Noravshan to’plamlarning arifmetikasi
![2 Kirish
Noravshan to’plam kons е psiyasi, Zad е ning fikricha, haqiqiy dunyoning
tizimlarida, ayniqsa odamlarni o’z ichiga olgan gumanistik tizimlarda noo’rin sun'iy
aniqlikka erishishni talab qilgan tizimlarning klassik nazariyasiga oid mat е matik
usullardan qoniqmaslik [35] hisobiga tug’ilgan.
Noravshan to’plamlar nazariyasi 1975 yilda amaliyotda qo’llanilgan bo’lib,
bunda Mamdani va Assilian (Mamdani and Assilian)lar oddiy bug’ dvigat е lini
boshqarish maqsadida birinchi noravshan hisoblagichni qurganlar [101,129].
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.
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.
(1.2.5-rasmga qarang ).
0 2 4 6 8 10 12 14 16 18 20 0
0.2
0.4
0.6
0.8
1
1.2
x
m(x)
1.2.5-rasm . Noravshan to’plamlarning algebraik yig’indisi](/data/documents/e0d58e53-bbfa-49a5-8448-3e00fa9f38ea/page_2.png)
![61.2.8.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.
( 1.2.9-rasmga qarang ).](/data/documents/e0d58e53-bbfa-49a5-8448-3e00fa9f38ea/page_6.png)
![80 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
x
m(x)1.2.10-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 .](/data/documents/e0d58e53-bbfa-49a5-8448-3e00fa9f38ea/page_8.png)
1Mavzu: Noravshan to’plamlarning arifmetikasi Reja: Kirish 1. Noravshan to’plamlarning algebraik yig’indisi 2. Noravshan to’plamlarning cheklangan yig’indisi Foydalanilgan adabiyotlar
2 Kirish Noravshan to’plam kons е psiyasi, Zad е ning fikricha, haqiqiy dunyoning tizimlarida, ayniqsa odamlarni o’z ichiga olgan gumanistik tizimlarda noo’rin sun'iy aniqlikka erishishni talab qilgan tizimlarning klassik nazariyasiga oid mat е matik usullardan qoniqmaslik [35] hisobiga tug’ilgan. Noravshan to’plamlar nazariyasi 1975 yilda amaliyotda qo’llanilgan bo’lib, bunda Mamdani va Assilian (Mamdani and Assilian)lar oddiy bug’ dvigat е lini boshqarish maqsadida birinchi noravshan hisoblagichni qurganlar [101,129]. 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. 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. (1.2.5-rasmga qarang ). 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 x m(x) 1.2.5-rasm . Noravshan to’plamlarning algebraik yig’indisi
3А 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. ( 1.2.6-rasmga qarang ). 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 x m(x) 1.2.6 -rasm. Noravshan to’plamlarning cheklangan yig’indisi 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+0.16/12+0.16/1 3+0.11/14+0/15+0/16+0/17+0/18+0/19. ( 1.2.7-rasmga qarang ) .
40 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 x m(x)1.2.7 -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,
5А∇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. (1.2.8. а va 1.2.8.b-rasmlarga qarang ). 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 x m(x) 1.2.8. а -rasm. Cheklangan ayirma 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 x m(x)