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Noravshan to’plamlar ustida amallar

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29.08.2023

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MAVZU :  Noravshan to’plamlar ustida amallar
REJA: 
1. Noravshan to’plamlarlarni to’ldirish.
2. Noravshan to’plamlarning kesishmasi, birlashmasi
3. Amallarning umumlashgan ta’riflari: t-norma, s-norma Klassik to’plamlar uchun quyidagi amallar kiritilgan: 
To’plamlarning  kesishmasi   –   A   va   B   to’plamlardagi    ham   A ,  ham   B
to’plamga tegishli elementlardan iborat bo’lgan  С  =  А     В   to’plamidir.
To’plamlarning birlashmasi -    A  va   B  to’plamlardagi yoki   A , yoki   B ,
yoki   ikkala   to’plamga   tegishli   elementlardan   iborat   bo’lgan   С   =   А  	
   В
to’plamidir.
To’plamlarning   inkori   -   universal   to’plamga   tegishli,   lekin   A
to’plamga   tegishli   bo’lmagan   elementlarni   o’z   ichida   mujassamlashtirgan   С
= 	
   А   to’plamidir . 
Zade   shu   to’plamlarning   tegishlilik   funksiyalari   amallari   yordamida
noravshan   to’plamlar   ustidagi   shu   kabi   amallar   majmuini   taklif   qildi     [35].
Shunday qilib,  A  to’plam   
А (u),  В  to’plam esa 	

В (u)  funksiya orqali berilgan
bo’lsa,   u   holda   natija   bo’lib  	

С (u)   tegishlilik   funksiyali   C   to’plam
hisoblanadi. 
Birlashma.
A   va   B   noravshan   to’plamlarning   birlashmasi   quyidagi   tarzda
aniqlanadi:	
∀	x∈	X	,μA∪B(x)=	max	{μA(x),μB(x)}
,
bu yerda 	
μA∪B  -  A  va  B  uchun tegishlilik funksiyasi.
Kesishma .  	
μA∩B
 tegishlilik funksiyasi quyidagicha aniqlanadi:	
∀	x∈	X	,μA∩B(x)=	min	{μA(x),μB(x)}
.
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan to’plam bo’lsin. Noravshan to’plamlar ustidagi birlashtirish amali
ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi:
A=0.07/2+0.2/3+0.4/4+0.63/5+0.87/6+1.0/7+0.89/8+0.5/9+
+0.2/10+0.07/11,
B=0.05/6+0.11/7+0.21/8+0.32/9+0.46/10+0.69/11+0.87/12+
+1.0/13+0.9/14+0.5/15+0.25/16+0.09/18, A∪	B=	0.07	/2+0.2/3+0.4/4+0.63	/5+0.87	/6+1.0/7+0.89	/8+	
+0.5/9+0.46	/10	+0.69	/11	+0.87	/12	+1.0/13	+0.9/14	+0.5/15	+	
+0.25	/16	+0.09	/18	.(11-rasmga qarang ).     
    	
Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)
   	
Основной
Основной
Основной	
x	
m(x)
11-rasm. Noravshan to’plamlarning birlashmasi
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   Noravshan   to’plamlar   ustidagi   kesishma   amali
ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi:
A= 0.15/2+0.41/3+0.66/4+0.85/5+0.97/6+1/7+0.9/8+0.6/9+
+0.42/10+0.3/11+0.18/12+0.1/13+0.03/14,
B=0.05/5+0.1/6+0.16/7+0.25/8+0.35/9+0.47/10+0.62/11+ 0.8/12+0.94/13+1/14+0.97/15+0.83/16+0.5/17+0.2/18+0.07/19,A∩	B
=0.05/5+0.1/6+0.16/7+0.25/8+0.35/9+0.42/10+0.3/11+
+0.18/12+0.1/13+0.03/14.
(12-rasmga qarang ).	
Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
Основной
ОсновнойОсновнойОсновной
x	
m(x)
12 -rasm. Noravshan to’plamlarning kesishmasi 
  To’ldirma .
A  to’plamning  	
¯А    to’ldirmasi quyidagicha aniqlanadi :	
∀	x∈	X	,	μ¯A(x)=	1−	μA(x)
.
