Luận văn thạc sĩ phân hoạch và hàm sinh

2 = 2 = 1 + 1
3 = 3 = 2 + 1 = 1 + 1 + 1
4 = 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1
X = ∅. X ×X
X × X = {(x, y)|x, y ∈ X}.
S X ×X
X. (x, y) ∈ S x S y xSy.
X = ∅ S = ∅
X. S X
x ∈ X xSx.
x, y ∈ X, xSy ySx.
x, y, z ∈ X, xSy ySz xSz.
S X ∼
S. C(x ) = {y ∈ X|y ∼ x } x
∼ X = ∅.
x ∈ X x ∈ C(x).
y, z ∈ C(x) y ∼ z y, z ∼ x.
x, y ∈ X, C(x ) ∩ C(y) = ∅ C(x ) = C(y).
X/ ∼
k
T n
k n k
n A
k
n
.
k n

A
k
n
= n
k
.
k
n
k n
k n
C
k
n
.
n n
1
n
2
n
s
n!
n
1
!n
2
! . . . n
s
!
·
a

1
, a
2
, . . . , a
s
, a
1
n
1
a
2
n
2
a
s
n
s
n
1
a
1
a
11
, . . . , a
1n
1
; n
2
a
2

a
21
, . . . , a
2n
2
; . . . ; n
s
a
s
a
s1
, . . . , a
sn
s
.
n = n
1
+ n
2
+ ··· + n
s
a
ij
n!
a
i1
, . . . , a
in
i
, i = 1,

n
1
a
11
, . . . , a
1n
1
n
1
!
n
1
! . . . n
s
!
n!
n
1
!n
2
! . . . n
s
!
· 
T n
k n
i
i i = 1, 2, . . . , k.
T =
n!

n
1
!n
2
! . . . n
k
!
·
C
n
1
n
n
1
n
C
n
2
n−n
1
n
2
n − n
1
C
n
k
n−n
1
−···−n

k−1
n
k
n −n
1
−···−n
k−1
T = C
n
1
n
C
n
2
n−n
1
. . . C
n
k
n−n
1
−···−n
k−1
.
T = C
n
1
n
C
n

2
n−n
1
. . . C
n
k
n−n
1
−···−n
k−1
.
T =
n!
n
1
!(n − n
1
)!
·
(n − n
1
)!
n
2
!(n − n
1
− n
2
)!
···

(n − n
1
− ··· − n
k−1
)!
n
k
!0!
=
n!
n
1
!n
2
! . . . n
k
!
·
T =
n!
n
1
!n
2
! . . . n
k
!
· 
m k n = mk.
T n k

m T =
n!
k!(m!)
k
·
k
T =
n!
m!m! . . . m!
=
n!
(m!)
k
·
k
k! T =
n!
k!(m!)
k
· 
s m
1
m
2
m
s
s,
n = m
1
n

1
+ m
2
n
2
+ ··· + m
s
n
s
i n
i
T
n s m
1
+ m
2
+ ··· + m
s
T =
(m
1
n
1
+ m
2
n
2
+ ··· + m
s
n

s
)!
m
1
!m
2
! . . . m
s
!(n
1
!)
m
1
(n
2
!)
m
2
···(n
s
!)
m
s
·
n s i
m
i
n
i
T =

n!
(m
1
n
1
)!(m
1
n
2
)! . . . (m
s
n
s
)!
·
i m
i
n
i
m
i
T
i
=
(m
i
n
i
)!
m

i
!(n
i
!)
m
i
·
T = S.T
1
.T
2
. . . T
s
T =
n!
(m
1
n
1
)!(m
1
n
2
)! . . . (m
s
n
s
)!
·
(m

1
n
1
)!
m
1
!(n
1
!)
m
1
···
(m
s
n
s
)!
m
s
!(n
s
!)
m
s
,
T =
(m
1
n
1

+ m
2
n
2
+ ··· + m
s
n
s
)!
m
1
!m
2
! . . . m
s
!(n
1
!)
m
1
(n
2
!)
m
2
···(n
s
!)
m
s

