Гладкий А.В. Введение в современную логику
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A ⊆ B
A
B A = ∅
A B
A B A B
A
B A B A ∪B
A B A B
A
B
A B A ∩B
A B
A
B A B A \B
A
B
'
&
$
%
'
&
$
%
A \BB\AA ∩B
A ∪B
AB
A = {1, 2, 3, 4} B = {2, 3, 5}
A ∪B = {1, 2, 3, 4, 5} A ∩B = {2, 3} A \B = {1, 4} B \A = {5}
X = {1, 3, 5,...} Y = {2, 4, 6,...}
X ∪Y = {1, 2, 3,...} X ∩Y = ∅ X \Y = X Y \X = Y
A ⊆ B A ∪ B = B A ∩ B = A
A \B = ∅
A ∩B = ∅ A B
A \B = A B \A = B
A A ∪ ∅ = A
A ∩∅ = ∅ A \∅ = A
A = {1, 2, 3,...} B = {n | n n<6}
A ∪B A ∩B A \B B \A
A B A ∪B A ∩B
A B A \B
A ∪B = B ∪A
A ∩B = B ∩A
A ∪(B ∪C)=(A ∪B) ∪C
A B ∪C
B ∪C B
C
A B C
A ∩(B ∩C)=(A ∩B) ∩C
A B ∩ C
B ∩C B
C
A B C
A ∪ B A ∩ B
A ∪ (B ∪ C)
(A ∪B) ∪C A ∪B ∪C
A ∩ (B ∪ C)=(A ∩B) ∪(A ∩C)
M N
M ⊆ N N ⊆ M
M ⊆ N x ∈ M x ∈ A x ∈ B ∪C
x ∈ B x ∈ C x ∈ B
x ∈ A x ∈ A ∩B x ∈ (A ∩B) ∪(A ∩C)=N
x ∈ C x ∈ A ∩C x ∈ N
N ⊆ M x ∈ N x ∈ A ∩ B x ∈
∈ A ∩C
x ∈ A x ∈ B x ∈ B x ∈ B ∪C
x ∈ A x ∈ A ∩(B ∪C)=M
A ∪(B ∩C)=(A ∪B) ∩(A ∪C)
M
N
M ⊆ N x ∈ M x ∈ A x ∈ B ∩C
x ∈ A ∪ B x ∈ A ∪ C x ∈ N
x ∈ B x ∈ A ∪ B x ∈ C
x ∈ A ∪C x ∈ A ∪B x ∈ A ∪C x ∈ N
N ⊆ M
A =(A \B) ∪(A ∩B)
A B
A ∩B
A \B
A \B A ∩B A
U
U
A
U U \ A CA
A U U
U
A U U
U
U
CCA = A
CCA
U CA
CA U
A
C(A ∪B)=CA∩CB
A B
U A
B CA CB
CA CB
C(A ∩B)=CA∪CB
A B
U
A B
CA CB
CA CB
C(A \B)=CA∪B
a
1
a
2
(a
1
,a
2
) (3, −1)
−1
(2, 2)
n n
a
1
a
2
n a
n
(a
1
,a
2
,...,a
n
) (2, 7, 6)
n {0, 1,...,9} n
k
k
2
k
k
3
k
2
k
A l
l
1
l
2
l
1
l
2
l
1
l
2
R x y R x
y R xRy
R
x y x y x<y
x = y
R R
R
P Q
R
x ∈ P y ∈ Q x y
R x y R
M
M
<
=
R P
Q (x, y) xRy
R
R P
Q P R
Q
R Q R
P R
= <
T
xT y x
y T T
T
T
T
S R xSy
yRx R
R
−1
R
R
−1
R
R M R
−1
R R
−1
R R
R
−1
R R
−1
l
A A l
S
{0, 1,...,9} { , ,..., } xSy
x y
S S
T
0
xT
0
y
x
y
T
0
T
0
T
0
x
x
x
y x y
x y
F
a F
b a F
aF b b F (a)
F a a
b
x y
F
x y
a
F
b = F (a)
a F
F F (x)
F x y = F (x) x
y x
0
F y
0
F
x
0
x
0
y
0
M M
Q
xQy y = x
2
xQy y = Q(x)
Q(2) = 4 Q(−2) = 4 Q(0) = 0
xMy
y x M
M
x (x)
T
y =
√
x
P
xP y
xy
P P
P
G
G(1) = 8
Magn
Magn(x)
x
Magn( )=
Magn( )=
Magn( )=