revaz grigolia
diskretuli struqturebi I
leqcia I
simravleebi, operaciebi simravleebze, venis diagramebi;
funqciebi: warmodgena, kompozicia
1.1 simravleebi, operaciebi simravleebze, venis diagramebi
intuiciurad, “simravle” aris elementTa erToblioba, xolo
“funqcia” – nebismieri wesi, romelic yovel elements erTi simravlidan
Seusabamebs Sesatyvis elements meore simravlidan.
simravlis magaliTebi: sibrtyeze yvela wrfeebis simravle, yvela
racionalur ricxvTa
Q
simravle, yvela kompleqsur ricxvTa C
simravle, yvela mTel ricxvTa Z simravle. sasruli raodenobis
sxvadasxva elementisagan Semdgari simravleebi SesaZlebelia aRiweros
yvela misi elementebisagan Semdgari siis saSualebiT, romelsac xSirad
aTavseben figurul frCxilebSi. maSasadame yvela luwi mTeli ricxvebis
simravle moTavsebuli
0-sa da
8-s Soris, am rcxvebis CaTvliT,
SesaZlebelia warmovadginoT rogorc
0,2,4,6,8, aseve
6-is yvela
dadebiTi gamyofebis simravle warmoidgineba Semdegi saxiT {1,2,3,6}. rigs,
romlis mixedviTac dalagebulia am simravlis elementebi, ar aqvs
mniSneloba: {1,2,3,6} = {1,3, 6,2}.
formalurad, “x  S ” aRniSnavs, rom “x aris S simravlis elementi”
an, eqvivalenturad, “x aris S simravlis wevri”
an
“x ekuTvnis S
simravles“. agreTve
“x  S “ niSnavs, rom “x ar aris S simravlis
elementi”. vinaidan simravle srulad ganisazRvreba misi elementebiT,
ori simravle S da T tolia maSin da mxolod maSin, roca maT gaaCniaT
erTi da igive elementebi; simboloebSi:
S = T  yoveli x-Tvis, x  S maSin da mxolod maSin, roca x  T.
(1)
(aq ormxrivi isari “” niSnavs “maSin da mxolod maSin”). S aris T-s
qvesimravle (an CarTulia T-Si), niSnavs, rom S–is yoveli elementi aris
T-s elementi, ase rom, Tu xS , maSin xT; simboloebSi :
S  T  nebismieri x-Tvis, xS  xT.
(aq, marjvena mxares, calmxrivi “ “ isari niSnavs “gamomdinareobs”). am
gansazRvrebis Tanaxmad, S  T da T  U -dan gamomdinareobs S  U. S da T
simravleebis toloba, rogorc es zemoT aris gansazRvruli, SeiZleba
gadaiweros Semdegnairad
S = T  S  T da T  S.
simravle S
carielia, Tu mas ar gaaCnia elementebi. toloba (1)-s
Tanaxmad, nebismieri ori carieli simravle tolia. carieli simravle
aRiniSneba  simboloTi. mas agreTve uwodeben nul simravles; is aris
nebismieri simravlis qvesimravle. S aris U simravlis sakuTrivi
qvesimravle, Tu
S   , S  U da S  U.
1
revaz grigolia
diskretuli struqturebi I
mocemuli U simravlis zogierTi qvesimravle xSirad aRiwereba
rogorc simravle yvela im x elementebisa
U-dan, romelTac gaaCniaT
raRac specifikuri Tviseba. ase rom yvela kompleqsuri z ricxvebis
qvesimravle, iseTi rom z 2  1 Caiwereba Semdegnairad: {z| zC da z 2  1 },
aseve formulebi
E ={x | x  Z da x = 2y romelime y Z}, N = { x | x  Z da x  0}
aRweren, Sesabamisad, yvela luw mTel ricxvTa simravle E-s da yvela
arauaryofiT mTel ricxvTa simravle N–s. sxvadasxva TvisebebiT
SesaZlebelia aRiweros erTi da igive qvesimravle; ase magaliTad
{n | n  Z da 0 < n < n} da
{n | n  Z da n 2  1 }
orive aRwers cariel simravles .
ganvixiloT TanakveTis da gaerTianebis operaciebi simravleebze.
Tu R da S mocemuli simravleebia, maTi TanakveTa R  S aris R da S
simravleebis yvela saerTo elementTa simravle:
R  S = {x | x  R da x  S},
maSin roca maTi gaerTianeba R  S aris yvela im elementTa simravle,
romlebic ekuTvnis an R an S (an orives) :
R  S = {x | x  R an x  S}.
es gansazRvrebebi Semdegnairad SeiZleba gamoiTqvas :
x  (R  S)  x  R da x  S,
x  (R  S)  x  R an x  S.
