-iy niiniaaie eae ia?a ia?aeeaeueiuo a?aiae,
oi i?e neeaeaaiee ii iieo/aaony a o?ao yecaiiey?ao.

Iie?uoey aoeaauo ooieoeee.

Iaaeaeo eoaaie ?acee/iie ?acia?iinoe, aoiaeyueo a eoae/aneee eiiieaen
K(f), nouanoaoao ioiioaiea aeeth/aiey eee iie?uoey. I?eiyoi aiai?eoue,
/oi eoa A iaiueoae ?acia?iinoe iie?uaaaony eoaii A aieueoae ?acia?iinoe,
anee eoa A aeeth/aaony a eoa A. Yoi icia/aao, /oi i?e ia?aciaaiee eoaa A
oioy au a iaeiii neeaeaaiee o/aanoaoao eoa A.

Ioiioaiea aeeth/aiey (iie?uoey) iaaeaeo eoaaie i?eiyoi iaicia/aoue A(A.
A oai?ee iiiaeanoa ioiioaiea aeeth/aiey naycuaaao iaaeaeo niaie
iaeioi?ia iiiaeanoai e aai iiaeiiiaeanoaa.

Aeey ?anniio?aiiiai i?eia?a ioiioaiey aeeth/aiey eiatho ianoi iaaeaeo
001(0o1; 011(x11(x1x… ethaie 1-eoa iie?uaaao 2 0-eoaa, 2-eoa – 4
0-eoaa e 2 1-eoaa, 3-eoa iie?uaaao 8 0-eoaia, 12 1-eoaia e 6 2-eoaia.

Iie?uoeai aoeaaie ooieoeee f iacuaaaony oaeia iiaeiiiaeanoai eoaia ec
eoae/aneiai eiiieaena K(f), eioi?ia iie?uaaao ana nouanoaaiiua aa?oeiu
ooieoeee.

A nayce n oai, /oi ethaiio eoao eiiieaena K(f) iiaeii iinoaaeoue a
niioaaonoaea eiiuthieoeaiue oa?i, aeey ethaiai iie?uoey iiaeii
i?aaenoaaeoue iaeioi?oth AeIO aoeaaie ooieoeee.

*anoiui neo/aai iie?uoey aoeaaie ooieoeee yaeyaony eoae/aneee eiiieaen
K0(f), iie?uoea c0(f)=K0(f). Yoiio iie?uoeth niioaaonoaoao EAeIO.

Aeey i?eia?a iie?uoeai yaeyaony oaeaea

|0x1

|01x

c1(f)=K1(f)=|x10

|x11

|11x

yoiio iie?uoeth niioaaonoaoao AeIO aeaea

_ _ _ _ _ _ _ _ _ _

f=x1x3vx1x2vx2x3vx2x3vx1x2 i?eaaaeaiiay AeIO ia yaeyaony ieieiaeueiie.

A ea/anoaa ieieiaeueiie aua iaeiiai iie?uoey iiaeii eniieueciaaoue
iiiaeanoai iaeneiaeueiuo eoaia

|0x1

c2(f)=|x1x

Aeaenoaeoaeueii, eoa 0o1 iie?uaaao nouanoaaiiua aa?oeiu 0o1((001, 011),
a eoa x1x((010, 011, 110, 111).

Iiiaeanoai iaeneiaeueiuo eoaia aoeaaie ooieoeee anaaaea yaeyaony aa
iie?uoeai.

Iie?uoea c2(f) niioaaonoaoao AeIO aeaea o1o3vx2. Yoa AeIO yaeyaony
ieieiaeueiie. Iie?uoea aoeaaie ooieoeee, eioi?ia niioaaonoaoao
ieieiaeueiie AeIO iacuaaaony ieieiaeueiui iie?uoeai.

Ieieiaeueiia iie?uoea aeieaeii ninoiyoue oieueei ec iaeneiaeueiuo
eoaia.

A /anoiii neo/aa ana iiiaeanoai iaeneiaeueiuo eoaia yaeyaony
ieieiaeueiui iie?uoeai. Yoi ni?aaaaeeeai aeey iaoaai i?eia?a. A iauai
neo/aa iiiaeanoai iaeneiaeueiuo eoaia yaeyaony ecauoi/iui e aeey
iieo/aiey ieieiaeueiiai iie?uoey aeinoaoi/ii acyoue iaeioi?ia aai
iiaeiiiaeanoai.

I?eia? : f3(x)=V(0,1,4,6,7)

(f=1)

|000 (1) |00x (1-2)

|001 (2) |x00 (1-3)

K0(f)=|100 (3) K1(f)=|1×0 (3-4)

|110 (4) |11x (4-5)

|111 (5)

Aeey aeaiiiai i?eia?a iiiaeanoai iaeneiaeueiuo eoaia niaiaaeaao n
eiiieaenii K1(f). Z(f)=K1(f)

Ieieiaeueiuie iie?uoeyie yaeythony

|00x |00x

n1(f)=|11x c2(f)=|11x

|x00 |1×0

Ec aiaeeca iie?uoey nouanoaaiiuo aa?oei iaeneiaeueiuie eoaaie ec
eiiieaena K1(f) neaaeoao :

Eoa 00o aeieaeai iaycaoaeueii aeeth/aoueny a iie?uoea, oae eae oieueei
ii iie?uaaao nouanoaaiioth aa?oeio 001, aiaeiae/ii 11o iie?uaaao 111.

Iiiaeanoai iaeneiaeueiuo eoaia aac eioi?uo ia iiaeao auoue ia?aciaaii
iie?uoea aoeaaie ooieoeee iacuaaaony yae?ii iie?uoey T(f)=|00x

|11x

Oae eae yae?ii iie?uoey e?iia nouanoaaiiuo aa?oei 001 e 111
iie?uaathony oaeaea nouanoaaiiua aa?oeiu 000 e 110, oi ia iie?uoie
yae?ii inoaaony oieueei nouanoaaiiay aa?oeia 100. Aeey aa iie?uoey
aeinoaoi/ii acyoue 1 ec inoaaoeony iaeneiaeueiuo eoaia (o00 eee 1o0).

Auaiaeu :

Caaea/a iieo/aiey ieieiaeueiie AeIO naiaeeony e caaea/a iieo/aiey
ieieiaeueiiai iie?uoey.

Iieo/aiea ieieiaeueiiai iie?uoey ?aaeecoaony a oaeii ii?yaeea : a)
Iaoiaeeony iiiaeanoai iaeneiaeueiuo eoaia a) Auaeaeyaony yae?i iie?uoey
a) Ec iaeneiaeueiuo eoaia, ia aioaaeoeo a yae?i, auae?aaony oaeia
ieieiaeueiia iiaeiiiaeanoai, eioi?ia iie?uaaao nouanoaaiiua aa?oeiu, ia
iie?uoua yae?ii.

Oeaia iie?uoey.

Oeaia r-eoaa i?aaenoaaeyao niaie eiee/anoai ianaycaiiuo eii?aeeiao.
Sr=n*r

Aeey ioeaiee ea/anoaa iie?uoey eniieuecotho aeaa aeaea oeaiu iie?uoey :

m

Sa=(SrNr, aaea Nr – eiee/anoai r-eoaia, aoiaeyueo a ii-

r=0

e?uoea, m – iaeneiaeueiay ?acia?iinoue eoaa. Oeaia Sa i?aaenoaaeyao
niaie noiio oeai eoaia, aoiaeyueo a iie?uoea.

Sb=Sa+k, aaea k – eiee/anoai eoaia, aoiaeyueo a iie?uoea

m m

Sa =((n-r) Nr ; Sb=((n-r)(Nr+1)

r=0 r=0

Iiae ieieiaeueiui iie?uoeai iiieiatho iie?uoea, iaeaaeathuaa
ieieiaeueiie oeaiie Sa ii n?aaiaieth n ethaui ae?oaei iie?uoeai yoie
ooieoeee.

Iiaeii iieacaoue, /oi iie?uoea, iaeaaeathuaa ieieiaeueiie oeaiie Sa
iaeaaeaao oaeaea e ieieiaeueiie oeaiie Sb.

I?eia? : f3(x)=V(0,1,4,6,7)

(f=1)

C0(f)=K0(f) ; Sa=5*3=15 ; Sb=Sa+5=20

C1(f)=K1(f) ; Sa=4*2=8 ; Sb=Sa+4=12

Cmin(f) : Sa=3*2=6 ; Sb=9

Oeaia iie?uoey Sa i?aaenoaaeyao niaie eiee/anoai aoea, aoiaeyueo a
AeIO, eioi?ay niioaaonoaoao aeaiiiio iie?uoeth.

