Гладкий А.В. Введение в современную логику
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Magn
A B C
C A B
k l
A
a b c
a + b = c
a/b = c/d
n n =2, 3,... n
n
R x
1
,x
2
,...,x
n
R
R
R
R
M
M
n R
n (x
1
,x
2
,...,x
n
) x
1
,x
2
,...,x
n
R
x y
z
x + y = z
(3, 2, 5) (2, 2, 4) (−7, 10, 3) (2, 3, 4) (2, 3, 6)
(2, 2, 9)
(n +1) (n =2, 3,...)
n (x
1
,x
2
,...,x
n
)
y x
1
,x
2
,...,x
n
,y
n
n =1 n x
1
(n +1)
n
n F
n (x
1
,x
2
,...,x
n
)
y x
1
,x
2
,...,x
n
,y F
n F y
n (x
1
,x
2
,...,x
n
)
x
1
,x
2
,...,x
n
,y F
x + y = z
x, y z
x + y = z
F x, y
F (x, y)=x + y F (3, 2) = 5 F (2, 2) = 4
V (x, y, z)=xyz
x y z
x, y, z, u, v
x
y z
u
u
Fl(x, y, z, u, v)
Fl( )=
Fl( ,z, )= z
Fl( ,y,z, )= y z
x
y n
x y z
x + y +1 = 2z
F
n
(x
1
,x
2
,...,x
n
) n 1, 2, 3,...,n
n
F
n
(x
1
,x
2
,...,x
n
)= x
1
,x
2
,...,x
n
F
n
(x
1
,x
2
,...,x
n
)= x
1
,x
2
,...,x
n
F
n
(x
1
,x
2
,...,x
n
)= x
1
,x
2
,...,x
n
F
5
( )= F
5
( )= F
5
( )=
F
4
(F
2
( ),F
3
( ),F
1
( ),F
1
( ))
F
3
(F
1
( ) F
1
( ) F
1
( ))
F
1
(F
3
( ))
F
100
(x
1
,x
2
,...,x
100
) x
1
,x
2
,...,x
100
M
M(x)
x
M(x) x
M( )= M ( )= M( )=
B
B(x, y)
(x, y)
B(x, y)
x y M( )= M( )=
M( )=
n
(n =2, 3,...) n
n n
{ , }
G(x)
x
M(x, y)
x<y
S(x, y, z)
x + y = z
x<y
x + y = z
n
n
F
M F (x)
M
M F
M
F ∀xF(x) M
F ∃xF (x) ∀x ∃x
x
M
F
∀xF (x) F (x)
∃xF (x)
F (x)
∀xF (x) ∃xF (x)
x
∀xF (x)
∀yF(y) ∀zF(z) ∃xF (x) ∃yF (y) ∃zF(z)
x C(x)
x
∀xC(x) (x)
x
∀x (x)
∃x (x)
x (x)
x
∃x (x)
∀xF (x) ∃xF (x)
∃xF (x) ∀xF (x)
∃xC(x) ∀x (x)
B(x, y)
x y x y
x
∀xB(x, y) y
y
∃xB(x, y)
y
y ∀yB(x, y) ∃yB(x, y)
x
x
∃x∀yB(x, y)
∃x∃yB(x, y)
M(x, y)
x<y ∀x∃yM(x, y)
∃y∀xM(x, y)
B(x, y)
M(x, y)
M
1
(x, y)
x y
M
1
(x, y)
M
1
(x, y)
B(x, y) y =
B(x, ) x
A B
A B A & B
A B
A B
& ∧
ABA & B
A B
A & B
P Q
P
Q
P & Q P & Q
P
& Q P
& Q