TanakveTisa da gaerTianebis operaciisa es ganmartebebi eTanadeba
logikur kavSirebs “da” da “an”.
ori A da B simravlis sxvaoba A  B ganisazRvreba Semdegnairad:
A  B = {x | x  A da x  B}.
Sesatyvisi Sesabamisoba logikuri kavSirisa “ara” aris “damatebis”
operacia: Tu
S-i U-s qvesimravlea, maSin S-is damateba S U-Si aris
yvela im elementebis simravle U-dan, romlebic ar ekuTvnian S :
 S= {x | x  U da x  S} = U  S.
TanakveTis, gaerTianebis da damatebis operaciebi akmayofileben
mravalnair “tolobebs”, romlebic samarTliania nebismieri simravleebisaTvis. erTerTi maTgani aris
R  (S  T) = (R  S)  (R  T),
2
(2)
revaz grigolia
diskretuli struqturebi I
romelic samarTliania nebismieri R, S da T simravleebisaTvis. (es
toloba
amtkicebs,
rom
“TanakveTis”
operacia
distribuciulia
“gaerTianebis” mimarT.) davamtkicoT es debuleba. zemoT moyvanili  da
 gansazRvrebis Tanaxmad,
x  [R  (S  T)]  x  R da x  S  T  x  R da (x  S an x  T ).
analogiurad,
x  [(R  S)  (R  T)]  (x  R da x  S) an (x  R da x  T).
Semdeg, “da”-s
da
“an”-is cnobili Tvisebebis Tanaxmad, es ori
sxvadasxva mtkicebulebebi x-is Sesaxeb, zemoT warmodenili marjvena
mxaris
Canawerebi
logikurad
eqvivalenturebia.
maSasadame
orive
warmodgenil simravles gaaCniaT erTi da igive elementebi da amitom
tolebia. analogiurad
R  (S  T) = (R  S)  (R  T).
(3)
or simravles, R da S-s, ewodeba TanaukveTi, Tu R  S = .
Tu gvaqvs nebismieri simravle U, P(U)-Ti aRvniSnavT U simravlis
yvela S qvesimravleebis simravles, romelsac vuwodebT U simravlis
buleans; e. i. , P(U) = {S | S  U}. magaliTad, Tu U or elementiania, maSin
mas gaaCnia oTxi sxvadasxva qvesimravle, romlebic arian P(U) –s oTxi
elementi. ufro naTlad, P({1,2}) = {{1,2}, {1}, {2}, }, sadac  aris carieli
simravle.
A
B
AB
A
B
AB
A
B
AB
3
revaz grigolia
diskretuli struqturebi I
A
1.2 funqciebi
nebismieri f funqcia
S simravlidan
T simravleSi
yovel s
elements S-dan aniWebs mniSvnelobas f(s)  T (anu s elementi gadahyavs f(s)
elemebntSi f(s)  T ). amas aRvniSvnaT Semdegnairad
s  S, f(s)  T.
s  f(s),
f(s) elementi SeiZleba agreTve Caiweros rogorc fs an fs, frCxilebis
gareSe; es aris f-is mniSvneloba s argumentze. simravle S-s ewodeba fis domeini, xolo T -s - kodomeini. isriani aRniSvna
f:ST
f
an
S T
miuTiTebs, rom
f aris funqcia S domeiniiT da T kodomeiniT. xSirad
funqcias uwodeben “asaxvas” an “transformacias”.
rom aRiweros konkretuli funqcia, naCvenebi unda iyos misi
domeini da misi kodomeini, da Cawerili unda iyos misi moqmedebis Sedegi
tipiur (“cvlad”) elementze domeinidan. ase rom kvadratSi ayvanis
funqcia f : R  R namdvil ricxvTa simravlisaTvis SeiZleba aRiweros
Semdegi nebismieri gziT: f(x) = x2 yoveli x namdvili ricxvisaTvis, an
rogorc funqcia (--)2, sadac – gaTvaliswinebulia adgili argumentisTvis,
an rogorc funqcia, romelsac gadahyavs yoveli xR x2-Si, an rogorc
funqcia, romelic mocemulia asaxviT x  x2 , sadac x R.