Oeaia Sb i?aaenoaaeyao aeey AeIO noiio eiee/anoaa aoea e eiee/anoaa
oa?iia.

Oeaia iie?uoey oi?ioi niaeanoaony n oeaiie noaiu ii Eaaeio, eioi?ay
no?ieony ii ii?iaeueiie oi?ia, niioaaonoaothuae yoiio iie?uoeth.

Aeey i?eaaaeaiiie noaiu oeaia ii Eaaeio SQ=9=Sb (9-/enei aoiaeia).

A i?eioeeia, iaaeaeo SQ e oeaiaie Sa e Sb nouanoaoao niioiioaiea Sa (
SQ ( Sb Yoi ia?aaainoai eiaao ianoi i?e neaaeothueo aeiiouaieyo ii
eiiaeiaoeeiiiie noaia :

Noaia no?ieony ii ii?iaeueiie oi?ia (AeIO eee EIO).

Noaia no?ieony ia yeaiaioao aoeaaiai aacena (E, EEE).

Ia aoiaeu noaiu iiaeii iiaeaaaoue eae i?yiua, oae e eiaa?niua cia/aiey
aoiaeiuo ia?aiaiiuo, i?aaenoaaeythuea niaie cia/aiey a?aoiaioia aoeaaie
ooieoeee (noaia n ia?aoaciuie aoiaeaie). Yeaiaiou IA (eiaa?oi?a a noaia
ionoonoaotho.

Ioeaaia iie?uoea aoeaaie ooieoeee e iieo/aiea ieieiaeueiie EIO.

Auoa auei ?anniio?aii iie?uoea aoeaaie ooieoeee ia iaai?ao a?aoiaioia
aeey eioi?uo ooieoeey ?aaia aaeeieoea.

Oaeea iie?uoey iiaeii iacaaoue aaeeie/iuie. Ia?yaeo n aaeeie/iuie
iie?uoeyie nouanoaotho e ioeaaua, aeey eioi?uo iie?uaathony iaai?u
a?aoiaioia, ia eioi?uo ooieoeey ?aaia ioeth, oi anoue iie?uoea
?aaeecoaony aeey nouanoaaiiuo aa?oei, ii ia naiie ooieoeee, a aa
io?eoeaiey (eiaa?nee).

Ioeaaia iie?uoea no?ieony oaeaea eae e aaeeie/iia, ii oieueei aeey
io?eoeaiey enoiaeiie ooieoeee.

f3(x)=V(0,1,4,6,7) f3(x)=&(2,3,5)

(f=1) (f=0) _ |010

K0( f )=|011

_ _ |101

C0( f )= K0( f ) Sa=9 Sb=12

_ _ _

K1( f )=|01x Z( f )=Cmin( f )=|01x Sa=5 Sb=7

|101

Oeaia ieieiaeueiiai ioeaaiai iie?uoey ieacaeanue iaiueoa oeaiu
ieieiaeueiiai aaeeie/iiai iie?uoey.

Oae eae ca?aiaa i?aaeneacaoue iaaiciiaeii, eaeia ec ieieiaeueiuo
iie?uoee aeaiiie ooieoeee, aaeeie/iia eee ioeaaia, aoaeao eiaoue
iaiueooth oeaio, oi aeey iino?iaiey noaiu, iaeaaeathuae ieieiaeueiie
oeaiie ii Eaaeio, oeaeaniia?acii ?aoaoue caaea/o ieieicaoeee a ioiioaiee
iaieo iie?uoee.

Eiieeeaiou aoeaaie ooieoeee.

Nenoaiu eiieeeaio.

?aoaiea caaea/e ieieiecaoeee aoeaaie ooieoeee iaoiaeii Eaaeia e
oniaa?oainoaiaaiiui iaoiaeii Eaaeia-IaeEeannee aace?oaony ia iiiyoeyo
eiieeeaio e eo nenoai.

Ii?aaeaeaiea : Aoeaaa ooieoeey g(x) iacuaaaony eiieeeaioie aoeaaie
ooieoeee f(x), anee aeey ethaiai iaai?a a?aoiaioia, ia eioi?uo g(x)=1,
f(x) oaeaea ?aaia aaeeieoea.

~ ~ ~

g( x )=1 => f( x )=1, aaea o – iaeioi?ue iaai? a?aoiaioia.

Naienoaa eiieeeaio :

Iaaeaeo eiieeeaioie e naiie ooieoeeae nouanoaoao ioiioaiea aeeth/aiey
g(x)(f(x).

Iiaeii ooaa?aeaeaoue, /oi aeey ethaiai iaai?a a?aoiaioia, ia eioi?ii
ooieoeey ?aaia ioeth, aa eiieeeaioa oaeaea ?aaia ioeth.

Anee g(x) e ((x) yaeythony eiieeeaioaie ooieoeee f(x), oi eo
aeecuthieoeey oaeaea yaeyaony eiieeeaioie yoie ooieoeee.

I?inoaeoeie i?eia?aie eiieeeaio iiaoo neoaeeoue eiiuthieoeaiua oa?iu,
aoiaeyuea a AeIO aeaiiie ooieoeee.

I?eia? : aeey f3(x)=V(0,1,4,6,7) (#)

(f=1) _ _ _

eiieeeaioaie yaeythony o1o2o3; o1o2o3; o1o2;…

I?iecaieueiay aeecuthieoeey yoeo oa?iia oaeaea yaeyaony eiieeeaioie
ooieoeee.

Ii?aaeaeaiea : I?inoie (ia?ae/iie) eiieeeaioie aoeaaie ooieoeee
iacuaaaony eiiuthieoeaiue oa?i, eioi?ue nai yaeyaony eiieeeaioie yoie
ooieoeee, ii ieeaeay aai nianoaaiiay /anoue oaea ia yaeyaony eiieeeaioie
yoie ooieoeee.

Iiae nianoaaiiie /anoueth oa?ia iiieiaaony iiaue oa?i, iieo/aiiue ec
enoiaeiiai, iooai au/a?eeaaiey i?iecaieueiiai /enea aoea.

Aeey aeaiiiai i?eia?a ooieoeee (#) i?inouie eiieeeaioaie yaeythony :
_ _ _

o1o2o3; o1o2o3; o1o2;…

Iiiaeanoao i?inouo eiieeeaio iiaeii iinoaaeoue a niioaaonoaea
iiiaeanoai iaeneiaeueiuo eoaia.

Ii?aaeaeaiea : Aeecuthieoeey anao i?inouo eiieeeaio aoeaaie ooieoeee
i?aaenoaaeyao niaie AeIO yoie ooieoeee, eioi?ay iacuaaaony nie?auaiiie –
NAeIO.

Aeey ooieoeee (#) ec i?eaaaeaiiiai i?eia?a

_ _ _ _ _

NAeIO : y= o1o2v o1o2vo2o3v o1o3

Iiiyoea «nie?auaiiia» i?enaiaii AeIO a oii niunea, /oi iia, eae
i?aaeei, niaea?aeeo iaiueoaa eiee/anoai aoea e oa?iia ii n?aaiaieth n
EAeIO. Aeey iaoaai i?eia?a EAeIO niaea?aeeo 15 aoea e 5 oa?iia, a NAeIO
– 8 aoea e 4 oa?ia.

Aiaeiaey iaaeaeo eiieeeaioaie e eoae/aneei i?aaenoaaeaieai Aoeaaie
ooieoeee

Ethaiio eoao ec E(f) iiaeii iinoaaeoue a niioaaonoaea eiiuthieoeaiue
oa?i eioi?ue iiaeii ?anniao?eaaoue eae eiieeeaioo aoeaaie ooieoeee
.Ethaie i?inoie eiieeeaioa aoeaaie ooieoeee niioaaonoaoao iaeneiaeueiue
eoa ,e a naith i/a?aaeue iiiaeanoai anao i?inouo eiieeeaio niioaaonoaoao
iiiaeanoao Z(f) anao iaeneiaeueiuo eoaia E(f).

Oaeei ia?acii iiaeii i?iaanoe iaeioi?oth aiaeiaeth iaaeaeo nie?auaiiie
NAeIO e Z(f).

A ioiioaiee eiieeeaio aoeaaie ooieoeee oaeaea eae e a ioiioaiee eoaia
niioaaonoaothueo ei nouanoaoao ioiioaiea iie?uoey.

