Логика. Множества. Вероятность. Лексаченко В.А. - 44 стр.

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10 . Privesti k KNF (ne uprowa):
1) (a
→ b)
∧a
→ b ,
2)
(
a
→ b
)
∧(c →
d
)
∧a
∧
c
→ b
∧
d ,
3) ((a
→ b
) ∨ (c → d
))∧
a∧c
→
b ∨ d ,
4)
(
a → b)∧
(c → b)
→ (a
∨ c → b) ,
5)
(a
∨ c →
b
)
∧
(a →
b)
∧(c
→
b) ,
6)
(
a
∧b
→ c
)
∧a
→
(
b
→ c
)
,
7) (a → (
b →
c))∧a
∧
b → c ,
8) (
a
∧b
→
c) →
(a
∧c
→
b
)
,
9) (
a
∧c
→
b
)
→ (
a∧b → c)
,
10)
(
a
→ b)
∧a∧c
→ b
∧
c .
R e x e n i e dl F =
(a
→ b)∧
(
c →
d)
→
(a ∨ c
→
b ∨ d
).
F
= (a ∨ b)∧
((
c ∨ d) ∨ a ∨ c ∨ b
∨ d = (
a ∨ b)∧
(
c ∨ d)∧
(a ∨
c)∧b∧d. J
11 . Dokazat~ osnovnye sootnoxeni bulevo$i algebry: 27, 28,
31, 33, 35, 37, 39, 41, 43, 45, 47.
12 . Dokazat~ s pomow~ sokrawennyh tablic istinnosti zako-
ny logiki teoremy 2.1. Ukazanie: sm. primer 2.1.
13 .
Dokazat~ teoremy 2.3, 2.4 i sledstvie teoremy 2.4.
14 . Dokazat~, qto sekvenci Γ
`
A
vypolnets togda i tol~ko
togda, kogda vypolnets sekvenci `
V
Γ →
A (pravila vvede-
ni i udaleni
∧
i →
).
15 .
Dokazat~ tavtologii trem sposobami: a) tabliqnym,
b) s pomow~ sokrawennyh tablic istinnosti (primer 2.1),
v) privod k tavtologii
|
=
A
∼
A s pomow~ teoremy 1.3.
1) |=
(A∧
B)
∨ A ∨
C
∼ ((A → B)∧A → C
) ,
2) |= (A ∼
B
) ∨ (
A∧
C) ∼
(
(A ∨ B
)
∧(B
→ A
)∧(
A
→ C)) ,
3)
|= (A → B)
∨
(
A
∼
C) ∼
(A ∨ B∧(A ∨ C)
∧
(
C
→
A)) ,
4) |= A ∨ B ∨ (C → A) ∼ (A → B∧A∧C) ,
5) |= (A → B) ∨ (A →
C
)
∼
(
A∧B
∧C) ,
6) |= ((
A →
B) ∨ A ∨ C)
∼
A∧B∧
(
A → C)
,
7) |
=
(A ∨
B
) ∨ (A ∼
C
)
∼
(A ∨ B
∧
(A →
C
)∧(C ∨ A
))
,
8) |= (A ∨ B
∨
A ∼ C
)
∼ (A → B
)
∧
(
A ∨ C
)
∧
(C
→
A) ,
9) |= (
A → B
∨
C → A
) ∼
(A →
B)∧
(
A ∨ C) ,
10)
|
= ((
A∧
B) ∨ A →
C
) ∼ A ∨
(
B∧
C
) .
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