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   Noravshan   to’plamlar   ustidagi   to’ldirish   amali
ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi:
A=0/1+0.05/2+0.14/3+0.27/4+0.5/5+0.76/6+0.93/7+1.0/8+0.96/9+0.84/10+
               +0.62/11+0.37/12+0.25/13+0.16/14+0.09/15+0.03/16+0/17, ¯A=1.0/1+0.95/2+0.86/3+0.73/4+0.5/5+0.24/6+0.07/7+0/8+0.04/9+0.16/10+
                +0.38/11+0.63/12+0.75/13+0.84/14+0.91/15+0.97/16+1.0/17.
(13-rasmga qarang).
Noravshan   to’plamlarning   birlashmasi   va   kesishmasi   uchun   boshqa
amallardan ham foydalanish mumkin.
Algebraik ko’paytma:	
∀	x∈	X	,μA⋅B(x)=	μA(x)⋅μB(x)
.
Cheklangan ko’paytma:	
∀	x∈	X	,μA⊗B(x)=	max	{0,μA(x)+μB(x)−	1}
.	
Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	
Осно...	Основной
Основной
Основной	
x	
m(x0
13-rasm. Noravshan to’plamning to’ldirmasi  Qat’iy (drastic) ko’paytma:μ	A	intersect	¿	B	(	x	)	=	¿	¿
¿	
¿	¿
Algebraik yig’indi:	
∀	x∈X	,μA+B(x)=	μA(x)+μB(x)−	μA(x)⋅μB(x)
.
Cheklangan yig’indi:	
∀	x∈X	,μA˙¿B(x)=min	{1,μA(x)+μB(x)}
.
Qat’iy (drastic) yig’ind:	
μ	A	∪
+	
B	(	x	)	=	¿	¿
¿	¿	¿
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   A   va   B   noravshan   to’plamlarning   algebraik
ko’paytmasi   amali  ularning tegishlilik  funksiyalariga qarab    quyidagi  tarzda
aniqlanadi:
A=0.1/1+0.24/2+0.4/3+0.63/4+0.82/5+0.94/6+1.0/7+0.98/8+0.91/9+0.76/10
              +0.57/11+0.35/12+0.2/13+0.1/14+0.04/15,
B=0.02/4+0.09/5+0.2/6+0.32/7+0.46/8+0.61/9+0.76/10+0.88/11+0.96/12+
               +1.0/13+0.96/14+0.85/15+0.62/16+0.37/17+0.2/18+0.09/19,	
А∗В
=0/3+0.01/4+0.07/5+0.19/6+0.32/7+0.45/8+0.55/9+0.58/10+0.5/11+
   +0.34/12+0.2/13+0.96/14+0.03/15+0/16.
(14-rasmga qarang ). Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)14-rasm. Noravshan to’plamlarning algebraik ko’paytmasi 
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   A   va   B   noravshan   to’plamlarning   algebraik
yig’indisi     amali   ularning   tegishlilik   funksiyalariga   qarab,   quyidagi   tarzda
aniqlanadi:
A=0.03/1+0.1/2+0.28/3+0.52/4+0.75/5+0.94/6+1/7+0.96/8+0.87/9+
               +0.71/10+0.55/11+0.4/12+0.28/13+0.19/14+0.12/15+0.06/16+0.02/
17,
B=0/1+0/2+0/3+0.02/4+0.06/5+0.12/6+0.17/7+0.25/8+0.35/9+0.5/10+
     +0.68/11+0.82/12+0.95/13+1/14+0.95/15+0.62/16+0.35/17+
     +0.17/18+0.06/19,
 	
A^+¿B¿ =0.03/1+0.1/2+0.28/3+0.52/4+0.75/5+0.94/6+1.0/7+0.96/8+
  +0.91/9+0.86/10+0.86/11+0.88/12+0.96/13+1.0/14+0.95/15+
   +0.62/16+0.35/17+0.17/18+0.06/19.
