Ilmiybaza.uz
To‘plamlarning dеkart ko‘paytmasi. To‘plamlar ustidagi amallarning
xossalari
Ma’ruza mashg’ulotining rejasi:
1. To‘plamlarning dekart ko‘paytmasi.
2. To‘plamlar ustidagi amallarning xossalari.
Ilmiybaza.uz
To‘plamlarning dekart (to‘g‘ri) ko‘paytmasi.
va
to‘plamlarning
to‘g‘ri ko‘paytmasi deb shunday to‘plamga aytiladiki, u to‘plam elementlari
tartiblangan
juftliklardan iborat bo‘lib, bu juftni birinchisi
to‘plamdan,
ikkinchisi esa
to‘plamdan olinadi. To‘g‘ri ko‘paytma
ko‘rinishda
belgilanadi.
Misol:
va
to‘plamlar berilgan bo‘lsin. U holda
va
to‘plamlarning to‘g‘ri ko‘paytmasi quyidagicha bo‘ladi:
Agar biz to‘g‘ri ko‘paytma elementi
dagi
ni biror nuqtani
abssissasi,
ni esa ordinatasi desak, u holda bu to‘g‘ri ko‘paytma tekislikdagi
nuqtalar to‘plamini ifodalaydi.
Boshqacha aytganda haqiqiy sonlar to‘plami
ni
ga to‘g‘ri ko‘paytmasi
ni tasvirlaydi.
Ta’rif. A va B to’plamlarning dekart ko’paytmasi deb, 1-elementi A
to’plamdan, 2-elementi B to’plamdan olingan (a; b) ko’rinishdagi barcha
tartiblangan juftliklar to’plamiga aytiladi. Dekart ko’paytma A ×B ko’rinishda
belgilanadi: A×B = {(a; b) | a∈A va b∈B}.
Masalan: A = {2; 3; 4; 5}, B = {a; b; c} bo’lsa, A × B = {(2; a),
(2; b),(2; c),(3; a),(3; b),(3; c),(4; a),(4; b),(4; c),(5; a), (5; b),(5; c)}
bo’ladi.
Sonli
to’plamlar
dekart
ko’paytmasini
koordinata
tekisligida
tasvirlash qulay. Masalan, A = {2; 3; 4},
B = {4; 5} bo’lsin, u holda A × B = {(2;
4), (2; 5), (3; 4), (3; 5); (4; 4), (4; 5)}
bo’ladi.
Koordinata tekisligida shunday
koordinatali nuqtalarni tasvirlaymizki,
bunda A to’plam Ox o’qida va B
to’plam Oy o’qida olinadi.
A
B
( , )
x y
A
B
B
A
}
7,5,4
A {
}
4,3,2,1
{
B
A
B
{( ;4 1),( 2;4 ),( 3;4 ),( 4;4 ),( ;5 1),( 2;5 ),( 3;5 ),( 4;5 ),( ;7 1),( 2;7 ),( 3;7 ),( 4;7 )}
B
A
( , )
x y
x
y
R
R
R
R
3
![Ilmiybaza.uz
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 2S 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 2S 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.](/media/image/toplamlarning-dekart-kopaytmasi-toplamlar-ustidagi-amallarning-xossalari1219page_3.png "Ilmiybaza.uz
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 2S 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 2S 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.")
Ilmiybaza.uz
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 2S 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 2S 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.
