Спекторський І.Я. Дискретна математика
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¿ À
hL, ∨, ∧i a, b, c ∈ L
hL, ∨, ∧i
hL, ∨, ∧i
(a ∧ b) ∨ (a ∧ c) = ((a ∧ b) ∨ a) ∧ ((a ∧ b) ∨ c) = a ∧ ((a ∧ b) ∨ c) =
= a ∧ ((a ∨ c) ∧ (b ∨ c)) = (a ∧ (a ∨ c)) ∧ (b ∨ c) = a ∧ (b ∨ c).
⇒
S
hS, ∪, ∩i
hL, 4i
a, b, c ∈ L
a b c 3! = 6
b c
b 4 c
a b c
a 4 b 4 c, b 4 a 4 c, b 4 c 4 a.
a 4 b 4 c
a ∨ (b ∧ c) = a ∨ b = b, (a ∨ b) ∧ (a ∨ c) = b ∧ c = b.
hL
n
, , i
n ∈ N
L
n
h{1, 2, . . . , n
j
}, 6i
max{x, y} min{x, y}
K
1
K
2
a
b
c
d
e
K
1
a
b
c
d
e
K
2
K
1
d ∨ (b ∧ c) = d ∨ e = d, (d ∨ b) ∧ (d ∨ c) = b ∧ a = b.
K
2
b ∨ (c ∧ d) = b ∨ e = b, (b ∨ c) ∧ (b ∨ d) = a ∧ a = a.
K
1
K
2
hL
1
, ∨
1
, ∧
1
i hL
2
, ∨
2
, ∧
2
i
f : L
1
→ L
2
f(a ∨
1
b) = f(a) ∨
2
f(b), f(a ∧
1
b) = f(a) ∧
2
f(b), a, b ∈ L
1
.
L
1
⊂ L
hL, ∨, ∧i L
1
¿
∨
À ¿
∧
À
L
L
1
⊂ L
hL, ∨, ∧i L
1
¿
∨
À ¿
∧
À
L
K
1
K
2
K
1
K
2
K
1
K
2
{a
1
, a
2
, a
3
, a
4
, a
6
} K
1
a
1
a
4
a
2
a
6
a
5
a
3
a
1
a
4
a
2
a
6
a
3
{a
1
, a
2
, a
3
, a
4
, a
6
} K
2
a
1
a
2
a
4
a
3
a
6
a
5
a
1
a
2
a
4
a
6
a
3
K
1
K
2
{a
1
, a
2
, a
3
, a
4
, a
6
}
K
1
h{a
1
, a
2
, a
3
, a
4
, a
6
}, 4i
a
2
∧ a
3
h{a
1
, a
2
, a
3
, a
4
, a
6
}, 4i
a
2
∧ a
3
= a
5
a
2
∧ a
3
= a
6
h{a
1
, a
2
, a
3
, a
4
, a
6
}, 4i ;
a
1
a
4
a
2
a
6
a
5
a
3
a
1
a
4
a
2
a
6
a
3
{a
1
, a
2
, a
3
, a
4
, a
6
} ⊂ L
L
a
2
∧ a
3
= a
5
/∈ {a
1
, a
2
, a
3
, a
4
, a
6
},
{a
1
, a
2
, a
3
, a
4
, a
6
} ∧
hL, ∨, ∧i a, x, y ∈ L
(
x ∨ a = y ∨ a,
x ∧ a = y ∧ a
⇒ x = y.
x ∨ a = y ∨ a x ∧ a = y ∧ a
x = x ∧ (x ∨ a) = x ∧ (y ∨ a) = (x ∧ y) ∨ (x ∧ a) =
= (x ∧ y) ∨ (y ∧ a) = y ∧ (x ∨ a) = y ∧ (y ∨ a) = y.
hL, ∨, ∧i L
1
L
= sup L L
0
L
= inf L
1
L
0
L
L
1
L
∈ L 0
L
∈ L
L L
1
L
0
L
1 = 1
L
, 0 = 0
L
.
L = {a
1
, a
2
, . . . , a
n
}
1 = a
1
∨ a
2
∨ · · · ∨ a
n
= sup{a
1
, a
2
, . . . , a
n
},
0 = a
1
∧ a
2
∧ · · · ∧ a
n
= inf{a
1
, a
2
, . . . , a
n
}.
hN, 6i 1 ∈ N
L
n
n ∈ N
1
L
n
= n, 0
L
n
= 1.
L
0
= N ∪ {0}
(a 4 b) ⇔ (b ·
·
·
a), a, b ∈ N;
∀ a ∈ N ∪ {0}: (a 4 0).
hL
n
, ·
·
·
i
n hL
0
, 4i
hL
0
, 4i
1
L
0
= 0, 0
L
0
= 1.
1 0
L
0
1 0 1 0
L
0
1 0
1 0 1
L
0
L
¿ À
hL, ∨, ∧i a ∈ L
a ∨ 0 = a, a ∧ 1 = a.
b ∈ L
a ∈ L
a ∨ b = 1, a ∧ b = 0.
b ∈ L a ∈ L a ∈ L b ∈ L
a b
0 1
hL, 4i 1 0
K
1
a e
c
b d
K
2
a e
b c d
{0, 1} 0 ≺ 1
hL, 4i
K
1
K
2
hL
n
, ·
·
·
i n = p
n
1
1
p
n
2
2
· · · p
n
m
m
p
1
p
2
p
m
n
1
, n
2
, . . . , n
m
∈ N
a = p
a
1
1
p
a
2
2
· · · p
a
m
m
a
k
∈ {0, n
k
} k = 1, 2, . . . , m
hL
n
, ·
·
·
i n
k
= 1
k = 1, 2, . . . , m n = p
1
p
2
· · · p
m
K
1
K
2
a
f
b
c
d
e
g
hL, ∨, ∧i
∀ x, y, z ∈ L: x 4 z ⇒ x ∨ (y ∧ z) = (x ∨ y) ∧ z.
∀ x, y, z ∈ L: x < z ⇒ x ∧ (y ∨ z) = (x ∧ y) ∨ z.
hS, ∪, ∩i
K
1
K
2
K
1
K
2
K
1
c 4 a, c ∨ (b ∧ a) = c ∨ 0 = c, (c ∨ b) ∧ a = 1 ∧ a = a,
K
1
K
2
K
2
x 4 z
x = 0 z = 1 x = y
x ∧ (y ∨ z) = (x ∧ y) ∨ z
0
b
c
a
1
K
1
a
b
c
1
0
K
2
X
L
X
X
¿
⊂
À
hL
X
, ⊂i
¿
⊂
À
A, B ∈ L
X
A ∩ B
A B
A + B A
B A ∪ B A ∪ B X
A, B ∈ L
X
{A, B}
A ∨ B = sup{A, B} = A + B, A ∧ B = inf{A, B} = A ∩ B.
hL
X
, ⊂i hL
X
, +, ∩i
hL
X
, ⊂i
X X
K
2
hG, ∗i
hG, ∗i
¿
⊂
À
hG, ∗i
¿
⊂
À
S
3
S
4
K
1
hL, ∨, ∧i
K
1