(15-rasmga qarang ). Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)15-rasm .  Noravshan to’plamlarning algebraik yig’indisi 
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   A   va   B   noravshan   to’plamlarning   chegaralangan
yig’indisi     amali   ularning   tegishlilik   funksiyalariga   qarab   quyidagi   tarzda
aniqlanadi:
A=0.06/1+0.17/2+0.31/3+0.5/4+0.67/5+0.82/6+0.93/7+1.0/8+0.98/9+
     +0.89/10+0.75/11+0.6/12+0.45/13+0.33/14+0.23/15+0.14/16+
     +0.08/17+0.03/18,
B=0.03/4+0.08/5+0.15/6+0.26/7+0.4/8+0.55/9+0.7/10+0.85/11+
+0.95/12+1/0/13+0.96/14+0.85/15+0.6/16+0.33/17+0.18/18+0.09/19,	
A	˙¿B
=0.06/1+0.17/2+0.31/3+0.53/4+0.75/5+0.97/6+1.0/7+1.0/8+1.0/9+
+1.0/10+1.0/11+1.0/12+1.0/13+1.0/14+1.0/15+0.64/16+0.41/17+
   +0.21/18+0.09/19.
( 16-rasmga qarang ). Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)16-rasm. Noravshan to’plamlarning cheklangan yig’indisi 
А   va   В   –   X   dagi mos   ravishda  	
μA   va   	μB   tegishlilik funksiyali ikkita
noravshan   to’plam   bo’lsin.   A   va   B   noravshan   to’plamlarning   cheklangan
ko’paytmasi   amali   ularning   tegishlilik   funksiyalariga   qarab,   quyidagi   tarzda
aniqlanadi.
A=0.03/1+0.15/2+0.5/3+0.77/4+0.93/5+1.0/6+0.96/7+0.85/8+0.71/9+
     +0.55/10+0.4/11+0.27/12+0.18/13+0.11/14+0.05/15+0.01/16,
B=0.04/5+0.1/6+0.17/7+0.28/8+0.4/9+0.55/10+0.71/11+0.89/12+0.98/13
+
     +1.0/14+0.93/15+0.65/16+0.2/17+0.06/18+0.01/19,	
A¿¿B
=0/1+0/2+0/3+0/4+0/5+0.1/6+0.13/7+0.13/8+0.11/9+0.1/10+0.11/11+0.16/1
2+0.16/13+0.11/14+0/15+0/16+0/17+0/18+0/19.
( 17-rasmga qarang ) . Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)17 -rasm.   A va B noravshan to’plamlarning cheklangan ko’paytmasi  
Cheklangan va simmetrik ayirmalar.
Norvshan   to’plamlarning   cheklangan   ayirmasi  	
|−|   quyidagi   formula
bilan aniqlanadi:	
∀	x∈X	,μA|−|B(x)=	max	(0,μA(x)−	μB(x))
.	
A|−|B
 elementlari  B  dan ko’ra  A  ga ko’proq tegishli bo’lgan noravshan
to’plam.
Noravshan to’plamlarning simmetrik ayirmasi – bu,   B   ga qaraganda   A
ga ko’proq tegishli bo’lgan 	
¿t  elementlarning noravshan to’plami:	
∀	x∈X	,μA∇B(x)=|μA(x)−	μB(x)|
.
A   va   B   noravshan   to’plamlarning   cheklangan   va   simmetrik
ayirmalariga misollar :
A=0.08/1+0.23/2+0.45/3+0.7/4+0.86/5+0.96/6+1/0/7+0.98/8+
     +0.92/9+0.82/10+0.67/11+0.47/12+0.3/13+0.13/14,
B=0.03/6+0.08/7+0.18/8+0.34/9+0.55/10+0.7/11+0.84/12+0.94/13+ +0.99/14+1.0/15+0.96/16+0.82/17+0.6/18+0.2/19,A|−|B
=0.08/1+0.23/2+0.45/3+0.7/4+0.86/5+0.93/6+0.92/7+0.8/8+
     +0.58/9+0.27/10+0/11,	
А∇В
=0.08/1+0.23/2+0.45/3+0.7/4+0.86/5+0.96/6+1.0/7+0.98/8+
       0.92/9+0.82/10+0.03/11+0.36/12+0.65/13+0.86/14+1.0/15+
       0.96/16+0.82/17+0.6/18+0.2/9.
