F n
F n
|A|
A A1 , A2 , . . . , An
Ai
∩A = ∅
i, j
∈ {1, 2, . . . , n}
j
|
n
i = j
n
Ai =
i=1
Ai .
|
i=1
AB
M Z C
∩ ∩ Z = Z ∩ ∩ C = C ∩ ∩ M = ∅,
M
|AB| = |M ∪ ∪ Z ∪ ∪ C | = |M | + |Z | + |C | = 2 + 3 + 2 = 7.7.
A1
× A2 × . . . × A A1 × A2 × . . . × A
A1 , A2 , . . . , An
n
n =
{(a1, a2, . . . , a
n)
: ai
∈ A } . i
A1 , A2 , . . . , An n
|A1 × A2 × . . . × A | = n
|
|
Ai .
i=1
A B
AD
AB
BC
CD
p ( p1 , p2 , p3 )
p1
∈ AB p2 ∈ BC p3 ∈ CD
∈ AD
|AD| = |AB × BC × × CD | = |AB| · |BC | · |C D| = 2 · 5 · 3 = 30. 30.
[Rj. [Rj. 24] n = p = p α pα 1 2 1
·
· ·
(α1 + 1) (α2 + 1) . . . (αk + 1)
·
2
αk k
· . . . · p
(x, y) x2 + y 2
≤ 5.
[Rj. [Rj. 8] x x
1 , 2 } ∈ {0, 1,
≥3
S i = (x, y)
2 , . . . , 100 X = 1, 2, , 100
{
S = = (a, b, c) c) : a, b, c,
{
∈ Z2 : x2 + y2 = i S i
, i
1 , . . . , 5} ∈ {0, 1,
}
∈ X, a < b, a < c} .
|S | \ \ {100}
a
X 100
−a
99
|S | =
99
(100
a=1
2
− a)
=
a a+1
b
c
99 100 199 = 328350 328350.. 6
·
a2 =
a=1
·
b c a 100 100
|S | =
(min b, c
{ } − 1) .
b=2 c=2
A S
A
⊆ S
|S \ \ A| = |S | − |A| . 10n 1, 2, 2 , . . . , n n
10n
10n
10 10n
−1
10n
9n
n
−1 (10n
n
n
− 1) − (9 − 1) = 10 − 9
n
.
S
P (S ) n
|S | = n ∈ N0
S 20 = 1 n
S
S n
∈N
|P (S )| = 2
S S = = a1 , a2 , . . . , an S i
{
}
ai S
n 2n
n
|P (S )| = 2
n
×n n j
b
i b
n
b
b
× j y (n − i + 1) 1) (n − j + j + 1)
x
i
n i+1 n j + j + 1
n
− −
n
n
i=1
j =1
−i+1 i × j
· · − · · − · − · · − · − n
[(n [(n
− i + 1)1) (n − j + j + 1)
i j] j ]
n
=
[(n [(n
i + 1) i]
[(n [(n
i=1
j =1
n
=
n
2
(n + 1) i
i
=
j =1
(n + 1)
2
n
2
i
i=1
j + j + 1) j] j ]
i2
(n + 1) i
i=1
n
· · −
i
i=1
=
n (n + 1) (n + 1) 2
=
n (n + 1) 1) (n + 2) 6
n (n + 1)(2n 1)(2n + 1) 6
2
2
.
r
(b1 , b2 , . . . , br ) r
n
{
A = a1 , a2 , . . . , an A
n
A
}
(b1 , b2 , . . . , br )
b1 b2 . . . br
{
A = a, b, c, d
}
A ab,ac,ad,ba,bc,bd,ca,cb,cd,da,db,dc
aa,bb,cc,dd ba = ab
r
n
· − 1) · . . . · 2 · 1
n
n! = n (n
n
0! = 1
7! = 5040 n n!
≈
√
2πn
n e
n
,
lim
n→∞
√ 2πnn!
r n
−1
r
n n e
= 1.
n
P rn n r + 1
r
n
−
− r)! = n! . · − 1) · . . . · (n − r + 1) = n · (n − 1) · . . . · (n − r + 1) · (n (n − r)! (n − r)!