·

n k,

n
r
1
, r
2
, . . . , r
k

=
n!
r
1
!r
2
! . . . r
k
!
,
r
i
∈ N r
1
+ r
2
+ ···+ r
k

= n,

n
r
1
, r
2
, . . . , r
k

=

n
r
1

n − r
1
r
2

···

n − r
1
− r
2
− ··· − r
k−1
r

k

.

n
r
1
, r
2
, . . . , r
k

=

n − 1
r
1
− 1, r
2
, . . . , r
k

+

n − 1
r
1
, r
2
− 1, . . . , r

k

+···+

n − 1
r
1
, r
2
, . . . , r
k
− 1

,
r
i
r
1
+ r
2
+ ··· + r
k
= n.
n
(x
1
+ x
2
+ ··· + x
k

)
n
(x
1
+ x
2
+ ··· + x
k
)
n
=

r
1
+···+r
k
=n

n
r
1
, r
2
, . . . , r
k

x
r
1
1

x
r
2
2
. . . x
r
k
k
,
r
1
, . . . , r
k
∈ N r
1
+ ··· + r
r
= n.
n. n = 1 n.
n + 1.
(x
1
+ ··· + x
k
)
n+1
=(x
1
+ ··· + x
k

)
n
(x
1
+ ··· + x
k
)
=

r
1
+r
2
+···+r
r
=n
n!
r
1
!r
2
! . . . r
k
!
x
r
1
1
x
r

2
2
. . . x
r
k
k
(x
1
+ ··· + x
k
)
=

r
∗
1
+···+r
∗
k
=n+1
[
n!
(r
∗
1
− 1)!r
∗
2
···r
∗

k
!
+
n!
r
∗
1
!(r
∗
2
− 1)! ···r
∗
k
!
+ ··· +
n!
r
∗
1
! . . . (r
∗
k
− 1)!
] x
r
∗
1
1
. . . x
r

∗
k
k
.
(x
1
+ ··· + x
k
)
n+1
=

r
∗
1
+···+r
∗
k
=n+1
n!(n + 1)
r
∗
1
! ···r
∗
k
!
x
r
∗

1
1
···x
r
∗
k
k
(x
1
+ ··· + x
k
)
n+1
=

r
∗
1
+···+r
∗
k
=n+1
(n + 1)!
r
∗
1
! ···r
∗
k
!

x
r
∗
1
1
. . . x
r
∗
k
k
,
r
∗
1
, ··· , r
∗
k
∈ N
k

i=1
r
∗
i
= n + 1.
n + 1. n. 
k
k
n
=


r
1
+···+r
k
=n

n
r
1
, r
2
, . . . , r
k

,
r
1
, . . . , r
k
∈ N r
1
+ ··· + r
r
= n.
k
n
= (1 + 1 + ··· + 1)
n
=


r
1
+···+r
k
=n

n
r
1
, r
2
, . . . , r
k


k = 2
(x
1
+ x
2
)
n
=

r
1
+r
2
=n


n
r
1
, r
2

x
r
1
1
x
r
2
2
.

m
0

n
k

+

m
1

n
k −1


+ ··· +

m
k

n
0

=

m + n
k

.
x
1
+···+x
k
= n

n + k −1
k −1

.
N
k
(n). N
1
(n) = 1. N

2
(n),
x
1
+ x
2
= n.
(0, n), (1, n −1), , (n, 0)
N
2
(n) = n + 1 =

n + 1
1

.
N
3
(n) x
1
+ x
2
+ x
3
= n. x
3
= 0, 1, 2, , n,
N
3
(n) = N

2
(n) + N
2
(n−1) +···+ N
2
(2) + N
2
(1) + N
2
(0) = (n+1) + ···+1.
N
3
(n) =

n + 2
2

. N
k
(n) =

n + k −1
k −1

N
k
(n) = N
k−1
(n) + N
k−1

(n − 1) + N
k−1
(n − 2) + ··· + N
k−1
(0).
N
k
(n) =

n + k −2
k −2

+

n + k −3
k −2

+ ··· +

k −2
k −2

=

n + k −1
k −1

.