(18. а  va 18.b-rasmlarga qarang ).
Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)
18. а -rasm. Cheklangan ayirma Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)18.b-rasm. Simmetrik ayirma
A noravshan to’plamning m-darajasi  quyidagiga teng:	
μAm(x)=	[μA(x)]m,	∀	x∈	X	,∀	m	∈R+
,
bu yerda 	
R+  - musbat aniqlangan  haqiqiy sonlar to’plami.
Noravshan to’plamlar konsentrasiyasi, kengaytmasi . 
A  quyidagi universumda noravshan to’plam bo’lsin:	
A=	{(x:μA(x))|x∈	X	}
.
U   holda  	
Con	m   konsentrasiyalash   amali   yordamida   darajaga   ko’tarish
natijasida hosil bo’ladigan noravshan to’plamlar 	
Con	mA=	{(x:(μA(x))m)|x∈	X	}
A   ning   konsentrasiyalari,   kengaytma   amali   yordamida   ildiz   olish	
dil	nA=	{(x:n√μA(x))|x∈X	}
 esa A ning kengaytmalari deyiladi.
Natija .  	
[μA(x)]
n≤	μA(x)≤n√μA(x)     ifoda   hamma  	x∈X   larda   haqiqiy
bo’lsa   va  	
n>1   bo’lsa,   u   holda  	Con	nA⊂A⊂dil	nA   qism   to’plamlarning
munosabati ham haqiqiy hisoblanadi. 
Noravshan to’plamning konsentrasiyasi:  n=2. A=0.03/1+0.1/2+0.21/3+0.37/4+0.57/5+0.8/6 +0,96/7+1.0/8+0.94/9+
     +0.7/10+ 0.42/11+0.27/12+0.17/13+0.09/14+0.03/15,
 А2 = 0.0009/1+0.01/2+0.044/3+0.137/4+0.325/5+0.64/6+0.92/7+1.0/8+
    +0.884/9+0.49/10+0.174/11+0.07/12+0/03/13+0/01/14+0/0009/15.
( 19-rasmga qarang ).	
Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)
19-rasm .  Noravshan to’plamlarning konsentrasiyasi 
Noravshan to’plamning kengaytmasi  n=2.
A=0.03/2+0.06/3+0.13/4+0.23/5+0.4/6+0.61/7+0.82/8+0.96/9+
+1.0/10+0.94/11+0.74/12+0.51/13+0.33/14+0.23/15+0.16/16+0.1/17+
     +0.05/18+0.02/19,	
A1/2
=0.17/2+0.25/3+0.36/4+0.48/5+0.63/6+0.78/7+0.9/8+0.98/9+1.0/10+
 +0.97/11+0.86/12+0.72/13+0.57/14+0.48/15+0.4/16+0.3/17+                  +0.22/18+0.15/19
   ( 20-rasmga qarang ).Основной	Основной	Основной	
Основной
Основной
Основной	
x	
m(x)
20-rasm .  Noravshan to’plamlarning kengaytmasi 
Noravshan   to’plamlarni   konsentrasiya   va   kengaytma   amallaridan
foydalangan holda almashtirish misollari quyida keltirilgan [6].
А
=	
∫	μA(x)/x
Juda  А
=	
∫	[μA(x)]
2/x
Juda juda   А
=	
∫	[μA(x)]
4/x
Bir muncha  А
=	
∫	√μA(x)/x
Ozgina  А
=	
∫	4√μA(x)/x
А  emas
=	
∫(1−μA(x))/x
Uncha  А  emas
=	
∫	(1−[μA(x)]
2)/x Noravshan nuqtalar, noravshan oraliqlar, noravshan sohalar .
Noravshan nuqta haqiqiy   R   to’g’ri chiziqning qavariq noravshan qism
to’plamidir.
[101]   da   ko’rsatilishicha,   noravshan   nuqtalar   shu   noaniqlikni
akslantiruvchi qismlarga nisbatan simmetrik oraliqlar yordamida tasvirlanadi
(giperpiramidal akslantirish holida).
