Klassik to’plamlar uchun quyidagi amallar kiritilgan

1.2. Noravshan to’plamlar ustida amallar

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:
,
bu yerda - A va B uchun tegishlilik funksiyasi.
Kesishma.
tegishlilik funksiyasi quyidagicha aniqlanadi:
.
А va В – X dagi mos ravishda va 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,

(1.2.1-rasmga qarang).

1.2.1-rasm. Noravshan to’plamlarning birlashmasi

А va В – X dagi mos ravishda va 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,
=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.
(1.2.2-rasmga qarang).

1.2.2-rasm. Noravshan to’plamlarning kesishmasi

To’ldirma.
A to’plamning to’ldirmasi quyidagicha aniqlanadi:
.
А va В – X dagi mos ravishda va 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,
=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.
(1.2.3-rasmga qarang).
Noravshan to’plamlarning birlashmasi va kesishmasi uchun boshqa amallardan ham foydalanish mumkin.


Algebraik ko’paytma:
.

Cheklangan ko’paytma:
.


1.2.3-rasm. Noravshan to’plamning to’ldirmasi
Qat’iy (drastic) ko’paytma:

Algebraik yig’indi:
.
Cheklangan yig’indi:
.
Qat’iy (drastic) yig’ind:


А va В – X dagi mos ravishda va 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,

=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.
(1.2.4-rasmga qarang).

1.2.4-rasm. Noravshan to’plamlarning algebraik ko’paytmasi


А va В – X dagi mos ravishda va 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,

=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).

1.2.5-rasm. Noravshan to’plamlarning algebraik yig’indisi

А va В – X dagi mos ravishda va 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,
=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).

1.2.6-rasm. Noravshan to’plamlarning cheklangan yig’indisi

А va В – X dagi mos ravishda va 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,

=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+0.16/13+0.11/14+0/15+0/16+0/17+0/18+0/19.

(1.2.7-rasmga qarang).

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:
.
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 elementlarning noravshan to’plami:
.
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,


=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,

=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).