SevniSnoT, rom simbolo rogoricaa f an g aRniSnavs funqcias,
maSin rodesac gamosaxuleba rogoricaa f(x) an g(x) aRniSnavs am funqciis
mniSvnelobas x elementze misi domeinidan. magaliTad
gamosaxuliba
“sin x” niSnavs romeliRac ricxvs, ase rom Cven vlaparakobT ara “ funqcia
sin x”, aramed funqcia sin : R  R .
4
revaz grigolia
diskretuli struqturebi I
ori funqcia f da g
tolia (aRniSneba, f = g ), roca maT
gaaCniaT erTi da igive domeini, erTi da igive kodomeini, da erTi da
igive mniSvneloba f(s) = g(s) yovel element s-ze maTi saerTo domeinidan.
magaliTad, asaxva x  x + 2 gansazRvravs funqcias f : Z  Z mTel ricxvTa
simravle Z-ze; namdvil ricxvTa simravle R–ze es asaxva agreTve
gansazRvravs funqcias g : R  R; magram es funqciebi sxvadasxva
funqciebia, vinaidan maT gaaCniaT sxvadasxva domeinebi.
funqciis f : S  T saxe image(f) aris yvela f (s) mniSvnelobaTa
simravle f (S), sadac s  S ; is yovelTvis aris f-is kodomeinis qvesimravle.
nebismieri
S
simravlisaTvis, igivuri funqcia 1S : S  S aris
funqcia s  s, romelic asaxavs yovel element s-s S-dan Tavis TavSi.
sxvadasxva simravleebs gaaCniaT sxvadasxva igivuri funqciebi.
Tu
S
aris U-s qvesimravle, maSin Cadgma
i : S  U aris funqcia S-dan U-Si,
romelsac gadahyavs nebismieri elementi
S-dan igive elementSi, xolo
ukve U-dan. SevniSnoT, rom “Cadgma” aris funqcia S  U , xolo “CarTva”
aris mimarTeba S  U; yoveli CarTvis mimarTeba warmoqmnis
Cadgmis
funqcias.
ori funqciis kompozicia f  g  fg aris iseTi funqcia, romelic
miiReba maTi TanmimdevrobiT moqmedebis saSualebiT; pirvelad g-si da
Semdeg f-is – romelic ar aris azrs moklebuli: e. i., f-is domeini aris
g-s kodomeini. ufro formalurad, mocemuli gvaqvs ra funqciebi
g : R  S,
f : S  T,
maTi kompozicia aris funqcia f  g : R  T rac niSnavs, rom
( f  g )(r)  f ( g(r)) ,
es
gansazRvreba
saSualebiT:
SegviZlia
yoveli
warmovadginoT
r  R.
“asaxvebis
(4)
diagramis”
f g
R
T
g
f
S
rom gadavideT R-dan f  g kompoziciiT T-Si es igivea rac mivaRwioT T-s
ori nabijiT S-is gavliT, pirvelad g-Ti da Semdeg
f-iT. am faqtis
aRsaniSnavad amboben agreTve, rom es samkuTxa diagrama “komutirebs”.
funqciebis kompozicia akmayofilebs
asociaciurobis kanons : ( f  g )  h  f  ( g  h),
roca maTSi Semavali kompoziciebi gansazRvrulia. es intuiciurad
( f  g)  h da f  ( g  h) gaaCniaT moqmedebis erTnairi
naTelia; orives
5
revaz grigolia
diskretuli struqturebi I
efeqti: jer h–i, Semdeg g, da bolos f-i. formalurad, mocemuli
funqciebis h : P  R, g : R  S da f : S  T , orive sameulis kompoziciebi
( f  g)  h da f  ( g  h) aris funqciebi P -dan T –Si, maSin rodesac
pirveli kompozicia yovel
pP aniWebs mniSvnelobas
[( fg)h] p  ( fg)(hp)  f ( g(hp))  f ((gh) p)  [ f ( gh)] p;
( fg ) h
fg
gh
f ( gh)
funqciebisTvis tolobis gansazRvris Tanaxmad mtkicdeba asociaciurobis
kanoni (fg)h = f(gh) : P  T. SevniSnoT, rom aq (da xSirad Semdgom)
mosaxerxebelia ar vweroT simbolo "" f  g -Si.
kompoziciis mimarT yoveli funqcia akmayofilebs
igiveobis kanons:
f  1S  f  1T  f : S  T .
pirveli toloba ( f  1S  f ) rom damtkicdes, (4)-s Tanaxmad SevniSnoT,
rom ( f 1S )s  f (1S s)  fs yoveli s  S ; maSasadame, f 1S  f funqciebis
tolobis gasazRvris Tanaxmad. analogiurad mtkicdeba meore toloba
( f  1T  f ).