Elliptik   giperparaboloid   holida   taqdim   etilgan   noaniqlik   fazoning
barcha yo’nalishlariga matematik statistikada  х0   kuzatilayotgan nuqta holiga
nisbatan   kovariasion   matrisaga   o’xshash   rolni   o’ynovchi   matrisa   yordamida
hisobga olinadi. 
Agar   oraliqning   chegaralari   normal   qavariq   noravshan   to’plamlar
bo’lsa, u holda u noravshan oraliq deyiladi. 
  Noravshan   oraliqlar   yadroni   shakllantiruvchi   ravshan   oraliqni   tanlash
yordamida   aniqlanib,   undan   boshlab   tegishlilik   funksiyalari   nolgacha
kamayib   boradi,   yoki   oraliqning   uchlari   sifatida   ikkita   noravshan   sonni
tanlash   orqali   aniqlanishi   mumkin.   Umuman   olganda,  	
Rk   fazoda   tegishlilik
funksiyalari   monoton   tarzda   nolgacha   kamayib   boruvchi   noravshan   o’tish
zonasi   bilan   qurshab   olingan   ravshan   hududni   tanlash   asosida   noravshan
hududni qurish mumkin. Noravshan sohani tasvirlashning ustuvor usuli - bu
uning   chegarasini  hosil   qiluvchi   noravshan   giperyuzani  aniqlashdir.  Bunday
noravshan giperyuza o’z yadrosining ravshan giperyuzasiga ega bo’lib, undan
uzoqlashib   borgan   sari   tegishlilik   funksiyalarining   qiymatlari   barcha
yo’nalishlar bo’ylab monoton kamayib boradi. 
t -normalarga asoslangan amallar 
t -norm   -   bu   [0,1]   dagi   binar   t   amal,   ya’ni   kommutativ,   assotsiativ   va
[0,1] da  monoton kamayuvchi [0,1] dan iborat  t  binar funksiya bo’lib, neytral
element   sifatida   1   ga   va   nol   element   holida   0   ga   egadir.     Bunda   t   uchun
ixtiyoriy  	
x,y,z,u,v∈[0,1	]   larda   t -normaga   nisbatan   quyidagi   shartlar
bajarilishi kerak [101]:
xty=ytx,
xt(ytz)=(xty)tz.
Agar 	
x≤	u  va 	y≤	v bo’lsa, u holda 	xty	≤	utv ;  х t1=x     va    xt0=0 . Har bir   t -normaga nisbatan noravshan to’plamlar ustidagi  ¿t   kesishma
amalini barcha 	
x∈X  uchun hosil qilib olish mumkin: 	
μA∩tB(x)=	μA(x)tμB(x)
.
Barcha   kesishma   amallari   mos   t -normalardan   huddi   shu   shaklda   hosil
qilinadi. 	
A∩	B   uchun  	t0  mos  t -norm   amali bo’lib, unda 	u,v∈[0,1	]  uchun:	
ut	0v=	min	{u,v}
.
Algebraik   ko’paytma  	
t1  	u,v∈[0,1	]   uchun   quyidagi   t -normadan   hosil
qilinadi:	
ut	1v=	uv
.
Cheklangan   ko’paytma  	
t2      	u,v∈[0,1	]   uchun   quyidagi   t -norma   bilan
xarakterlanadi:	
ut	2v=[u+v−1]+
.
Qat’iy (drastic)  ko’paytma 	
t3  quyidagi  t -norma yordamida hosil qilinadi:  	
ut3v=¿{min	{u,v},agar	u=1	yoki	v=1,u,v∈[0,1	]uchun	,¿¿¿¿
To’ldirma   amalini  	
¿t   kesishma   amali   bilan   qo’llab,   ikkilamchi   t -
normaga asoslangan 	
¿t  birlashma amalini hosil qilish mumkin:	
A∪tB=(¯A∩t¯B)−
.
t -norma   asosidagi   kesishma   va   birlashma   amallarining   asosiy   g’oyasi
min   amalini   t -norma   bilan   almashtirishdan   iboratdir.   Bu   g’oya   noravshan
kartezian ko’pyatmaga nisbatan ham qo’llanilishi mumkin. Bunda   t -normaga
asoslangan kartezian ko’paytmadan foydalaniladi: 	
μA⊗tB(u,v)=	μA(u)tμB(v),∀	u,v∈	X
.