(x

1
+ x
2
+ ··· + x
k
)
n
n

n + k −1
k −1

.
(x
1
+ x
2
+ ···+ x
k
)
n
x
r
1
1
x
r
2
2
. . . x

r
k
k
r
1
+ ···+r
k
= n.

n + k −1
k −1


x
i
k x
1
x
2
. . . x
k
r
i
r
i
T (r
1
, r
2
, . . . , r

k
)
(r
1
, r
2
, . . . , r
k
) r
1
+ r
2
+ ··· + r
k
= n
A = {x
1
, x
2
, . . . , x
k
} k
i r
i
A k k
(r
1
, r
2
, . . . , r

k
) r
1
+ r
2
+ ··· + r
k
= n
T (r
1
, r
2
, . . . , r
k
) =

n
r
1
, r
2
, . . . , r
k

.
k (r
1
, r
2
, . . . , r

k
)
r
1
+ r
2
+ ··· + r
k
= n x
r
1
1
x
r
2
2
. . . x
r
k
k
(x
1
+ x
2
+ ··· + x
k
)
n
. T (r
1

, r
2
, . . . , r
k
) =

n
r
1
, r
2
, . . . , r
k

. 
a, b, c.
n a.
2r a C
2r
n
n − 2r 2
n−2r
b c.
C
2r
n
2
n−2r
n


r=0
C
2r
n
2
n−2r
.
(1 + x)
n
+ (1 − x)
n
= 2
n

r=0
C
2r
n
x
2r
n

r=0
C
2r
n
2
n−2r
=
2

n
2

(1 +
1
2
)
n
+ (1 −
1
2
)
n

=
3
n
+ 1
2
·

f(k) g(k) f(n) =
n

k=0
C
k
n
g(k)
n ∈ N.

g(n) =
n

k=0
(−1)
k
C
k
n
f(n − k).
f(k) g(k) f
k
g
k
.
f
n
=
n

k=0
C
k
n
g
k
f
n
= (g + 1)
n

∀ n ∈ N.
(f + x)
n
= (g + 1 + x)
n
, ∀ x.
f
k
g
k
f
k
g
k
.
x = −1 g
n
= (f − 1)
n
g(n) =
n

k=0
(−1)
k
C
k
n
f(n − k). 
F

0
= F
1
= 1, F
n+1
= F
n
+ F
n−1
, n  1.
F
n
=
n

j=1
(−1)
j

n
j

(F
2j
− 1).
x
1
=
1 −
√

5
2
, x
2
=
1 +
√
5
2
F
n
=
1
√
5
(x
n+1
2
− x
n+1
1
).
n

j=0

n
j

F

j
=
1
√
5
n

j=0

n
j

(x
j+1
2
− x
j+1
1
)
=
1
√
5
[x
2
(1 + x
2
)
n
− x

1
(1 + x
1
)
n
]
=
1
√
5
[x
2
x
2n
2
− x
1
x
2n
1
],
1 + x
2
= x
2
2
, 1 + x
1
= x
2

1
.
n

j=0

n
j

F
j
=
1
√
5
(x
2n+1
2
− x
2n+1
1
) = F
2n
.
F
n
=
n

j=0

(−1)
j

n
j

g(j) =
n

j=0
(−1)
j

n
j

F
2j
f(j) = F
j
, g(n) = F
2n
. F
0
= 1 = −
n

j=1
(−1)
j


n
j

F
n
=
n

j=1
(−1)
j

n
j

(F
2j
− 1).

L
0
= 2, L
1
= 1 L
n+2
= L
n+1
+ L
n

n  1.
L
n
= x
n
1
+ x
n
2
x
1
=
1 −
√
5
2
, x
2
=
1 +
√
5
2
·
L
2n
=
n

j=0


n
j

L
j
.
L
n
=
n

j=1
(−1)
j

n
j

(L
2j
− 2).
n, x
1
=
1 −
√
5
2
, x

2
=
1 +
√
5
2
L
n
= x
n
2
+ x
n
1
1 + x
2
= x
2
2
, 1 + x
1
= x
2
1
n

j=0

n
j


L
j
=
n

j=0

n
j

(x
j
2
+ x
j
1
) = (1 + x
2
)
n
+ (1 + x
1
)
n
= x
2n
2
+ x
2n

1
= L
2n
.
L
2n
=
n

j=0

n
j

L
j
.
L
n
=
n

j=0
(−1)
j

n
j

g(j) =

n

j=0
(−1)
j

n
j

L
2j
f(j) = L
j
, g(n) = L
2n
. L
0
= 2 = −2
n

j=1
(−1)
j

n
j

L
n
=

n

j=1
(−1)
j

n
j

(L
2j
− 2).


