Noravshan to’plamlar ustida amallar
![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](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_1.png)
![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,](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_2.png)
![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 ).
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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+](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_3.png)
![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 ).
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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,](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_4.png)
![¯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}
.
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13-rasm. Noravshan to’plamning to’ldirmasi](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_5.png)
![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 ).](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_6.png)
![Основной Основной Основной
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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 ).](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_7.png)
![Основной Основной Основной
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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 ).](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_8.png)
![Основной Основной Основной
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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 ) .](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_9.png)
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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+](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_10.png)
![+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 ).
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18. а -rasm. Cheklangan ayirma](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_11.png)
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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.](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_12.png)
![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 ).
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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+](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_13.png)
![+0.22/18+0.15/19
( 20-rasmga qarang ).Основной Основной Основной
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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](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_14.png)
![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 .](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_15.png)
![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;](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_16.png)
![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 .](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_17.png)
![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)
.](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_18.png)
![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)
.](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_19.png)
![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.](/data/documents/b4ea43eb-9cbb-4f76-9142-d26e1475fb6c/page_20.png)
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