Ko’rinib   turganidek,   noravshan   to’plamlar   ustida   olib   boriladigan
amallarga   mo’ljallangan   keng   qamrovli   operatorlar   spektri   mavjud.   Qanday
hollarda qanaqa operatorlardan foydalanish masalasi katta qiziqish tug’diradi.
[5] da mos operatorlarni tanlashning 8 ta mezoni keltiriladi: aksiomatik kuch; empirik saqlash; moslashish imkoni; hisoblash samaradorligi; o’rnini bosish;
o’rnini   bosish   chegaralari;   amalning   hatti-xarakati;   tegishlilik   funksiyalarini
shkalashtirishning zaruriy darajasi . 
F-to’plamlar.
F -to’plamlar   deb   ixtiyoriy   X   to’plamning   F(X)   noravshan   qism
to’plamlariga   aytiladi,   ularning   tegishlilik   funksiyalarini   esa   F -funksiyalar
deb atashadi.   Odatda tegishlilik funksiyasi μA   deganda  X  to’plamni 	σ(A)  ga
qisqartirish   tushuniladi,   bu   yerda  	
σ(A)     noravshan   qism   to’plamning
tashuvchisidir:	
σ(A)=	{x|μA(x)>0}
.
F -to’plamni belgilash uchun quyidagi ko’rinishdagi yozuv qo’llaniladi:	
A=	⟨μA,σ(A)⟩
.
Masalan,	
A=	⟨exp	{−(x−	a)2},[c,d]⟩
,   	B=	⟨sin	x,[0,x]⟩ .
Ravshan   to’plamlarning   birlashmasi   va   kesishmasi   kommutativ,
assotsiativ     bo’lib,   shuningdek   bir-biriga   nisbatan   distributiv   xossalarga
egadirlar.   F -to’plamlarning   shu   kabi   xossalarini   aniqlash   quyidagi
funksiyalarini tahlil qilishga keltiriladi [20,21]:	
f(α,β)=	max	(α,β)
,	
g(α,β)=	min	(α,β)
,
bu yerda	
α=	μA(x),β=	μB(x),A	,B	∈F	(X	)
.
Quyidagi munosabatlar  f  va  g  funksiyalar xossalarining natijalaridir.
Bu yerda   	
A	,B	,C	,A1,...,An∈F(X	) .
1.	
A∪	A=	A	,A∩	A=	A .
2.	
A∪	B=	B	∪	A	,A∩	B	=	B	∩	A .
3.	
A∪	(B	∪	C	)=	(A∪	B	)∪	C	,A∩	(B∩	C	)=	(A∩	B	)∩	C . 4.A∪	(B∩	C	)=	(A∪	B	)∩	(A∪	C	);A∩	(B∪	C	)=	(A∩	B	)∪	(A∩	C	) .
5.	
A1=	A∪	B	,A2=	A∪	C	,B⊆C	⇒	A1⊆	A2 .
6.	
A1=	A∩	B	,A2=	A∩	C	,B⊆C	⇒	A1⊆	A2 .
7.	
A∪	∅	=	A .
8.	
A∩	∅	=	∅ .
9.	
A∪	X	=	X .
10.	
A∩	X	=	A .
Qabul   qilingan   belgilashlarda   quyidagi   to ’ rtta   turlar   F - to ’ plamlarning
kesishmasini   hamda   birlashmasini   ifodalaydilar  [30,54,100,106]:	
f1(α	,β)=	α∧	β
,                            	g1(α	,β)=	α∨	β ,	
f2(α,β)=	α⋅β
,                              	g2(α	,β)=	1−	(1−	α)(1−	β) ,	
f3(α	,β)=	√α⋅β
,                            	g3(α,β)=	1−	√(1−	α)(1−	β) ,	
f4(α,β)=	αβ	+√αβ	(1−	α)(1−	β)
,	
g4(α,β)=	1−	[(1−	α)(1−	β)+√αβ	(1−	α)(1−	β)]
.