I?eiyoi n/eoaoue ,/oi eiieeeaioa iie?uaaao iaeioi?oth nouanoaaiioth
aa?oeio eee a iauai neo/aa iaeioi?ue eoa ec E(f) ,anee cia/aiea
eiieeeaiou ia iaai?a a?aoiaioia i?aaenoaaeythuai aeaiioth nouanoaaiioth
aa?oeio ?aaii 1 eee a iauai neo/aa cia/aiea eiieeeaiou ?aaii 1 aeey anao
nouanoaaiiuo aa?oei iie?uaaaiuo eoaii ec E(f).

I?EIA? : eiieeeaioa o1o2 iie?uaaao nouanoaaiiua aa?oeiu (110,111) e a
naith i/a?aaeue iie?uaaao eoa 11o.

Ii?aaeaeaiea :iiiaeanoai eiieeeaio aoeaaie ooieoeee ia?acoao iieioth
nenoaio eiieeeaio ,anee ethaay nouanoaaiiay aa?oeia aoeaaie ooieoeee
iie?uaaaony oioy au iaeiie eiieeeaioie yoiai iiiaeanoaa.

Anee n/eoaoue ,/oi a iieioth nenoaio eiieeeaio aeeth/athony eiieeeaiou
oieueei a aeaea eiiuthieoeaiuo oa?iia e ia aeeth/aao eiieeeaiou a aeaea
aeecuthieoeaiuo oa?iia ,oi iieiie nenoaia eiieeeaio iiaeii iinoaaeoue a
niioaaonoaea iaeioi?ia iiiaeanoai eoaia ec E(f) ia?acothueo iie?uoea
aoeaaie ooieoeee f .

Oae iai?eia? ,eoaai ec eoae/aneiai eiiieaena E((f) niioaaonoaoao iieiay
nenoaia eiieeeaio ,i?aaenoaaeythuay niaie iiiaeanoai eiinoeooaio 1
aeaiiie ooieoeee f. A naith i/a?aaeue iiiaeanoao iaeneiaeueiuo eoaia
Z(f) ,anoanoaaiii ia?acothueo iie?uoea aoeaaie ooieoeee ,niioaaonoaoao
iieiay nenoaia i?inouo eiieeeaio.

Ii?aaeaeaiea :nenoaia i?inouo eiieeeaio iacuaaaony i?eaaaeaiiie ,anee
iia yaeyaony iieiie ,a ieeaeay aa nianoaaiiay /anoue oaea ia ia?acoao
iieioth nenoaio eiieeeaio.

3 (*)

nenoaia i?inouo eiieeeaio

3 (

2(

Ii?aaeaeaiea: Aeecuthieoeey anao i?inouo eiieeeaio ,ia?acothueo
iaeioi?oth i?eaaaeaiioth nenoaio iacuaaaony ooieeiaie AeIO aoeaaie
ooieoeee eee OAeIO

Aeey ooieoeee (*) nouanoaotho aeaa OAeIO

3

3

A aeaiiii neo/aa iie niaiaaeatho n ieieiaeueiie AeIO. Ii a iauai neo/aa
yoi ooaa?aeaeaiea ia ni?aaaaeeeai. O.a. ieieiaeueiay AeIO iaycaoaeueii
yaeyaony OAeIO ii ia ethaay OAeIO yaeyaony IAeIO. Oaeei ia?acii
iiiaeanoai IAeIO yaeyaony iiaeiiiaeanoaii OAeIO.

Ii?aaeaeaiea: i?inoay eiieeeaioa aoeaaie ooieoeee iacuaaaony
nouanoaaiiie anee iia e oieueei iia iie?uaaao iaeioi?oth nouanoaaiioth
aa?oeio yoie ooieoeee.

Iiiaeanoai nouanoaaiiuo eiieeeaio niioaaonoaoao iaeneiaeueiui eoaai
ia?acothuei yae?i iie?uoey.

IINEAAeIAAOAEUeIINOUe aeaenoaee aeey ?aoaiey eaiiie/aneie caaea/e
ieieiecaoeee iaoiaeii Eaaeia-Iae-Eeanee.

1)Iaoiaeaeaiea iiiaeanoaa iaeneiaeueiuo eoaia eee i?inouo eiieeeaio
ooieoeee.

2)Auaeaeaiea yae?a iie?uoey.

3)Aeiiieiaiea iiiaeanoaa eoaia ,i?eiaaeeaaeaueo yae?o iie?uoey oaeei
ieieiaeueiui iiaeiiiaeanoaii ec iaeneiaeueiuo eoaia ,ia aoiaeyueo a
yae?i iie?uoey ,aeey iieo/aiey iie?uoey n ieieiaeueiie oeaiie.

N oi/ee c?aiey iineaaeiaaoaeueiiai i?aia?aciaaiey AeIO aoeaaie ooieoeee
n oeaeueth eo oi?iuaiey eaiiie/aneay caaea/a ieieiecaoeee iiaeao auoue
i?aaenoaaeaia a aeaea EAeIO.

EAeIO(NAeIO((OAeIO(((IAeIO(

?ani?ino?aiaiea oa?ieiieiaee a ioiioaiee ioeaaiai iie?uoey
ii?aaeaeyaony ia iiiyoee eiieeeaioa eae niioaaonoaea eiieeeaioa e ia
nenoaia eiieeeaio.

I?EIA?: (ieieiecaoeey aoeaaie ooieoeee iaoiaeii Eaaeia-Iae-Eeanee)

1) f4(x)=V(0,1,5,7,8,10,12,14,15)

(f=1)

f4(x)=&(2,3,4,6,9,11,13)

(f=0)

Ia yoaia iieo/aiey iiiaeanoaa iaeneiaeueiuo eoaia oeaeaniia?acii
?acaeaeeoue iiiaeanoai iieue – eoaia (E((f)) ia ?yae iiaeiiiaeanoa
,ioee/athueony eiee/anoaii aaeeieoe .

A iia?aoeeth neeaeaaiey a yoii neo/aa iiaoo anooiaoue oieueei eoau
,ioiinyueany e ninaaeiei iiaeiiiaeanoaai ,oi anoue ioee/athueany ia
aaeeieoeo

( 000X (

( X000 ( K((f)=C((f)

( 0X01 (

Z(f)= ( 01X1 ( S(=36

( X111 ( S(=45

( 111X (

( 1XX0 (

K1(f)=C1(f) S(=10(3=30

S(=40

Z(f)=C3(f) S(=20

S(=27

I?e ieieiecaoeee ia iieiinoueth ii?aaeaeaiiie aoeaaie ooieoeee
iiiaeanoai iaeneiaeueiuo eoaia ii?aaeaeyaony ia iauaaeeiaiee iiiaeanoaa
nouanoaaiiuo aa?oei e aac?acee/iuo iaai?ia a oeaeyo iieo/aiey eoaia
iaeaieueoae ?acia?iinoe .

Ii?aaeaeaiea yae?a iie?uoey .

Auiieiaiea yoiai yoaia ?aaeecoaony n iiiiuueth oaaeeoeu iie?uoee .

Kaaeaeay no?iea oaaeeoeu – iaeneiaeueiue eoa(i?inoay eiieeeaioa).

Eaaeaeue noieaaoe – nouanoaaiiay aa?oeia aoeaaie ooieoeee (aac?acee/iua
iaai?u ia aeeth/athony).

Yeaiaiou yoie oaaeeoeu io?aaeatho ioiioaiea iie?uoey ,oi anoue ia
ia?ana/aiee i-ie no?iee e j-iai noieaoea noaaeony iaeioi?ay ioiaoea a
oii neo/aa anee i-ue iaeneiaeueiue eoa iie?uaaao j-oth aa?oeio .

Oaaeeoeo iie?uoee eiiaaea iacuaatho eiieeeaioiie oaaeeoeae n o/aoii
oiai ,/oi eaaeaeue iaeneiaeueiue eoa niioaaonoaoao i?inoie eiieeeaioa a
nouanoaaiiua aa?oeiu eiinoeooaioai

aaeeieoeu(ioey).

Nouanoaaiiua aa?oeiu

iaen.

