Логика. Множества. Вероятность. Лексаченко В.А. - 54 стр.

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9. Iz A
×B
6= ∅
sleduet
A 6= ∅, B
6
=
∅
. Po teoremam
I.2.1.20), I
.3.1.13)
[(A
×B
)⊆(
C
×D)] ∼ [(
∀
x
)(
∀
y)((((x
∈ A
)
→ (
x ∈
C))∨
∨
(
y /∈
B))∧
(((
y ∈ B) →
(
y
∈
D
))∨(
x /∈
A
)))] ∼
[((∀
x)((x
∈
A) → (x
∈
C
))
∨
∨(∀
y
)(y /
∈
B
))
∧
((
∀
x)(
x /∈
A
)
∨(∀y
)((
y ∈ B
)
→(y
∈ D)))]∼
[(A
⊆
B
)∧
(B⊆
D
)]
.
10. Poloit~
D = B
v 9). 11. Poloit~
C
=
A v 9). J
Opredelenie 1.5 (Proekcii mnoestv).
Pust~
C⊆A
×
B. Mnoes-
tva pr
1
C
{x
∈
A : (∃
y)((
x, y) ∈C)}
, pr
2
C
{y
∈B : (∃x)((x, y
) ∈C)}
nazyvats 1-$i i 2-$i proekcimi mnoestva C .
Teorema 1.9 (Svo$istva proekci$i).
Dl
C
⊆
A
×
B, D
⊆
A×
B, i = 1
,
2 :
1) C
⊆(pr
1
C
)
×
(pr
2
C
)
,
2) pr
i
(
C ∪
D) = (pr
i
C) ∪ (pr
i
D) ,
3) pr
i
(
C ∩
D)⊆(pr
i
C
)
∩ (pr
i
D) , 4) esli C
⊆
D, to
(pr
i
C)
⊆
(pr
i
D)
,
5) pr
1
C =
S
(x,y
)
∈C
{
x
}, 6) pr
2
C =
S
(
x,y)∈
C
{
y
}
.
D o k a z a t e l ~ s t v o . Dokaem tol~ko sootnoxeni 1 — 4.
1. Iz
(x, y)
∈ C
po pravilu
∃-vvedeni poluqim sootnoxe-
ni (
∃
y ∈
B)((x, y)
∈
C
)
i (∃x
∈ A)((
x, y
) ∈
C)
, oznaqawie, qto
x
∈
pr
1
C, y ∈ pr
2
C . Otsda sleduet, qto (x, y
)
∈
(pr
1
C)
×(pr
2
C) i,
znaqit, C
⊆(pr
1
C
)×(pr
2
C
).
2. Ispol~zu p. 2 teoremy I.3.1, poluqim [
x
∈
pr
1
(
C
∪
D)]
∼
∼
[(∃
y
∈
B
)((x, y
)
∈ C
∪ D)] ∼
[(
∃
y
∈ B)(((x, y
) ∈ C
)
∨ ((
x, y)
∈
D))] ∼
∼[(∃
y
∈ B)((x, y
) ∈
B) ∨ (∃
y
∈
B
)((x, y)
∈
D)]∼[(
x ∈
pr
1
C
) ∨ (
x ∈
pr
1
D
)]
∼
∼
[
x ∈
(pr
1
C
)
∪(pr
1
D)]
, otkuda sleduet
pr
1
(
C ∪D
) = (pr
1
C) ∪
(pr
1
D
)
.
Analogiqno dokazyvaets ravenstvo
pr
2
(
C ∪ D) = (pr
2
C
)
∪
(pr
2
D
).
3. Ispol~zu p. 4 teoremy
I
.3.1, poluqim [
x
∈
pr
1
(C ∩ D
)] ∼
∼
[(∃
y
∈
B
)((
x, y)
∈
C
∩ D)] ∼ [(∃y
∈
B)(((
x, y
)
∈
C
)∧
((
x, y
) ∈ D))] →
→
[(
∃
y
∈ B
)((x, y)
∈
C
)∧
(
∃
y ∈
B
)((x, y) ∈
D
)]
∼[(x
∈
pr
1
C)
∧(x
∈ pr
1
D
)] ∼
∼[
x ∈ (pr
1
C) ∩
(pr
1
D
)]
, otkuda
pr
1
(
C ∩ D
)
⊆(pr
1
C)
∩ (pr
1
D)
. Analo-
giqno dokazyvaets vklqenie pr
2
(
C ∩
D)
⊆
(pr
2
C)
∩
(pr
2
D
).
4. Poskol~ku
(
C
⊆
D) ∼ (
D = C
∪
D)
, to po svo$istvu 2 iz C
⊆
D
sleduet
pr
i
D
= (pr
i
C
) ∪
(pr
i
D)
, t. e.
(pr
i
C)⊆
(pr
i
D)
, i
= 1, 2
.
J
2. Binarnye otnoxeni
Opredelenie 2.1 (Binarnye otnoxeni). Binarnym otnoxeni-
em medu mnoestvami A i B
nazyvaets tro$ika mno-
estv (
R, A, B
), gde R⊆A
×
B
— lboe podmnoestvo mnoes-
tva
A
×
B , nazyvaemoe grafikom otnoxeni (R, A, B)
. Binarnoe
otnoxenie
(
R, A, A) nazyvaets otnoxeniem v mnoestve A.
Oblast~ opredeleni
(R, A, B) nazyvaets
Dom R
pr
1
R, oblas-
t~ znaqeni$i — Im R pr
2
R
. Pust~
X⊆
A, R
|
X
R ∩
(X×B
)
.
54

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