Логика. Множества. Вероятность. Лексаченко В.А. - 72 стр.

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6) [A
∪ B
= A ∩
B
] ∼ [A =
B
]
,
7) [A
⊆B] →
[(
A
\C)
⊆
(
B
\
C
)] ,
8) [
A
⊆(B
∩ C
)]
∼ [(
A⊆
B)∧
(
A⊆C)] ,
9) [(
B
\
A)⊆
C
] ∼ [A⊆
(
B ∪
C
)] ,
10) [
A⊆B
] →
[(
A
∪ C
)⊆(
B
∪
C)]
.
R e x e n i e dl [(A\
B
) ∪ B
) = A] ∼
[B
⊆
A
].
[(
A\
B)
∪ B
=
A
]
∼ [(
A
∩
B
)
∪
B = A
] ∼ [(A
∪ B
)
∩
(B
∪
B
) = A]
∼
∼
[(A
∪
B
)
∩
Ω =
A
] ∼
[A
∪
B
= A] ∼
(
B⊆A
)
. J
3 . Dokazat~ ravenstva:
1) (A
\B)×(C
∩ D) = (A×
C) ∩
(B
×D
) ,
2) (A
\
B)×
(C
\
D) = (A
×
C
)
∩
(B
×
D
)
,
3) (A ∩ B
)
×(
C
∩
D
) = (
A
×
C)
∩ (
B
×
D) ,
4)
A×
B
∪
C
= (
A
×
B
)
∩
(
A
×
C
)
,
5) A
∩
B
×C
= (A
×
C
) ∪
(B
×
C
) ,
6) A×(B
\
C) = (A
×
B)
∩
(A
×
C) ,
7) A
∪
B×C
= (
A
×
C) ∩
(
B
×
C) ,
8) (A\
B
)×C
= (
A×C
) ∩ (
B
×C
)
,
9) A×
(
B
\
C
) = (A×
B)
\
(A
×C
) ,
10) (
A\
B
)×C
= (A
×
C)\
(
B
×
C
)
.
R e x e n i e dl
(
A ∩ B)
×
(
C\D
) = (
A
×
C
)
∩ (B×
D)
.
(
x, y) ∈ (A ∩
B
)×
×(C\D
) = (
x ∈ A
∩
B)
∧
(y
∈
C
\
D) = (x ∈ A
)
∧(x
∈
B)
∧
(y
∈
C
)
∧(y /∈ D
) =
= ((x, y
) ∈ A×
B
)∧((
x, y
) ∈ B
×D
) = (x, y)
∈ (A×
C)
∩
(
B×D
). J
4
.
Dl sleduwih binarnyh otnoxeni$i na
R postroit~ gra-
fiki, na$iti oblast~ opredeleni i oblast~ znaqeni$i.
1)
x
6
y,
2) x
= y,
3)
x < y, 4)
x
2
+ y
2
61
, 5) x =
y
2
,
6) x
2
= y, 7)
x
2
= y
2
, 8) |
x − y
|
= 1, 9) y6
log
2
x,
10) tg
x = 1.
Kakie iz tih otnoxeni$i vlts otnoxenimi kvivalentnos-
ti, pordka, funkcional~nymi otnoxenimi?
5 . Pust~
f : M
→ R
— nepreryvna de$istvitel~na funkci,
opredelenna na
M
= [
−
a
1
, b
1
]×···×[
−a
n
, b
n
]. Dokazat~, qto otno-
xenie
R
f
= {
(
x
,
y )
∈ M
2
: f(x)
>
f
(
y )} vlets otnoxeniem pred-
pordka v
M .
6 .
Opredelit~ qislo razliqnyh otobraeni$i f : A
→
B koneq-
nyh mnoestv A, B . Pri kakih uslovih suwestvut srek-
cii, inekcii i biekcii? Opredelit~ qislo biekci$i?
7 . Dokazat~, qto intervaly
(a; b
) i (
c
; d
) pri koneqnyh
a, b, c, d (
a < b, c < d
) ravnomowny.
72

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