Eoau

0000 0001 0101 0111 1000 1010 1100 1110 1111

A 000X * *

| | | |

B X000 *

| * | | |

C 0X01

* *

| | | |

D 01X1

* * | | | |

E X111

* | | | | *

F 111X

| | | | *

1XX0

| * | * | * | *

a b c d | | | |
e

Aeey iieiinoueth ii?aaeaeaiiie aoeaaie ooieoeee eiee/anoai iaoie a
eaaeaeie no?iea ?aaii /eneo iieue – eoaia iie?uaaaiuo eoaii aeaiiie
no?iee .Aeey ia iieiinoueth ii?aaeaeaiiie ooieoeee eiee/anoai iaoie a
no?iea caaeneo io eiee/anoaa aac?acee/iuo iaai?ia iie?uaaaiuo aeaiiui
eoaii .Aeey iaoiaeaeaiey eoaia ,i?eiaaeeaaeaueo yae?o iie?uoey a
oaaeeoea euoony noieaoeu n aaeeinoaaiiie iaoeie .No?iea ,eioi?ie
i?eiaaeeaaeeo yoa iaoea ii?aaeaeyao eoa yae?a .

O(f)=(1XX0(

Ii?aaeaeaiea iiiaeanoaa ieieiaeueiuo iie?uoee .

Ia yoii yoaia ec iiiaeanoaa iaeneiaeueiuo eoaia ia i?eiaaeeaaeaueo
yae?o iie?uoey ,auaeaeythony oaeea ieieiaeueiua iiaeiiiaeanoaa ,n
iiiiuueth eaaeaeiai ec eioi?uo iie?uaathony inoaaoeany aa?oeiu (ia
iie?uoua yae?ii) .

?aaeecaoeeth yoiai yoaia oeaeaniia?acii i?iecaiaeeoue n eniieueciaaieai
oi?iuaiiie oaaeeoeu.

A oi?iuaiiie oaaeeoea au/a?eioou ana eoau i?eiaaeeaaeauea yae?o e
aa?oeiu iie?uaaaiua yae?ii.

Aeey ?aoaiey caaea/e 3-ai yoaia iiaeii eniieueciaaoue iaeei ec 3-o
iaoiaeia eee eo eiiaeiaoeeth:

Iaoiae i?inoiai ia?aai?a

Iaoiae Iao?eea

Aeaeueiaeoaa oi?iuaiea.

1)Ia aeaiiii yoaia oeaeaniia?acii aaanoe iaicia/aiea iaeneiaeueiuo
eoaia e nouanoaaiiuo aa?oei.

Iaeneiaeueiua eoau iaicia/aiu a oaaeeoea A…F

1-ue iaoiae oeaeaniia?acii i?eiaiyoue aeey oi?iuaiiie oaaeeoeu
iaaieueoiai iauaia .Yoio iaoiae ia aeaao aa?aioee iieo/aiey anao
iaeneiaeueiuo iie?uoee.

Aeey iaoaai i?eia?a ana eoau aoiaeyuea a oi?iuaiioth oaaeeoeo iie?uoee
iaeaaeatho iaeiie ?acia?iinoueth(oi anoue iaiaoiaeeii aua?aoue
ieieiaeueiia eiee/anoai yoeo eoaia aeey iie?uoey anao inoaaoeony
nouanoaaiiuo aa?oei).

Ec oaaeeoeu aeaeii ,/oi ieieiaeueiia /enei eoaia ?aaii o?ai.E
aiciiaeiui aa?eaioai iie?uoee ioiinyony:

( T( ( T(

C min1(f)= (A( C min 1(f)= (B( ,…

(C( (C(

(E ( (E (

2)Aeinoieinoai yoiai iaoiaea-iieo/aiea anao ieieiaeueiuo iie?uoee.

Iaoiae aace?oaony ia ninoaaeaiee eiae/aneiai au?aaeaiey ,
i?aaenoaaeythuaai niaie oneiaea iie?uoey anao aa?oei ec oi?iuaiiie
oaaeeoeu iie?uoee e i?aia?aciaaiey yoiai au?aaeaiey .

Y=(AvB)(AvC)(CvD)(DvE)(EvF)=(AvBC)(DvCE)(EvF)=

=(AvBC)(DEvCEvDFvCEF)=

=(ADEvACEvADFvBCDEvBCEvBCDF)

Eaaeaeue ec iyoe eiiuthieoeaiuo oa?iia niioaaonoaoao iie?uoeth aoeaaie
ooieoeee(n o/aoii aeiiieiaiey yae?ii),eaaeaeiio ec eioi?uo iiaeii
iinoaaeoue a niioaaonoaea ooieeiaoth AeIO.

Iineaaeiee oa?i ia niioaaonoaoao ieieiaeueiiio iie?uoeth ,oi anoue
aeaiiay ooieoeey eiaao /aou?a ieieiaeueiuo iie?uoey.

Aeaeueiaeoaa oi?iuaiea ninoieo a i?eiaiaiea aeaoo iia?aoeee :
a)Au/a?eeaaiea eeoieo no?ie.

a)Au/a?eeaaiea eeoieo noieaoeia.

Anee iiiaeanoai iaoie i-e no?iee yaeyaony iiaeiiiaeanoaii iaoie j-e
no?iee e eoa i eiaao iaaieueooth ?acia?iinoue, /ai eoa j, oi ec oaaeeoeu
iiaeii au/a?eiooue i-th no?ieo oae eae nouanoaaiiua aa?oeiu iie?uaaaiua
i-i eoaii aoaeoo n aa?aioeae iie?uou j-i eoaii.

A aeaeueiaeoai ?aeiiaiaeoaony iino?ieoue iiaoth oi?ioaiioth oaaeeoeo.

A ioiioaiee iiaie oaaeeoeu iiaeii eniieueciaaoue iaeei ec o?ao
iaoiaeia: 1) Iaoiae i?inoiai ia?aai?a.

2)Iaoiae Iao?eea.

3)Aeaeueiaeoaa oi?iuaiea.

Ooieoeeiiaeueiay iieiioa nenoaiu aoeaauo ooieoeee.

Nenoaia aoeaauo ooieoeee S=(y1,y2,…,ym(iacuaaaony ooieoeeiiaeueii
iieiie ,anee n iiiiuueth ooieoeee yoie nenoaiu iiaeii au?aceoue ethaoth
neieue oaiaeii neiaeioth aoeaao ooieoeeth n eniieueciaaieai iaoiaea
noia?iiceoeee, aiciiaeii iiiaie?aoii.

Iiae noia?iiceoeeae a ioiioaiee aoeaauo ooieoeee iiieiaaony
iiaenoaiiaea iaeieo ooieoeee a ae?oaea aianoi eo a?aoiaioa.

I?eia?aie iieiuo nenoai yaeythony :

1)S1 =((,&,(((aoeaa aacen)

Iainiiaaiiinoue ooaa?aeaeaiey i ooieoeeiiaeueiie iieiioa yoie nenoaiu
aace?oaony ia aiciiaeiinoe i?aaenoaaeaiey ethaie aoeaaie ooieoeee a
ii?iaeueiie oi?ia ,eioi?ay yaeyaony eiiaeiaoeeae iia?aoeee io?eoeaiey
,eiiuthieoeee e aeecuthieoeee, i?eiaieoaeueii e a?aoiaioo yoie ooieoeee.

Nenoaia S1 =((,&,(( yaeyaony ecauoi/iie oae eae ec iaa iiaeii oaeaeeoue
iaeio ec ooieoeee (& eee () aac ia?ooaiey ooieoeeiiaeueiie iieiiou.

Iieo/aaiua i?e yoii nenoaiu S2 ={(,&}(e S3((,&,((iau/ii iacuaatho
nie?auaiiui aoeaaui aacenii .

Iaaeinoathuea iia?aoeee( ( a nenoaia S2 e & a nenoaia S3 ) iiaoo auoue
au?aaeaiu n iiiiuueth neaaenoaee ec caeiiia

____

2

_____

2

Ooieoeeiiaeueiay iieiioa nenoaiu aoeaauo ooieoeee iacuaaaony
ieieiaeueiie ,anee oaeaeaiea ec iaa eaeie-eeai ooieoeee i?eaiaeeo e
ia?ooaieth naienoaa ooieoeeiiaeueiie iieiiou.

Nenoaiu ec iaeiie ooieoeee S4=((no?aeea Ie?na)

S5=|(oo?eo Oaooa?a)

eioi?ua i?eiyoi iacuaaoue oieaa?naeueiui aacenii.