P rn = n (n
P 325
P 25
P 325 P 25 P 325 = 5 4 25 24 23 = 276000.
·
· · · ·
3!
7!
·
3! 7!
3! 7! 8 3! 7! = 3! 8!
· ·
·
7!
· · ·
7! 6 5 4
{2, 3, . . . , 6 } ◦
· ·
· · · ·
8 7 6 ◦
· ·
3 4 6 7 8
· · · · 6 · 7 · 8 · (3 · 4 + 2 · 5) = 6 · 7 · 8 · 22 = 7392
8 7 6
2 5 6 7 8
[Rj. 8!]
2
Rj. (8!)
n
(n
≥ 2) · − 1)!] · (n − 1)!]
[Rj. n! (n [Rj. 2 n
(2n)! 2n (2n (m
− 1)!
− 1)! (2n
m
− 1)! 2n
·
2 (2n
− 2)! n
(n
−2
− 1)! n! · (n − 1)!
(2n
− 2)! n!
(n
− 1)!
2n (n
· − 1)!
100! p
n!
100! n! 100!
100 100 + = 24 5 25 p
n!
{1, 2, . . . , n}
2
p
p
2
p p
p
3
p
∞
n!
n pk
k =1
∞
k=1
A
n pk
k
n = 0 pk
p >n
A = a1 , a2 , . . . , a8 ai < bi i
} B = {b1 , b2 , . . . , b8 } ∈ { 1, 2, . . . , 8}
{
∪B
16!
ai < bi
28
i
i
16! 28
∈
A [8]
∪B
{1, 3, 5, 7 }
S
|S |
k
[Rj. 64] n
[Rj. 117856]
n∈S
S n
− ·
3n 2 Rj. 100% n2
k
A
{a, b, c, d} {a, b} , {a, c} , {a, d} , {b, c} , {b, d} , {c, d} {a, b} = {b, a} k
k
A
{a, a}
n
n n! = r (n r)! r!
k
− ·
P kn =
n
k
r!
n! (n−k )!
k
n = k (n
n! . k)! k!
− ·
r>n
n = 0 r
11 5
4 + 7 = 11
4 2
◦
◦
7 3
· · · 4 3
7 = 4 2
4 4
7 = 7 1
7 = 84 2
· 4 2
7 3
9 5
· 240 80
160 80
· 240 80
160 80
3!
n r
n
≥r
− − − n = r
n = r
n r
−
n n! = = r r! (n r)! (n
· −
− − n r
−
1 n 1 + 1 r
= = = =
(n r
− r)
n
n
r
.
1 n 1 + . 1 r
n! r)! [n (n
− · − − r)]!
=
n
n
−r
(n 1)! (n 1)! + 1)! (n r)! r! (n r 1)! (n 1)! 1 1 + 1)! (n r 1)! n r r (n 1)! n 1)! (n r 1)! r (n r) n! n = . (n r)! r
− − · − · − − (r − − (r − · − − · − − · − − · · − (r −
r!
.
· −
r n
n (n
− r)
− n = n r
n = r r
∈
x S S
n n
−
1 = 1 1 n S
r
• •
n r
A = {X ⊆ S : |X | = r, x ∈ X } x B = {X ⊆ S : | X | = r, x ∈ / X } A n−1 S |B| = r
x
−
|A| = nr − 11 n−1 r
x
S
−1 A B r
n k k
n
n
≥k
n
−k
n k
Rj.
n
m
∈N
− −
n
m
←− ←−
.
..
←−
.
∈N
..
←− −→
1 n+m 1 m 1
−
m n x1 + x2 + . . . + xm = n.
Rj.
(x1 , x2 , x3 , x4 )
∈ N4
−
n+m 1 m 1
−
x1 x2 x3 x4 = 9000
9000 = 23 32 53
· ·
xi = 2α
βi
·3 ·5
i
x1 x2 x3 x4 = 9000
2
∈ {0, 1, 2, 3} β ∈ {0, 1, 2 } i ∈ {1, 2, 3, 4 }
αi , γ i
i
α1 + α2 + α3 + α4
= 3,
β 1 + β 2 + β 3 + β 4 γ 1 + γ 2 + γ 3 + γ 4
= 2, = 3.