F - to ’ plamning   qayd   etilgan   kesishma   va   birlashma   variantlari   min   va   max
funksiyalari   orqali   ifodalangan   ta ’ rifni   ma ’ lum   darajadagina   qanoatlantiradi .
F ( X )   dan   olingan   A   va   B   to ’ plamlarning   ayirmasi   deb   quyidagi   ko ’ rinishdagi
F   funksiyali   С=A\B   to ’ plamga   aytiladi :	
μC(x)=	μA(x)−	νA∩B(x)=	
¿μA(x)−	min
x∈X
(μA(x),μB(x))=	
¿max
x∈X
(0,μA(x)−	μB(x)).
Х \ А     ayirma   A   to’plamning   F -to’ldiruvchisi   deb   ataladi   va   A’   bilan
belgilanadi. 	
μA=	1−	μA(x)
. F(X)  dan olingan  A  va  B  uchun quyidagi munosbatlar o’rinli:
1.A	¿=	∅ .
2.	
A	¿	⊆	A .
3.	
A	¿	¿	¿ .
4.	
A	⊆	B	⇔	A	¿=	∅ .
5.	
A	∩	B	=	∅	⇔	A	¿=	∅ .
6.	
(A∪	B)=	A	∪	B .
7.	
(A∩	B)=	A	∩	B .
8.	
A	⊆	B	⇔	B	⊆	A .
6   va   7   tengliklar   de   Morgan   qoidalari   deb   ataladilar   va   mos   ravishda
quyidagi ayniyatlardan kelib chiqadilar:	
1−	max	(μA,μB)=	min	(1−	μA,1−	μB)
;	
1−	min	(μA,μB)=	max	(1−	μA,1−	μB)
. Foydalanilgan adabiyotlar
1. Muhamediyeva   D.T.   Noravshan     axborot   holatida   sust   shakllangan
jarayonlarni modellashtirish.   Toshkent: O’zR FA   matematika va axborot
texnologiyalar instituti, 2010. 37 ta jadval, 87 ta rasm, 155 ta bibl.atama, 400
bet. 
2. Артикова   С.,   Мухамедиева   Д.Т.     Информатизация   регулирования
развития экономики Республики // Известия ВУЗов. –Т., 2000. №3.
3. Артикова С., Мухамедиева Д.Т. Реализация моделей принятия решений
с учетом информационных ситуаций //Узбекский журнал энергетики и
информатики.-Т. ,2000. №3.
4. Ахмедов   Т.М.   Мухамедиева   Д.Т.   Шодмонова   У.А.   Рациональное
управление   распределением   и   использованием   ресурсов   в   условиях
рыночной   экономики.   Доклады   международной   конференции
«Устойчивое   экономическое   развитие   и   эффективное   управление
ресурсами   в   Центральной   Азии».   ТГЭУ   и   Ноттенгемский   Трент
Университет (Великобритания). Ташкент-Ноттенгем. 2001. –С.14-17.

MAVZU : Noravshan to’plamlar ustida amallar REJA: 1. Noravshan to’plamlarlarni to’ldirish. 2. Noravshan to’plamlarning kesishmasi, birlashmasi 3. Amallarning umumlashgan ta’riflari: t-norma, s-norma

Klassik to’plamlar uchun quyidagi amallar kiritilgan: To’plamlarning kesishmasi – A va B to’plamlardagi ham A , ham B to’plamga tegishli elementlardan iborat bo’lgan С = А  В to’plamidir. To’plamlarning birlashmasi - A va B to’plamlardagi yoki A , yoki B , yoki ikkala to’plamga tegishli elementlardan iborat bo’lgan С = А  В to’plamidir. To’plamlarning inkori - universal to’plamga tegishli, lekin A to’plamga tegishli bo’lmagan elementlarni o’z ichida mujassamlashtirgan С =  А to’plamidir . Zade shu to’plamlarning tegishlilik funksiyalari amallari yordamida noravshan to’plamlar ustidagi shu kabi amallar majmuini taklif qildi [35]. Shunday qilib, A to’plam  А (u), В