2)Aacen AEaaaeeeia S6= {&, (, 1}

Iiiyoea ooieoeeiiaeueiie iieiiou nenoaiu aoeaauo ooieoeee naycaii n
aiaeiae/iui iiiyoeai aeey nenoaiu eiae/aneeo yeaiaioia.

Yoa naycue caeeth/aaony a neaaeothuai :

Anee eaaeaeie ooieoeee ec iaeioi?ie ooieoeeiiaeueii iieiie nenoaiu
niiinoaaeoue eiae/aneee yeaiaio, ?aaeecothuee yoo ooieoeeth ,oi nenoaia
eiae/aneeo yeaiaioia niioaaonoaothuay iaeioi?ie ooieoeeiiaeueii iieiie
nenoaia aoeaauo ooieoeee anoanoaaiiui ia?acii ieacuaaaony oiaea
ooieoeeiiaeueii iieiie .

Caaea/a neioaca eiiaeiaoeeiiiuo noai n eniieueciaaieai ooieoeeiiaeueii
iieiie nenoaiu eiae/aneeo yeaiaioia iiaeii iino?ieoue eiiaeiaoeeiiioth
noaio ?aaeecothuoth ethaoth iaia?aae caaeaiioth ,neieue oaiaeii
neiaeioth aoeaao ooieoeeth.

Aeieacaoaeuenoai ooieoeeiiaeueiie iieiiou iaeioi?ie nenoaiu

aoeaauo ooieoeee iiaeii inouanoaeyoue iaeiei ec aeaoo niiniaia:

N eniieueciaaieai oai?aiu i ooieoeeiiaeueiie iieiioa .

N eniieueciaaieai eiino?oeoeaiiai iiaeoiaea .

Oai?aia i ooieoeeiiaeueiie iieiioa (Iino – ssaeiineiai).

Aeey oiai, /oiau nenoaia aoeaauo ooieoeee auea ooieoeeiiaeueii iieiie
iaiaoiaeeii e aeinoaoi/ii /oiau iia niaea?aeaea oioy au iaeio ooieoeeth
ia:

cio?aiythuoth eiinoaioo iieue

cio?aiythuoth eiinoaioo aaeeieoea

eeiaeioth ooieoeeth

iiiioiiioth ooieoeeth

naiiaeaienoaaiioth ooieoeeth.

Caia/aoaeueiua eeannu aoeaauo ooieoeee.

Aoeaaa ooieoeey iacuaaaony nio?aiythuae eiinoaioo iieue , anee ia
ioeaaii iaai?a a?aoiaioia iia i?eieiaao cia/aiea ?aaiia ioeth, oi anoue
f(0,0,0,…,0) = 0;

A i?ioeaiii neo/aa ooieoeey ioiineony e eeanno ia cio?aiythueo
eiinoaioo iieue.

E ooieoeeyi ,nio?aiythuei eiinoaioo iieue ioiinyony

f(x1,x2)= x1 v x2

f(x1,x2)= x1 * x2

E ooieoeeyi ia cio?aiythuei eiinoaioo iieue ioiinyony

e f(x1,x2)=x1(x2

2.Aoeaaa ooieoeey iacuaaaony nio?aiythuae eiinoaioo aaeeieoea , anee ia
aaeeie/iii iaai?a a?aoiaioia iia i?eieiaao cia/aiea ?aaiia aaeeieoea,
oi anoue f(1,1,1,…,1)= 1;

A i?ioeaiii neo/aa ooieoeey ioiineony e eeanno ia cio?aiythueo
eiinoaioo aaeeieoea.

E ooieoeeyi ,nio?aiythuei eiinoaioo aaeeieoea ioiinyony

f(x1,x2)= x1 v x2

f(x1,x2)= x1 * x2

E ooieoeeyi ia cio?aiythuei eiinoaioo aaeeieoea ioiinyony

e f(x1,x2)=x1(x2

3. Aoeaaa ooieoeey iacuaaaony eeiaeiie anee iia i?aaenoaaeia iieeiiiii
AEaaaeeeia ia?aie noaiaie.

A aoeaaie aeaaa?a aeieacuaaaony oai?aia i aiciiaeiinoe i?aaenoaaeaiey
ethaie aoeaaie ooieoeee io n ia?aiaiiuo n iiiiuueth iieeiiia AEaaaeeeia
n-ie noaiaie.

A iauai neo/aa iieeiii eiaao aeae :

fn (x) = K0 (K1x1 (…(Kn xn (…

…(Kn+1x1x2 (Kn+2x1x3 (…(Kn+lxn-1xn (…

…(Kn+mx1x2…xn

K0 ,K1 ,Kn+m -yaeythony eiyooeoeeaioaie e i?aaenoaaeytho niaie
eiae/aneea eiinoaiou ioey eee aaeeieoeu.

A aeaaa?a AEaaaeeeia iaeiieiaiiie iieeiii iiaeii n/eoaoue eaiiie/aneie
ii?iaeueiie oi?iie aeey aoeaaie aeaaa?u.

Iieeiii AEaaaeeeia yaeyaony eeiaeiui (1-ie noaiaie) anee ana
eiyooeoeeaiou iauaai iieeiiia ,ia/eiay n Kn+1=Kn+2 =…=Kn+m =0

A ioiioaiee ooieoeee io 2-o ia?aiaiiuo iieeiii AEaaaeeeia eiaao aeae
(eeiaeiue): f2(x)=K0(K1x1(K2x2

I?eia?aie eeiaeiuo ooieoeee yaeythony:

y= x1(x2 (K0=0,K1=K2=1)

_____

y= x1(x2=x1(x2=1(x1(x2 (K0=K1=K2)

=1(x1 (K0=K1=1 ,K2=0)

I?eia?u iaeeiaeiuo ooieoeee:

y= x1(x2

____

y= x1(x2 =x1(x2=1(x1(x2

4.Aoeaaa ooieoeey iacuaaaony iiiioiiiie anee i?e aic?anoaiee iaai?ia
a?aoiaioia iia i?eieiaao iaoauaathuea cia/aiey.

A=(a1,a2,…,an)>B=(b1,b2,…,bn)

f(A)(f(B)

Iaaeaeo iaai?aie a?aoiaioia A e A eiaao ianoi ioiioaiea aic?anoaiey a
oii e oieueei oii neo/aa , anee eiaao ianoi ioiioaiea ia oauaaiey aeey
anao eiiiiiaio yoiai iaai?a:

___

ai(bi (i=1, n )

e ii e?aeiae ia?a aeey iaeiie eiiiiiaiou eiaao ianoi ioiioaiea
aic?anoaiey.

I?eia?u iaai?ia ,aeey eioi?uo eiaao ianoi ioiioaiea aic?anoaiey:
(1011)>(0011)

(1011)>(0001)

(0001)>(0000)

I?eia? ianiiinoaaeiuo iaai?ia (1011) e (0111)

A ioiioaiee ooieoeee io 2-o ia?aiaiiuo ianiiinoaaeiuie yaeythony iaai?u
(01) e (10)

y= x1(x2

5.Aeaa aoeaau ooieoeee fn(x) e gn(x) iacuaathony aeaienoaaiiuie anee
aeey ethauo iaai?ia a?aoiaioia auiieiyaony ?aaainoai

____

i?eieiaao i?ioeaiiieiaeiua cia/aiey .

Aeaa iaai?a a?aoiaioia iacuaathony i?ioeaiiieiaeiuie anee ethaay ec eo
eiiiiiaio i?eieiaao i?ioeaiiieiaeiua cia/aiey.

=(1010)

Aoeaaa ooieoeey iacuaaaony naiiaeaienoaaiiie anee iia yaeyaony
aeaienoaaiiie ii ioiioaieth e naiie naaa oi anoue i?eieiaao
i?ioeaiiieiaeiua cia/aiey ia i?ioeaiiieiaeiuo iaai?ao a?aoiaioia.

I?eia?u ia naiiaeaienoaaiiuo ooieoeee: o=o1(o2

o=o1vo2

o=o1(o2

I?eiaaeeaaeiinoue aaciauo aoeaauo ooieoeee e eiae/aneeo eiinoaio e
caia/aoaeueiui eeannai i?aaenoaaeaia oaaeeoeae.