· 6 3
γ i
6 3 5 = 4000 2
5 3
6 3
(0, 0)
( p, q )
p
∈ N, p < m q ∈ N, q < n
(m, n)
∈ N2
[( p, q ), ( p + 1, q )]
(x0 , y0 ) (x0 + 6, y0 + 6)
∈
∈
∈ N0
y
m
(0, 0) m
m+n m
n
m+n m p + q p
m
m
(m, n)
·
− p + n − q m − p
− ·
(x0 , y0 )
( p, q )
∈ N, p < m − 1 q ∈ N, q < n
x N0 (x0 + x, y0 + y)
2 N0
m+n
m+n m
p
p + q p
m
(0, 0) ( p, q ) p + q m p p m
·
− p − 1 + n − q m − p − 1
− p + n − q m − p ( p + 1, q )
p + q p
(m, n)
− − 1 + n − q − p − 1
n
n
A
2n
A
A
∈N A
(2n)!
A 2n n!
·
A
·
2n
2n
n
i=0
n!
A
(2n)! 2n n!
A
n−1
2n
2n−2i 2
n!
1 = n!
·
n−1
i=0
(2n
−2
− 2i) (2n − 2i − 1) = 2
(2n)! . 2n n!
·
r
(x1 , x2 , . . . , xr )
S
r
{
}
A = a, b, c
A
(a, a) (a, b) (a, c) (b, a) (b, b) (b, c) (c, a) (c, b) (c, c) r
n
nr r
n
r ...
n
r
n
n nr
r
Rj. 6 10
M m(x)
x
x∈S
S
∈ S
m(x)
M = (S, m)
(S, m) x r m (xi )
xi
m : S
→ N0
(x1 , x2 , . . . , xr ) , xi r
∈ S, ∀i ∈ {1, 2, . . . , r}
m(x) = r
x∈S
M = a,a,b,b,c,c,c = a2 , b2 , c3
{
} {
}
M
7! a
M
a 7! 2! 2! 3!
· ·
7 2
a
· · 5 3
c
2 2
b
7 2
5 3
2 7! = 2 2! 3! 2!
· ·
k
m1
{x1
· − · · − N m1
N
m1
m2
...
N
m2
, x2 , . . . ,
m1
k xm k
}
− m2 − . . . − m mk
mi = N
i=1
k−1
=
N ! m1 ! m2 ! . . . mk !
·
· ·
·
10! Rj. 2! 3! 5!
·
74 , 2 2 , 4 1 , 6 1
◦
· · · · · · · 74 , 2 1
74 , 4 1
5! 4!
#1 = 3
◦
◦
73 , 6 1 , 4 1
73 , 2 2
5! = 10 2! 3!
72 , 2 2 , 6 1
#4 = 2 ◦
5! 4!
= 15
73 , 2 1 , 4 1 73 , 2 1 , 6 1 5! #2 = 3 = 60 3! #3 =
◦
74 , 6 1
5! = 60 2! 2! 1
72 , 2 1 , 4 1 , 6 1 71 , 2 2 , 4 1 , 6 1 5! #5 = 2 = 120 2!
5
#=
#i = 265
i=1
23n 3n
4n! (4n)! 4!·4!·...·4!
=
(4n)! 23n ·3n
M = a41 , a42 , . . . , a4n
∈ N
n! r!(n r)!