D
n
=
n

k=0
(−1)
k

n
k

(n − k)!

n  1.
n! = 1 +

n

k=2

n
k

D
k
n  2.
(−1)
n
D
n
=
n

k=0
(−1)
n−k

n
n − k

(n − k)! =
n

k=0
(−1)
k


n
k

k!
n  1. f
k
= (− 1)
k
D
k
g
k
= (− 1)
k
k!.
f(n) =
n

k=0

n
k

g
k
.
g
n
=

n

k=0
(−1)
n−k

n
k

f
k
=
n

k=0
(−1)
n

n
k

D
k
(−1)
n
n! =
n

k=0
(−1)

n

n
k

D
k
.
D
0
= 1 D
1
= 0 n! = 1 +
n

k=2
(−1)
n

n
k

D
k
. 
(A ∪ B) = (A) + (B) − (A ∩B).
A |A|.
A
1
, . . . , A

n
A. s
k
s
0
= |A|
s
1
= |A
1
| + |A
2
| + ··· + |A
n
|
s
2
= |A
1
∩ A
2
| + |A
1
∩ A
3
| + ··· + |A
n−1
∩ A
n
|

···
s
k
=

1i
1
<i
2
<···<i
k
n
|A
i
1
∩ A
i
2
∩ ··· ∩ A
i
k
|
···
s
n
= |A
1
∩ A
2
∩ ··· ∩ A

n
|
e
k
e
k
A
k A
1
, . . . , A
n
.
s
i
e
k
.
A
1
, . . . , A
n
.





n

i=1

A
i





=
n

i=1
|A
i
| −

1i<jn
|A
i
∩ A
j
| +

1i<j<kn
|A
i
∩ A
j
∩ A
k
|

− ··· + (−1)
n+1





n

i=1
A
i





.
n = 1 n = 2, A, B
A ∩ B = ∅ |A ∪ B| = |A| + |B|.
C = A ∩B = ∅ A ∪ B = (A \ B) ∪ C ∪ (B \ A).
|A ∪ B| = |A \ B| + |C| + |B \ A|.
|A\B| = |A|−|C| |B \A| = |B|−|C|, |A∪B| = |A|+|B|−|C|, (1).
n > 2 n − 1 A =
n−1

i=1
A
i
.

(1)





n

i=1
A
i





= |A ∪ A
n
| = |A| + |A
n
| − |A ∩ A
n
|.
|A| |A ∩ A
n
| 
n ∈ [1, 2005]
A = {1, 2, . . . , 2005} A
i
A

i. |A
2
| =

2005
2

= 1002, |A
3
| =

2005
3

= 668, |A
11
| =

2005
11

= 182
|A
13
| =

2005
13

= 154;

|A
2
∩A
3
| =

2005
6

= 334, |A
2
∩A
11
| =

2005
22

= 91, |A
2
∩A
13
| =

2005
26

=
77, |A
3

∩ A
11
| =

2005
33

= 60, |A
3
∩ A
13
| =

2005
39

= 51, |A
11
∩ A
13
| =

2005
143

= 14.
|A
2
∩ A
3

∩ A
11
| =

2005
66

= 30, |A
2
∩ A
3
∩ A
13
| =

2005
78

= 25, |A
2
∩A
11
∩A
13
| =

2005
286

= 7, |A

3
∩A
11
∩A
13
| =

2005
429

= 4,
|A
2
∩ A
3
∩ A
11
∩ A
13
| =

2005
858

= 2.
T A
2, 3, 11, 13
T = |A| − |A
2
∪ A

3
∪ A
11
∪ A
13
| = 2005 − (1002 + 668 + 182 + 154) +
+ (334 + 91 + 77 + 60 + 51 + 14) − (30 + 25 + 7 + 4) + 2 = 562.
T = 562. 
n > 1
n = p
α
1
1
p
α
2
2
. . . p
α
s
s
ϕ(n) = n
s