funqcias r : S  T ewodeba f : U  V funqciis SezRudva, roca S 
U, T  V da r(s)=f(s) yoveli s  S. (agreTve SegviZlia vTqvaT, rom f aris
r funqciis gafarToeba). magaliTad, mocemuli qvesimravlesTvis S  U,
Cadgma i: S  U aris igivueri funqciis 1U : U  U SezRudva.
savarjiSoebi
Cven vgulisxmobT, rom am paragrafis amocanebSi yvela simravleebi
aris romeliRac universaluri U simravlis qvesimravle. sxvaobas U  A
ewodeba A simravlis damateba da aRiniSneba rogorc  A.
A da B simravleebis simetriul sxvaobas uwodeben simravles A
 B=
(A  B )  (B  A ). gavixsenoT, rom P(A) = { X : X  A }.
1. daamtkiceT:
(a) A  A (refleqsuroba);
(b) Tu A  B da B  C, maSin A  C (tranzituloba);
(c) A  B  A  A  B;
(d) A  B  B  A  B;
(e) A  B  A.
6
revaz grigolia
diskretuli struqturebi I
2. daamtkiceT, rom Tu A aris
x2  7x + 6 = 0
gantolebis fesvebis simravle da B = {1,6}, maSin A = B.
3. daamtkiceT, rom   {}.
4. daamtkiceT, rom {{1,2}, {2,3}}  {1,2,3}.
5. daamtkiceT, rom nebismieri A simravlisaTvis:
(a)   A  U;
(b) Tu A  , maSin A = ; Tu U  A, maSin A = U;
(c) A   = A, A   =   A  B;
6. arsebobs Tu ara simravleebi A, B da C, iseTi, rom
A  B  , A  C = , (A  B)  C =  ?
7. daamtkiceT, rom
A  B  A  B = B  A  B = A  A  B =   ( A)  B = U.
8. daamtkiceT Semdegi igiveobebi:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
A  A = A = A  A;
A  B = B  A;
A  B = B  A;
A  (B  C ) = (A  B)  C;
A  (B  C ) = (A  B)  C;
A  (B  C ) = (A  B)  ( A  C);
A  (B C ) = (A  B)  ( A C);
(A  B)  ( C  D) = ( A  C)  (B  C)  (A  D)  (B  D);
(A  B)  A = (A  B)  A = A.
9. daamtkiceT Semdegi igiveobebi:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(k)
(l)
(A B ) = (A)  ( B);
(A  B ) = (A)  ( B);
A  (B  C) = (A B )  (A  C );
A  (B  C) = (A B )  (A  C );
A  (A  B) = A  B;
A  B = A  (A  B);
A  (B  C ) = (A  B)  (A  C) = (A  B )  C;
(A  B)  C = (A  C )  (B  C );
A B = A  (B  A );
 ( A) = A;
A  ( A) = U;
7
revaz grigolia
diskretuli struqturebi I
(m) A  ( A) = ;
(n) (A  B)  (A  B) = (A  B)  (A  B) = A;
(o) A  (B  A) = ;
10. daamtkiceT, rom
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
A B  C  A  C da B  C;
A  B  C  A  C da B  C;
A  B  C  A  ( B )  C;
A  B  C  A  ( B )  C;
(A  B)  B = A  B  A;
(A  B)  C = A  (B  C )  C  A;
A  B  A  C  B  C;
A  B  B  A;
A  B = A  B  A = B;
A = B  A  B =  da A  B = U.
11. daamtkiceT Semdegi igiveobebi:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
A B = B  A ;
A  (B  C)  ( A  B)  C ;
A  (B 
 C)  ( A  B)  ( A  C) ;
A  B = A
 ( A  B) ;
A  B = A
 B  ( A  B) ;
A  ( A  B)  B ;
A    A ;
A  A   ;
A  U   A ;
A  B  ( A  B)  ( A  B) .
12. daamtkiceT, rom
(a) A
 B =   A = B;
(b) A
 B =   A B = A B ;
(c) A
 B = C  B  C  A  C  A  B
13. moZebneT , {}, {x}, {1,2} simravleebis yvela qvesimravle.
14. daamtkiceT, rom n
elementian simravles gaaCnia
qvesimravleebi.
15. daamtkiceT, rom
(a) P (A  B) = P (A)  P (B);
(b) P (A  B) = {C  D : C  P (A) da D  P (B)}.
8
2n
odenobis