Логика. Множества. Вероятность. Лексаченко В.А. - 53 стр.

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V P
(Ω) vypolnts osnovnye sootnoxeni 21, 26 — 47 s
uqetom togo, qto znakam
∨
,
∧, ⊕ sootvetstvut znaki ∪
, ∩,
M
v
P(Ω), a znaki →, ∼, ↓, | (obyqno ne upotreblemye dl
mnoestv) opredelts ravenstvami 20, 22, 23, 25.
Sledstvie (Otnoxenie ⊆ v P(Ω)).
1. A
⊆B ravnosil~no lbomu
iz ravenstv:
A=
A
∩
B, B =
A ∪ B,
A ∪
B
=Ω
, A\B
=
∅
.
2
. Esli A⊆
B , to (
A
∩ C
)⊆
(
B ∩ C
),
(A ∪ C)⊆
(B ∪ C)
,
B⊆A.
Dokazatel~stvo predlagaets qitatel kak upraneni.
Teorema 1.7 (Osnovnoe svo$istvo upordoqenno$i pary).
`
(c, d
)=(a, b)
∼
(c
=
a)
∧(d
=
b
)
, gde
a, b, c, d
— termy.
D o k a z a t e l ~ s t v o . sno, qto `(c = a
)∧(
d = b) → (c, d) = (
a, b)
.
Imeem
[(c, d) = (a, b
)] →[{c
}∈(a, b
)] →[(c = a)∨
((c
= a)
∧(c
= b
))] ∼
c
=
a
.
Potomu
[(
c, d
) = (
a, b
)]
→
{a, d
} =
{
a, b
}]
→
[d = b
]
i
[(c, d) = (
a, b
)]
→
→[(c=
a)∧
(d=b)]. J
Sledstvie. `
(
x
1
, . . . , x
n
)=(y
1
, . . .
, y
n
)
∼
V
n
i=1
(x
i
=y
i
)
.
Teorema 1.8 (Svo$istva dekartovyh proizvedeni$i). Pust~
A, B, C, D ∈ P
(Ω)
. Togda:
1) (
A
∪ B)×
C
= (
A×C
)
∪
(
B
×
C),
2)
A×
(B ∪ C) = (
A×B
) ∪ (A×C),
3) (A×
B
)
∩
(
C×D
) = (A
∩
C)
×(
B ∩ D
)
,
4) (
A
∩
B)
×C
= (
A
×C) ∩
(
B×C
)
,
5) A×
(
B ∩ C
) = (A
×B)
∩
(A×
C)
,
6) (
A\B
)×C
= (A
×C
)\
(
B
×
C)
,
7) A×
(
B\C
) = (
A×B
)\(A
×C
)
,
8) esli A
=
∅
ili
B = ∅,
to A×
B
= ∅
,
9)
esli A
×
B
6
= ∅, to
(A×
B
)⊆
(C
×
D) ∼
(A
⊆C)∧(B
⊆D),
10) esli
C
6= ∅,
to
(
A⊆B
)
∼
(
A
×
C)⊆(B×
C
),
11) esli A 6=
∅,
to (B
⊆C) ∼
(A×B)⊆(A×
C)
.
D o k a z a t e l ~ s t v o . 1.
[(
x, y
)∈
((
A ∪B)
×
C
)]∼[((
x
∈
A
)
∨
(x∈B))∧
(
y ∈
∈C)]∼[((
x, y)
∈A×
C)∨((
x, y)∈B×C)]∼[(
x, y)
∈(A
×C) ∪
(B×C)].
2. Analogiqno dokazatel~stvu punkta 1.
3.
[(
x, y) ∈
(A
×
B)
∩ (
C×D
)]
∼ [(x ∈ A
)
∧
(
y
∈ B)
∧(
x
∈ C
)∧
(y ∈
D
)] ∼
∼
[(x, y
)∈
(
A ∩ C)
×
(B ∩
D)].
4. Poloit~ D = B v 3). 5. Poloit~ C = A v 3).
6. [(x, y
)
∈
(
A
×C)
\(
B
×C
)] ∼
[(x ∈
A
)∧(y ∈
C
)
∧
(
x∈B
)∧(
y ∈C
)] ∼
∼[(
x
∈A)
∧(
x∈
B
)∧(
y ∈
C
)]∼
[(x, y
)∈(
A
\B)×C
].
7. Analogiqno dokazatel~stvu punkta 6.
8. Iz
A
=
∅ (ili
B
= ∅
) sleduet, qto
((x, y
)
∈ A
×
B) = 0
i
potomu
A×B
= ∅
.
53

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