i=1

1 −
1
p
i


,
ϕ(n)
k ∈ {1, 2, . . . , n} (k, n) = 1.
1  i  s A
i
= {p
i
, 2p
i
, . . . , np
i
}.
A
i
T = {1, 2, . . . , n} |A
i
| =
n
p
i
·
i, j, i = j, A
i
∩ A
j
T
p
i
p

j
|A
i
∩ A
j
| =
n
p
i
p
j
·
ϕ(n) = n −





n

i=1
A
i





= n −
s


i=1
n
p
i
+

1i<js
n
p
i
p
j
−

1i<j<ks
n
p
i
p
j
p
k
+ ··· + (−1)
s
n
p
1
p
2

. . . p
s
·
ϕ : ϕ(n) = n
s

i=1

1 −
1
p
i

· 
K[[x]] = {a
0
+ a
1
x + a
2
x
2
+ ··· | a
i
∈ K} =

∞

i=0
a

i
x
i
| a
i
∈ K

.
f ∈ K[[x]], f =
∞

i=0
a
i
x
i
x
0
= 1,
x K. K[[x]]
f =
∞

i=0
a
i
x
i
, g =
∞


i=0
b
i
x
i
∈ K[[x]]
f = g a
i
= b
i
i = 0, 1, . . .
f + g =
∞

i=0
(a
i
+ b
i
)x
i
, fg =
∞

i=0

i

j=0

a
i−j
b
j

x
i
.
K[[x]]

f p(x), q(x) ∈ K[x]
f(x) =
p(x)
q(x)
f(x)q(x) = p(x) K[[x]]. q(0) = 1, f(x)
deg f(x) := deg p(x) −deg q(x).
F (x) f (x) = F (x) F (x)
f(x). f
f(x) =
∞

i=0
a
i
x
i
f(α) =
∞

i=0

a
i
α
i
lim
n→+∞
n

i=0
a
i
α
i
A A
f(x) =
∞

i=0
a
i
x
i
x = α . A
∞

i=0
a
i
α
i

.
K[[x]]
f =
∞

i=0
a
i
x
i
f
′
=
∞

i=1
ia
i
x
i−1
.
f(x) x = a,
f(x) =
∞

n=0
f
(n)
(a)
n!

(x −a)
n
K[[x]]
1
1 − x
= 1 + x + x
2
+ x
3
+ ··· + x
n
+ ···
(1 ± x)
1/2
= 1 ±
1
2
x −
1.1
2.4
x
2
±
1.1.3
2.4.6
x
3
−
1.1.3.5
2.4.6.8

x
4
± ···
(1 ± x)
−1/2
= 1 ∓
1
2
x +
1.1
2.4
x
2
∓
1.1.3
2.4.6
x
3
+
1.1.3.5
2.4.6.8
x
4
∓ ···
e
x
= 1 +
x
1!
+

x
2
2!
+
x
3
3!
+ ··· +
x
n
n!
+ ···
ln(1 + x) = x −
x
2
2
+
x
3
3
−
x
4
4
+ ··· + (−1)
n−1
x
n
n
+ ···

π
2
6
= 1 +
1
2
2
+
1
3
2
+ ··· +
1
n
2
+ ··· .
(a
n
) a
1
a
2
. . . a
n
. . . =
∞

n=1
a
n

.
A
k
=
k

n=1
a
n
.
lim
n→+∞
A
n
= A A
∞

n=1
a
n
A =
∞

n=1
a
n
.
∞

n=1

(1 + a
n
)
(a
n
).
(a
n
)
(b
n
).
∞

i=1
(1 + a
n
)
∞

i=1
a
n
∞

i=1
(1 + b
n
)
∞


i=1
b
n
∞

n=1
(1 − x
n
)
|x| < 1.
∞

n=0
x
n
|x| < 1
∞

n=1
(1 −x
n
)
|x| < 1 
∞

n=2

1 −
1

n
2

=
1
2
·
P
k
=
k

n=2

1 −
1
n
2

·
lim
k→+∞
P
k
= lim
k→+∞
k + 1
2k
=
1

2
∞

n=2

1 −
1
n
2

=
1
2
·

S = ∅. S
k 1  k  n, S
1
, . . . , S
k







k

i=1

S
i
= S,
S
i
= ∅
i,
S
i
∩ S
j
= ∅ i, j, i = j.
A
k k
A a, b a
2
+ b
2
k k = 8.
T 2, 4, 6, 8, 10, 12, 14, 16
a, b ∈ T, a = b a
2
+ b
2
k < 8
U ⊂ T k  9.
{a, b |a
2
+ b
2

