4-Ta’rif. to‘plamning o‘zi va to‘plam shu to‘plamning xosmas qism to‘plami deyiladi.
5-Ta’rif. Agar A1, A2,..., An to’plamlar A to’plamning qism to’plami bo’lsa, A to’plam A1, A2,..., An to’plamlar uchun universal to’plam deyiladi.
Z to`plam R to`plamning xos qism to`plami ekan Z R, ko`rinishda belgilanadi. Xuddi shunday munosabatni barcha kompleks sonlar to`plami C va ratsional sonlar to`plami Q, haqiqiy sonlar to`plami R uchun ham o`rnatish mumkin:
Z ⊆ Q ⊆ R ⊆ C
Geometriyadan misol keltirsak, R3 – uch o`lchovli fazo bo`lsa, П – R3 fazodagi tekislik, L – П tekislikdagi chiziq bo`lsa, quyidagi munosabat o`rinli bo`ladi: L П ⊂ R3 yoki L ⊆ П ⊆ R. Bu yerda R3 ning boshqa ko`p qism to`plamlari ham mavjudligini hisobga olish kerak.
Since Z is a subset of R we have the familiar notation Z ⊆ R; if we wish to emphasize that they’re different sets (or that Z is properly
contained in R), we write Z ⊂ R (some authors write Z ⊆ R). Likewise, if we let C be the set of all complex numbers, and consider also the set Q of all rational numbers, then we obviously have
Z ⊆ Q ⊆ R ⊆ C.
As a more geometrical sort of example, let us consider the set R3 of all points in Cartesian 3-dimensional space. There are certain naturally defined subsets of R3, the lines and the planes. Thus, if П is a plane in R3, and if L is a line contained in П, then of course we may write either L ⊂ П ⊂ R3 or L ⊆ П ⊆ R3. Note, of course, that R3 has far more subsets that just the subsets of lines and planes!1
Universal to’plam, odatda, J yoki U harflari bilan belgilanadi. Uuniversal to‘plamning barcha qism to‘plamlari orasida ikkita xosmas qism to‘plam mavjud bo‘lib, ulardan biri ning o‘zi, ikkinchisi esa bo‘sh to‘plam, qolganlari esa xos qism to‘plamlar bo‘ladi.Masalan, N — barcha natural sonlar to’plami; Z— barcha butun sonlar to’plami; Q — barcha ratsional sonlar to’plami; R— barcha haqiqiy sonlar to’plami bo’lib, NZQRshartlar bajariladi va R qolgan sonli to’plamlar uchun universal to’plam vazifasini bajaradi.
R to’plamning to’plam ostisini koordinatalar o’qida tasvirlash mumkin. Agar va a bo’lsa, quyidagi bеlgilashni kiritish mumkin.
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