E0 + nio?aiyao eiinoaioo iieue ,- ia nio?aiyao eiinoaioo iieue

E1 + nio?aiyao eiinoaioo aaeeieoea ,- ia nio?aiyao eiinoaioo

Ee + eeiaeiay ,- iaeeiaeiay

Ei + iiiioiiiay , – ia iiiioiiiay

En + naiiaeaienoaaiiay ,- ia naiiaeaienoaaiiay

o1(o2 + + – + –

o1vo2 + +

+ –

o1(o2 + – + – –

o1(o2 –

+

o1(o2 +

o1(o2 –

o1(o2 –

o1(o2 –

Eiino?oeoeaiue iiaeoiae e aeieacaoaeuenoao ooieoeeiiaeueiie iieiiou
iaeioi?ie nenoaiu aoeaauo ooieoeee.

Iiaeoiae iniiaai ia aeieacaoaeuenoaa ?aaeecoaiinoe ooieoeee aoeaaa
aacena n iiiiuueth ooieoeee yoie nenoaiu.

I?e yoii anoanoaaiii i?aaeiieaaaoue ,e yoi aeaenoaeoaeueii oae, /oi
aoeaa aacen ia?acoao ooieoeeiiaeueii iieioth nenoaio.

I?eia? :S5=(((

_ ____

x =x ( x= x(x

((((

x1(x2 = x1(x2 =( x1(x2)(( x1(x2)

______

2 =( x1(x1)(( x2(x2)

Neioac eiiaeiaoeeiiiuo noai.

Iiiyoea eiae/aneiai yeaiaioa.

Oeiiaua eiae/aneea yeaiaiou e eo iaicia/aiey ia ooieoeeiiaeueiuo noaiao.

Ii?aaeaeaiea: eae i?aaeei ,iiae eiae/aneei yeaiaioii iiieiaaony
eiiaeiaoeeiiiay noaia ,?aaeecothuay iaeioi?oth yeaiaioa?ioth aoeaao
ooieoeeth.

Ethaie eiae/aneee yeaiaio oa?aeoa?ecoaony :

Iaee/eai iaeiiai eee ianeieueeeo aoiaeia ia eioi?ua iiaeathony aoiaeiua
neaiaeu( aoiaeiua ia?aiaiiua).

Iaee/eai auoiaea ,ia eioi?ii oi?ie?oaony auoiaeiie neaiae

(auoiaeiay ia?aiaiiay).

Ii?aaeaeaiiie ooieoeeae ,eioi?ay ioia?aaeaao caaeneiinoue auoiaeiiai
neaiaea io aoiaeiuo.

E iniiaiui oeiai eiae/aneeo yeaiaioia ioiinyony:

Eiaa?oi?( IA)

Aeecuthieoi? (EEE)

Eiiuthieoi? (E)

Aeecuthieoi? n io?eoeaieai (EEE – IA)

Eiiuthieoi? n io?eoeaieai (E – IA)

Eneeth/eoaeueiia EEE

(aaeeie/iue neaiae ia auoiaea eiaao ianoi a oii e oieueei oii neo/aa
anee ia

iaeiii e oieueei iaeiii aoiaea i?enoonoaoao aaeeie/iue neaiae)

Noiiaoi? ii iiaeoeth 2

1)Yeaiaiou 1,2,3 ia?acotho aoeaa aacen.

2)Yeaiaiou 1 e 2 eee 1 e 3 ia?acoao nie?auaiiue(iaiieiue)

aoeaa aacen.

3)Yeaiaiou 4 eee 5 ia?acotho oieaa?naeueiue aacen.

4)Yeaiaiou 3 e 7 ia?acotho aacen AEaaaeeeia.

Ooieoeee yeaiaioia 6 e 7 niaiaaeatho i?e iaee/ee oieueei aeaoo aoiaeia.

Iiiyoea aeaie/iiai neaiaea.

Niiniau aai eiaee?iaaiey.

A nayce n eniieueciaaieai aeaoo cia/aiee eiaeee a eiae/aneeo noaiao eae
aoiaeiua ,oae e auoiaeiua neaiaeu a yoeo noaiao i?aaenoaaeythony n
iiiiuueth oae iacuaaaiiai aeaie/iiai neaiaea – iniaaiiinoueth eioi?iai
yaeyaony iaee/ea aeaoo /aoei ?acee/eiuo o?iaiae ,ioiaeaeanoaeyaiuo n
ioeai e aaeeieoeae.

A caaeneiinoe io oiai ,eaeie o?iaaiue neaiaea niiinoaaeyaony n
eiae/aneei ioeai a eaeie n eiae/aneie aaeeieoeae ?acee/atho aeaa niiniaa
eiaee?iaaiey aeaie/iuo neaiaeia:

1)Iiceoeaiia eiaee?iaaiea (iieiaeeoaeueiia)

aunoee o?iaaiue neaiaea – 1 ,iecoee – 0

2)Iaaaoeaiia eiaee?iaaiea (io?eoeaoaeueiia)

aunoee o?iaaiue neaiaea – 0 ,iecoee – 1

I?e eciaiaiee niiniaa eiaee?iaaiey aeaie/iiai neaiaea ooieoeey iaeiie e
oie aea yeaeo?iiiie noaiu ,?aaeecothuae iaeioi?ue eiae/aneee yeaiaio
iaiyaony ia i?ioeaiiieiaeioth.

Iiiyoea eiae/aneie nenoaiu.

Oeiu eiae/aneeo nenoai.

Eiae/aneay noaia i?aaenoaaeyao niaie niaieoiiinoue eiae/aneeo yeaiaioia
e naycae iaaeaeo ieie.

Niaaeeiaiey eiae/aneeo yeaiaioia a ?aieao aaeeiie eiae/aneie nenoaiu
aeieaeiu oaeiaeaoai?youe neaaeothuei i?aaeeai:

1)E ethaiio aoiaeo eiae/aneiai yeaiaioa iiaoo auoue iiaeeeth/aiu:

a) auoiae ethaiai ae?oaiai eiae/aneiai yeaiaioa( a /anoiii neo/aa
,oiai aea naiiai)

a) aoiaeiie neaiae (aoiaeiay ia?aiaiiay)

a) eiae/aneay eiinoaioa(0 eee 1)

A ?aaeueiuo yeaeo?iiiuo noaiao iiaea/a eiae/aneie eiinoaiou ia aoiae
yeaiaioa ?aaeecoaony eeai cacaieaieai eeai iiaeeeth/aieai yoiai aoiaea
iaycaoaeueii /a?ac ?acenoi? e oeia ieoaiey.

2)Auoiae ethaiai eiae/aneiai yeaiaioa noaiu iiaeao auoue iiaeeeth/ai e
aoiaeo ae?oaiai eiae/aneiai yeaiaioa eee i?aaenoaaeyoue niaie auoiaeiie
neaiae noaiu .A /anoiii neo/aa aiciiaeia eiiaeiaoeey oiai e ae?oaiai.

Eiae/aneea noaiu ?acaeaeythony ia aeaa oeia :

1)Eiiaeiaoeeiiiua

2)Iineaaeiaaoaeueiiniua

A eiiaeiaoeeiiiuo noaiao cia/aiea auoiaeiiai neaiaea a ethaie iiiaio
a?aiaie caaeneo oieueei io eiiaeiaoeee aoiaeiuo neaiaeia (a yoio aea
iiiaio a?aiaie n o/aoii caaea?aeee ?ani?ino?aiaiey neaiaea ii yeaiaioai
noaiu)

N o/aoii yoie caaea?aeee cia/aiea auoiaeiiai neaiaea ii a?aiaie
caiacaeuaaao ia a?aiy caaea?aeee ii n?aaiaieth n iiiaioii eciaiaiey
aoiaeiuo neaiaeia.

Ooieoeeiie?iaaiea eiiaeiaoeeiiiie noaiu iiaeao auoue iienaii aoeaaie
ooieoeeae, io?aaeathuae caaeneiinoue auoiaeiiai neaiaea noaiu, eae
ooieoeee io aoiaeiuo neaiaeia , eae a?aoiaio yoie ooieoeee.

Aeey eiiaeiaoeeiiiuo noai n ianeieueeeie auoiaeaie yoa caaeneiinoue
io?aaeaaony nenoaiie aoeaauo ooieoeee.

I?eia? eiiaeiaoeeiiiie noaiu ia yeaiaioao aoeaaa aacena :

A iineaaeiaaoaeueiiniuo noaiao auoiaeiua neaiaeu a ethaie iiiaio a?aiaie
caaenyo ia oieueei io eiiaeiaoeee aoiaeiuo neaiaeia a aeaiiue iiiaio
a?aiaie ,ii e io i?aaeunoi?ee eo eciaiaiey ,oi anoue io
iineaaeiaaoaeueiinoe aoiaeiuo neaiaeia ai a?aiaie. Eae i?aaeei
iineaaeiaaoaeueiiniua noaiu oa?aeoa?ecothony iaeioi?ui aioo?aiiei
no?iaieai ,io eioi?iai caaeneo cia/aiea auoiaeiiai neaiaea(ia).