} :
A = {1, 4}∪{2, 3}∪{5, 8}∪{6, 11}∪{7, 10}∪{9, 16}∪{12, 13}∪{14, 15}.
A
A a, b
a
2
+ b
2
k = 9. 
n k
n
k
n A = {a
1
, a
2
, . . . , a
n
}.
k  2
n−1
. a
1
A
i
A
A
i
a
1

. A
i
{a
2
, a
3
, . . . , a
n
} 2
n−1
. 2
n−1
A
i
A
A
i
∩ A
j
= {a
1
, . . .} = ∅ i = j k  2
n−1
.
k  2
n−1
. k > 2
n−1
A
i

A
A
i
, A
j
A
2
n
A 2
n−1
A
i
B
i
= A\A
i
i.
k > 2
n−1
A
r
, A
s
{A
r
, A
s
} = {A
i
, B

i
}
A
i
∩ B
i
= A
r
∩ A
s
= ∅ : k = 2
n−1
. 
k
X = {2012, 2012 + 1, 2012 + 2, . . . , 2012 + k}
A B A ∩ B = ∅ , A ∪ B = X
A B.
k. A B
s A.
B s X 2s.
4s = 2[2012 + (2012 + 1) + (2012 + 2) + ··· + (2012 + k)]
= 4024(k + 1) + k(k + 1).
k(k + 1) k ≡ 3( mod 4)
k ≡ 0( mod 4).
k ≡ 3( mod 4).
X n, n +1, n +2, n +3
n + n + 3 = n + 1 + n + 2 {n, n + 3} ∩ {n + 1, n + 2} = ∅.
X
k ≡ 0( mod 4).
X X A B

A ∩ B = ∅. (A) > (B). k = 4m
m.
(A)  2m + 1, (B)  2m.
s  2012 + (2012 + 1) + ··· + (2012 + 2m)
s < (2012 + 2m + 1) + ··· + (2012 + 4m).
2012+(2012+1)+···+(2012+2m)  s < (2012+2m+1)+···+(2012+4m)
2012 < 2m.2m m  23 k = 4m  23.4 = 92.
k = 92 :
A
1
= {2012, 2012 + 1, . . . , 2012 + 46}
a
1
= 2012 + (2012 + 1) + ··· + (2012 + 46);
B
1
= {(2012 + 47) + ··· + (2012 + 92)
b
1
= (2012 + 47) + ··· + (2012 + 92).
b
1
− a
1
= 46. 46 −2012 = 104.
2012+52 B
1
A
1
2012+52.

A = A
1
\ {2012} ∪ {2012 + 52} B = B
1
\ {2012 + 52} ∪ {2012}
k ≡ 0( mod 4) k > 92 :
X = {2012, 2012 + 1, . . . , 2012 + 92}∪ {2012 + 93, . . . , 2012 + 4m}.
X
1
= {2012, 2012 + 1, . . . , 2012 + 92}
A B
X
2
= {2012 + 93, . . . , 2012 + 4m}
C D
C ∩ D = ∅ C ∪ D = X
2
C D
A
0
= A ∪ C, B
0
= B ∪ D
k ≡ 3( mod 4) k ≡ 0( mod 4) k  92. 
X n (A, B),
A, B ⊆ X, A B.
X 2
n
.
(A, A) 2

n
. k B k
A B 2
k
. (A, B),
A, B ⊆ X, A B
n

k=0

n
k

2
k
− 2
n
= 3
n
− 2
n
.
(A, B), A, B ⊆ X, A
B 2
n
.2
n
− [3
n
− 2

n
] = 4
n
− 3
n
+ 2
n
. 
a, b, (a, b) = 1,
a−1

i=1

bi
a

+
b−1

j=1

aj
b

= (a − 1)(b −1).
A = {(i, j)|1  i  a − 1, 1  j  b −1}