To'plamlarning to'g'ri ko'paytmasining xossalari Reja: To‘plamlarning dekart ko‘paytmasi

3
Koordinata tekisligida shunday koordinatali nuqtalarni tasvirlaymizki, bunda A to’plam Ox o’qida va B to’plam Oy o’qida olinadi.

A={-2;2}; B=R A=[-2;4]; B=R



Dekart ko’paytmaning xossalari:

1°. A×B≠B×A.

2°.A ×(B∪C) = (A×B)∪(A×C).

3°. A×(B∩C) = (A×B)∩(A×C).

We have already encountered an elementary construction on a given set: that of the power set. That is, if S is a set, then 2^S is the set of all subsets of the set S. Furthermore, we saw in the theorem on page 189 that if S is a finite set containing n elements, then the power set 2^S contains 2ⁿ elements (which motivates the notation in the first place!). Next, let A and B be sets. We form the Cartesian product A × B to be the set of all ordered pairs of elements (a, b) formed by elements of A and B, respectively. More formally,

A × B = {(a, b) | a ∈ A and b ∈ B}.

From the above, we see that we can regard the Cartesian plane R² as the Cartesian product of the real line R with itself: R² = R × R. Similarly, Cartesian 3-space R³ is just R × R × R.
Ikkitadan ortiq to’plamlarning dekart ko’paytmasini ham qarash mumkin. Umumiy holda A₁ A₂ ..., A_n to’plamlar berilgan bo’lsin. Ularning dekart ko’paytmasi A₁×A₂×...,×A_n= {(a₁ a₂; ..., a_n) | a₁∈A₁,a₂∈A₂, ..., a_n∈A_n dan iborat bo’ladi. (a₁; a₂; ..., a_n) tartiblangan n lik deyiladi. (Masalan, uchlik, to’rtlik va h.k.). bunday tartiblangan n lik n o’rinli kortej deb ham ataladi. Yana n o’rinli kortejlar faqat bitta to’plam elementlaridan tuzilgan bo’lishi ham mumkin, bu holda u to’plamni o’z-o’ziga n marta dekart ko’paytmasi elementidan iborat bo’ladi.

Yuqorida aytilganlardan xulosa qilsak, Dekart koordinata tekisligini haqiqiy sonlar to’plami R ni o’ziga-o’zining dekart ko’paytmasi R²=R×R, koordinata fazosini R³ =R×R×R deb qarash mumkinligi kelib chiqadi.¹
A va B to’plamlarning to’g’ri (dekart) ko’paytmasi ko’rinishida belgilanib, u quyidagicha aniqlanadi:

.

Masalan,1. .

2. ²



to’plamlarning to’g’ri (dekart) ko’paytmasi esa quyidagicha aniqlanadi:

.

Agar bu to’plamlar bir-biriga teng bo’lsa, ni ko’rinishida yozishimiz mumkin, ya’ni , shuningdek n=1 hol uchun

tenglikka ega bo’lamiz. Agar dagi binar munosabat f uchun va dan kelib chiqsa, u holda A to’plamni B to’plamga o’tkazuvchi funktsiya (akslantirsh) berilgan deyiladi. Odatda ni ko’rinishda belgilaymiz. ³