Aioo?aiiaa ninoiyiea oaeie noaiu nio?aiyaony ia caiiieiathueo yeaiaioao
(o?eaaa?ao) ,a nayce n /ai ,noaiu yoiai oeia iacuaathony noaiaie n
iaiyoueth.

A iauai neo/aa iinaaeiaaoaeueiiniay noaia i?aaenoaaeyao niaie iaeioi?ue
oeeo?iaie aaoiiao.

I?eia? iineaaeiaaoaeueiiniie noaiu: (oieaa?naeueiue aacen E-IA)

Iineaaeiaaoaeueiiniua noaiu oa?aeoa?ecothony iaee/eai oae iacuaaaiuo
iaoaeue ,ii eioi?ui auoiae iaeioi?iai yeaiaioa niaaeeiyaony ni aoiaeii
yoiai aea naiiai yeaiaioa (/a?ac ae?oaea yeaiaiou noaiu).

Iniiaiua ia?aiao?u eiiaeiaoeeiiiie noaiu.

Iniiaiuie ia?aiao?aie eiiaeiaoeeiiiuo noai (EN) yaeyaony noieiinoue e
auno?iaeaenoaea ,eae i?aaeei i?e iino?iaiee aano?aeoiuo EN ia
i?eaycaiiuo e eiie?aoiie nenoaia yeaiaioia oeaia noaiu ii?aaeaeyaony a
niunea Eaaeia. Auno?iaeaenoaea noaiu ,eae i?aaeei ioeaieaaaony
caaea?aeeie ?ani?ino?aiaiey neaiaeia io aoiaeia noaiu e aa auoiaeo. Aeey
aano?aeoiuo EN yoo caaea?aeeo i?eiyoi n/eoaoue a aeaea : O=e(
,(-caaea?aeea ia iaeiii eiae/aneii yeaiaioa,e-iaeneiaeueiia eiee/anoai
eiae/aneeo yeaiaioia ,/a?ac eioi?ua i?ioiaeeo neaiae io aoiaeia e
auoiaeo.

Eae i?aaeei caaea?aeea noaiu niiinoaaeyaony n /eneii o?iaiae yoie noaiu.
Aeey yoie oeaee ana yeaiaiou noaiu ?ani?aaeaeythony ii o?iaiyi. O?iaaiue
yeaiaioa ,ia auoiaea eioi?iai oi?ie?oaony auoiaeiie neaiae noaiu
niaiaaeaao n eiee/anoaii o?iaiae noaia e neaaeiaaoaeueii n aa
caaea?aeeie.

Aeey i?eaaaeaiiie noaiu yeaiaiou 1,2,3 ioiinyony e ia?aiio o?iaith.

Yeaiaiou 4,5 ei aoi?iio o?iaith.

Yeaiaio 6 e o?aoueaio o?iaith.

Yeaiaio 7 e /aoaa?oiio o?iaith.

Caaea/e aiaeeca e neioaca eiiaeiaoeeiiiuo noai.

A iauai aeaea caaea/a aiaeeca ,eiiaeiaoeeiiiuo noai naiaeeony e
ii?aaeaeaieth ooieoeee ,?aaeecoaiie caaeaiiie noaiie ,a /anoiii neo/aa
caaea/a aiaeeca ninoieo a ii?aaeaeaiee ?aaeoeee caaeaiiie noaiu ia
ii?aaeaeaiioth eiiaeiaoeeth aoiaeiuo neaiaeia.

Aeey ii?aaeaeaiey ooieoeee noaiu oeaeaniia?acii eniieueciaaoue iaoiae
iiaenoaiiaee ,aai eaeay ninoieo a neaaeothuai: Auoiaeu eiae/aneeo
yeaiaioia iaicia/athony iineaaeiaaoaeueii i?iaeaeaaynue io auoiaea noaiu
e aoiaeai, inouanoaeytho iiaenoaiiaeo a auoiaeioth ooieoeeth
i?iiaaeooi/iuo ia?aiaiiuo, eae a?aoiaio, aei oao ii? ,iiea a au?aaeaiee
ooieoeee ana i?iiaaeooi/iua ia?aiaiiua ia aoaeoo caiaiaiu ia aoiaeiua
ia?aiaiiua:

__

4v y3y6=x1x2v(y4v y5)x4x5=

___

3)x4x5

Ii?aaeaeei ?aaeoeeth noaiu ia aoiaeiie iaai?.

Iai?eia? (00000) o=1

Caaea/a neioaca ninoieo a iino?iaiee eiiaeiaoeeiiiie noaiu ii caaeaiiiio
caeiio ooieoeeiie?iaaiey.

I?e ?aoaiee yoie caaea/e iaiaoiaeeii o/eouaaoue neaaeothuea iiiaiou:

Neioace?oaiay noaia aeieaeia ii aiciiaeiinoe niaea?aeaoue ieieioi
iai?oaeiaaiey. A nayce n yoei aeooaeueiie caaea/ae yaeyaony ieieiecaoeey
caaeaiiie aoeaaie ooieoeee. I?e ?aoaiee yoie caaea/e oeaeaniia?acii
iieo/eoue eae IAeIO oae e IEIO.

Eae i?aaeei ,neioace?oaiay noaia no?ieony ia eiae/aneeo yeaiaioao
,i?eiaaeeaaeaueo iaeioi?iio aaceno. Anoanoaaiii ,/oi eniieuecoaiay
nenoaia yeaiaioia aeieaeia iaeaaeaoue naienoaii ooieoeeiiaeueiie iieiiou
,oi anoue auoue aeinoaoi/iie aeey iino?iaiey ia aa iniiaa
eiiaeiaoeeiiiie noaiu ,?aaeecothuoth ethaoth iaia?aae caaeaiioth aoeaao
ooieoeeth. Oaeeie ooieoeeiiaeueii iieiuie nenoaiaie eiae/aneeo yeaiaioia
yaeythony: 1.(E,EEE,IA( 2.(E,IA( 3.(EEE,IA( 4.(E-IA(
5.(EEE-IA( 6.(E,I2(

Eae i?aaeei i?e ?aoaiee caaea/e neioaca noa?athony aeiaeoueny
yeno?aiaeueiiai cia/aiey iaeiiai ec ia?aiao?ia noaiu :ieieioi oeaiu eee
iaeneioi auno?iaeaenoaey (ieieioi caaea?aeee).A oao neo/ayo ,eiaaea
e?eoa?eai yooaeoeaiinoe noaiu yaeyaony ieieioi oeaiu ii Eaaeio iaae
ieieiaeueiuie oi?iaie i?iaiaeyo aeiiieieoaeueiua i?aia?aciaaiey ,iooai
?aoaiey caaea/ oaeoi?ecaoeee e aiciiaeii aeaeiiiiceoeee aoeaaie
ooieoeee. Eae i?aaeei ieieiaeueiay oi?ia ia aeaao aaniethoiiai ieieioia
noieiinoe ,/aai iiaeii aeiaeoueny ?aoaieai caaea/ oaeoi?ecaoeee e
aeaeiiiiceoeee. Anee e?eoa?eai yooaeoeaiinoe noaiu yaeyaony ieieiaeueiay
caaea?aeea ,oi neaaeoao eiaoue a aeaeo ,/oi oaeoi?iia i?aia?aciaaiea e
aeaeiiiiceoeey aoeaaie ooieoeee a iauai neo/aa oiaiueoaao oeaio noaiu e
oaaee/eaaao aa caaea?aeeo. A aieaa neiaeiii neo/aa noaia iioeiece?oaony
ii iaeiiio ec iieacaoaeae i?e iaee/ee ia?aie/aiey ia aoi?ie. I?eia?ii
iiaeiaiie iinoaiiaee caaea/e neioaca yaeyaony: Neioace?iaaoue noaio n
ieieiaeueiie oeaiie ii Eaaeio ,/oiau aa caaea?aeea ia i?aauoaea 4(.

