To’plamlar va ular ustida amallar. Reja: Тўпламлар устида амаллар. Тўпламларни солиштириш. Математик индукция усули

Z to`plam R to`plamning xos qism to`plami ekan Z <sub> </sub> 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 <sub> </sub> П ⊂ R<sup>3</sup> yoki L ⊆ П ⊆ R. Bu yerda R<sup>3</sup> 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, N_{ }Z_{ }Q_{ }Rshartlar 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.