to’plam esa  В (u) funksiya orqali berilgan bo’lsa, u holda natija bo’lib  С (u) tegishlilik funksiyali C to’plam hisoblanadi. Birlashma. A va B noravshan to’plamlarning birlashmasi quyidagi tarzda aniqlanadi: ∀ x∈ X ,μA∪B(x)= max {μA(x),μB(x)} , bu yerda μA∪B - A va B uchun tegishlilik funksiyasi. Kesishma . μA∩B tegishlilik funksiyasi quyidagicha aniqlanadi: ∀ x∈ X ,μA∩B(x)= min {μA(x),μB(x)} . А va В – X dagi mos ravishda μA va μB tegishlilik funksiyali ikkita noravshan to’plam bo’lsin. Noravshan to’plamlar ustidagi birlashtirish amali ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi: A=0.07/2+0.2/3+0.4/4+0.63/5+0.87/6+1.0/7+0.89/8+0.5/9+ +0.2/10+0.07/11, B=0.05/6+0.11/7+0.21/8+0.32/9+0.46/10+0.69/11+0.87/12+ +1.0/13+0.9/14+0.5/15+0.25/16+0.09/18,

A∪ B= 0.07 /2+0.2/3+0.4/4+0.63 /5+0.87 /6+1.0/7+0.89 /8+ +0.5/9+0.46 /10 +0.69 /11 +0.87 /12 +1.0/13 +0.9/14 +0.5/15 + +0.25 /16 +0.09 /18 .(11-rasmga qarang ). Основной Основной Основной Основной Основной Основной x m(x) Основной Основной Основной x m(x) 11-rasm. Noravshan to’plamlarning birlashmasi А va В – X dagi mos ravishda μA va μB tegishlilik funksiyali ikkita noravshan to’plam bo’lsin. Noravshan to’plamlar ustidagi kesishma amali ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi: A= 0.15/2+0.41/3+0.66/4+0.85/5+0.97/6+1/7+0.9/8+0.6/9+ +0.42/10+0.3/11+0.18/12+0.1/13+0.03/14, B=0.05/5+0.1/6+0.16/7+0.25/8+0.35/9+0.47/10+0.62/11+

0.8/12+0.94/13+1/14+0.97/15+0.83/16+0.5/17+0.2/18+0.07/19,A∩ B =0.05/5+0.1/6+0.16/7+0.25/8+0.35/9+0.42/10+0.3/11+ +0.18/12+0.1/13+0.03/14. (12-rasmga qarang ). Основной Основной Основной Основной Основной Основной x m(x) Основной Основной Основной Основной Основной Основной Основной Основной Основной Основной Основной Основной Основной ОсновнойОсновнойОсновной x m(x) 12 -rasm. Noravshan to’plamlarning kesishmasi To’ldirma . A to’plamning ¯А to’ldirmasi quyidagicha aniqlanadi : ∀ x∈ X , μ¯A(x)= 1− μA(x) . А va В – X dagi mos ravishda μA va μB tegishlilik funksiyali ikkita noravshan to’plam bo’lsin. Noravshan to’plamlar ustidagi to’ldirish amali ularning tegishlilik funksiyalariga qarab quyidagi tarzda aniqlanadi: A=0/1+0.05/2+0.14/3+0.27/4+0.5/5+0.76/6+0.93/7+1.0/8+0.96/9+0.84/10+ +0.62/11+0.37/12+0.25/13+0.16/14+0.09/15+0.03/16+0/17,

¯A=1.0/1+0.95/2+0.86/3+0.73/4+0.5/5+0.24/6+0.07/7+0/8+0.04/9+0.16/10+ +0.38/11+0.63/12+0.75/13+0.84/14+0.91/15+0.97/16+1.0/17. (13-rasmga qarang). Noravshan to’plamlarning birlashmasi va kesishmasi uchun boshqa amallardan ham foydalanish mumkin. Algebraik ko’paytma: ∀ x∈ X ,μA⋅B(x)= μA(x)⋅μB(x) . Cheklangan ko’paytma: ∀ x∈ X ,μA⊗B(x)= max {0,μA(x)+μB(x)− 1} . Основной Основной Основной Основной Основной Основной x m(x) Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Осно... Основной Основной Основной x m(x0 13-rasm. Noravshan to’plamning to’ldirmasi