Iaiaoiaeeii o/eouaaoue ,a eaeii aeaea i?aaenoaaeaiu aoiaeiua neaiaeu
noaiu: oieueei a i?yiii eee e a i?yiii e a ia?aoiii. A ia?aii neo/aa
no?ieony eiiaeiaoeeiiiay noaia n iaeiioaciuie aoiaeaie. Ai aoi?ii neo/aa
n ia?aoaciuie. A ?aaeueiuo eiiaeiaoeeiiiuo noaiao aoiaeiua neaiaeu
i?aaenoaaeytho niaie cia/aiea auoiaeia ?aaeno?ia. Iai?eia? i?e
iino?iaiee eiiaeiaoeeiiiiai noiiaoi?a aoiaeiua neaiaeu nieiathony n
?aaeno?ia neaaaaiiai. I?e eioaa?aeueiie ?aaeecaoeee ?aaeno?ia a aeaea
NEN a oeaeyo ieieiecaoeee /enea auoiaeia auoiaeiua neaiaeu ?aaeno?ia eae
i?aaeei i?aaenoaaeythony oieueei a i?yiii aeaea ,/oi aeaeaao
aeooaeueiuie noaiu n iaeiioaciuie aoiaeaie.

I?e iino?iaiee noai a ?aaeueiie nenoaia yeaiaioia iaiaoiaeeii o/eouaaoue
?yae eiino?oeoeaiuo ia?aie/aiee ,iniiaiuie ec eioi?uo yaeythony:

a) Eiyooeoeeaio iauaaeeiaiey ii aoiaeo, eioi?ue i?aaenoaaeyao niaie
ia?aie/aiea ia /enei aoiaeia a yeaiaio. Iiaeao i?eieiaoue cia/aiey
2,3,4,8,16.

a) Eiyooeoeeaio ?acaaoaeaiey ii auoiaeai eioi?ue ii?aaeaeyao
iaeneiaeueiia /enei eiae/aneeo yeaiaioia, eioi?ua iiaeii iiaeeeth/eoue e
auoiaeo yeaiaioa a oneiaeyo aai ii?iaeueiiai ooieoeeiie?iaaiey. Yoio
eiyooeoeeaio ii?aaeaeyao iaa?oci/ioth niiniaiinoue. Aa?uee?oaony io 10
aei 30.

A ?aaeueiuo nenoaiao yeaiaioia iaeiioeiiua yeaiaiou iauaaeeiythony a
iiaeoee ,?aaeecoaiua iaeiie eioaa?aeueiie noaiie n iaeui o?iaiai
eioaa?aoeee(IEN). A nayce n yoei i?e iino?iaiee noai a ?aaeueiie nenoaia
yeaiaioia iaiaoiaeeii ieieiece?iaaoue ia noieueei /enei yeaiaioia e
aoiaeia a ieo neieueei /enei iiaeoeae ,ec eioi?uo eiiiiioaony noaia.

Eae i?aaeei a aieueoeinoaa ?aaeueiuo nenoai yeaiaioia ia?yaeo n i?inouie
eiae/aneeie yeaiaioaie eniieuecothony oaeaea naeaiaiiua yeaiaiou
?aaeecothuea ninoaaioth aoeaao ooieoeeth. Oeie/iui i?eia?ii iiaeao
neoaeeoue yeaiaio E-EEE-IA.

A ?aaeueiuo nenoaiao yeaiaioia eae i?aaeei eniieuecoaony cia/eoaeueiia
?aciiia?acea eiae/aneeo yeaiaioia, ioiinyueony e ?aciui aacenai. Oai ia
iaiaa iino?iaiea noaiu a ?aieao ii?aaeaeaiiiai aacena yaeyaony
aeinoaoi/ii aeooaeueiie caaea/ae, oae eae iicaieyao oiaiueoeoue
iiiaieeaoo?o eniieuecoaiuo yeaiaioia.

Iino?iaiea eiiaeiaoeeiiiuo noai (EN) ii ieieiaeueiui ii?iaeueiui oi?iai
a ?acee/iuo aacenao.

Aoeaa Aacen (E, EEE, IA)

_ _ _ _ _ _ _

y=x1x2x3vx1x2x4vx1x5vx6 (IAeIO)

——– ——- —–

e (3) e (3) e (2)

Noaia n ia?aoaciuie aoiaeaie

SQ=3+3+2=12 Sa

Au/eoaieaIia?aoeey au/eoaiey /enae n ieaaathuae caiyoie naiaeeony e neiaeaieth iooai i?aaeaa?eoaeueiiai eciaiaiey ciaea aoi?iai iia?aiaea ia i?ioeaiiieiaeiue.A nayce n oai, /oi iaioenna /enea i?aaenoaaeyaony a i?yiii eiaea, i?e eciaiaiee ciaea /enea iaiyaony oieueei ciaeiaue ?ac?yae, a iaioenna inoaaony i?aaeiae.Iia?aoeey oiiiaeaiey oeaeuo /enae ei?eioeeiu aa ?aaeecaoeee a YAIIniiaiua iieiaeaiey aeaie/iiai oiiiaeaieyA=13 x1101A=11 1011---------1101+ 11011101-------------10001111=(143)10Ae?oaie iaoiae:x11011011--------1101+ 11011101------------------10001111=(143)10Oiiiaeaiea aeaie/iuo /enae (eae e aeanyoe/iuo) ninoieo a iineaaeiaaoaeueiii oiiiaeaiee iiiaeeiiai ia ioaeaeueiua ?ac?yaeu iiiaeeoaey n noiie?iaaieai ?acoeueoaoia oiiiaeaiey. ?acoeueoao oiiiaeaiey iiiaeeiiai ia iaeei ?ac?yae iiiaeeoaey i?eiyoi iacuaaoue /anoe/iui i?iecaaaeaieai. ?acoeueoao oiiiaeaiey aeaoo /enae i?aaenoaaeyao niaie noiio anao /anoiuo i?iecaaaeaiee.Eaaeaeia /anoiia i?iecaaaeaiea eeai niaiaaeaao n iiiaeeiui, eeai ?aaii 0.Oi?ie?oaiua /anoiua i?iecaaaeaiey aeieaeiu auoue ii?aaeaeaiiui ia?acii naeaeioou ae?oa ioiineoaeueii ae?oaa aeey eo iineaaeothuaai noiie?iaaiey.*anoiua i?iecaaaeaiey iiaeii oi?ie?iaaoue eae ia/eiay io ieaaeoeo, oae ia/eiay e io noa?oeo ?ac?yaeia iiiaeeoaey.A iauai neo/aa aeey ?acoeueoaoa oiiiaeaiey o?aaoaony eiee/anoai oeeo? ?aaiia noiia eiee/anoaa oeeo? iia?aiaeia. I?e iaeeiaeiaie ?ac?yaeiinoe iia?aiaeia: 2n - ?ac?yaeia.Iniaaiiinoe ?aaeecaoeee oiiiaeaiey a YAI1. A iia?aoeeiiiii ono?ienoaa aeey oiiiaeaiey aeaie/iuo /enae aeieaeai eniieueciaaoueny iiiai?ac?yaeiue aeaie/iue noiiaoi?Iia?aoeey aeaeaiey e aa ?aaeecaoeey a YAIIniaaiiinoe aeaie/iiai aeaeaieyI?eia?: 130/10A= 10000010 | 10101010 |--------------------- |0110101100101010-------------0010101010-----------10101010Iia?aoeey aeaie/iiai aeaeaiey naiaeeony e iineaaeiaaoaeueiiio au/eoaieth aeaeeoaey ec aeaeeiiai e inoaoeia ia iineaaeothueo oaaao.1) Iiae oaeouei inoaoeii iiieiaaony ?acoeoao iieo/aiue i?e au/eoaiee aeaeeoaey ec aeaeeiiai ia ia?aii oaaa eee ec i?aaeeaeouaai inoaoea ia iineaaeothueo oaaao.2) Ia i?aaeaa?eoaeueiii oaaa aeaeeoaeue niaiauaaony ni noa?oeie ?ac?yaeaie aeaeeiiai ,a caoai ia eaaeaeii oaaa naeaeaaaony ai?aai ia iaeio oeeo?o ioiineoaeueii iaiiaeaeaeiiai inoaoea Ia iineaaeiai oaaa aeaeeoaeue niaiauaaony n ieaaeoeie ?ac?yaeaie inoaoea.3) Oeeo?u /anoiiai au?aaaouaaaiua eaaeaeii oaaa ii?aaeaeythony ciaeii oaeouaai inoaoea ,aeey inoaoea >= 0 oeeo?a /anoiiai ?aaia
aaeeieoea ,aeey inoaoea

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