−
r
x∈S
S
m(x) = r
M = (S, m)
S = a, b, c
{
r
}
{a, b} , {a, c} , {b, c} , {a, a} , {b, b} , {c, c}
M = a∞ , b∞ , c∞
{
r xi
ai
}
S = a1 , a2 , . . . , an
{
n r
}
x1 + x2 + . . . + xn = r.
r
−
n+r 1 = n 1
n
−
n+r r
−1
S xi
∞
{S x1 + x2 + x3 = 6
, T ∞ , V ∞ 6+3 1 = 3 1
−
} − 8 2
x1 + x2 + x3 + x4 + x5 = 50
≥ 0 2 ≤ x3 ≤ 7 x2 ≥ 2
x1 x4 x5
− 2
− 2
y1 = x 1 y2 = x 2
y3 = x 3
y4 = x 4 y5 = x 5
y1 + y2 + y3 + y4 + y5 = 46
≥ 0 i ∈ {1, 2, . . . , 5 }
yi
y3 > 5 i = 1, 2, 4, 5
y3
≥
y3 6
≤ 5
y3
≤ 5
x
≥ c c ∈ N
x
≥ 0
y3
y3 6 z1 + z2 + z3 + z4 + z5 = 40 46 + 5 1 40 + 5 1 = 46 40
z3 = y3
− 6
≥
− −
−
−
50 4
44 4
≤ 5 zi = y i
{1, 2 , . . . , n}
r b1 b2
br
1
a1 = b 1 a2 = b 2 + 1 a1 . . . ar
≤ b1 < b2 < ··· < b ≤ n − r + 1 a3 = b 3 + 2 a = b + r − 1 r
r
1
r
≤ a1 < a2 < .. . < a ≤ n r
(a1 , a2 , . . . , ar ) (b1 , b2 , . . . , br ) r
− n
n
r + 1 r
− r + 1
m
n
· − 1) · . . . · (n − m + 1).
n (n
nm . r1 k k
n!
r1 !
··· r ! ,
n =
k
ri .
i=1
n (n + 1) (n + 2) . . . (n + m
·
·
· ·
m
n
n . m
n+m m
−1
.
− 1).
. . . r k
r1 + r2 + . . . + rn = m, si = r i
− 1 ⇒ s1 + . . . + s = m − n + 1 m−n+n−1 m−1 = n−1 n−1 n
≥ 1
ri
.
n n
Bn
n n
Bn =
B0 = 1
−− k=1
n k
1 Bn−k . 1 S (n, k)
S (n, k) =
S (n + 1, k) = S (n, k
S (n, 1) = S (n, n) = 1, n k
\
∀n ∈ N
n k
− 1) + kS (n, k) .
k
n
A=B
A
B
S
| |
A = S S
S
| |
A A = B
B = S B = S
||
P rn = nP rn−−11 S S
r
X
n r
x1
\ {x1} − 1)
X = X (n
(r
− 1)
n n
P rn−−11
A
−1
r
−1
P rn+1 = P rn + rP rn−1 S
r
X
n + 1
S
x
A
r
A
x rP nr−1 n
x
− 1)
x
(r x
r
n
\ { } \ { }
X x X x
n
− n m
m = r
n r
n m
X
n
−
r r
S
|Z | = r, |Y | = m
Y Y
⊆ X Z ⊆ Y
|Y | = m |Z | = r
(Y, Z ) (Y , Z ) Y = Y
\ Z
⊆ X |Z | = r Y ⊆ X ⊆ |Z | = r |Y | = m − r
Z
⊆ X Y ⊆ X ⊆ Z
Z
Z
(Y, Z )
| | −
n m m r S Z Y = m n r
n
r
n−r m−r
S
m
r
n
r
r
n
−r
m
− k=0
m k
n
r
k
⊆ Y ⊆ X
Z
−r
m+n r
=
S m m + n
n
r
S 0, 1, 2, . . . , n
r
k
S
m k
k
F n
r S X Y Z
⊆ X ∪ Y
|Z | = r
S
(P, R)
P
⊆ X, R ⊆ Y
i
i=0
n r −k
|X | = n |Y | = m S k ∈ {0, 1, . . . , n} |R| = k, |P | = r − k
S S
k
P = Z ∩ X, R = Z ∩ Y n
−k
Z = P
∪
S k R
n = n2n−1 i n n
n
i
−1
n
i n
X
∈ X Y ⊆ X \ {x} Z = Y ∪ {x}, Y = Z \{x} x
S S i n
n i
(x, Z )
i i S Z
0
n (x, Y ) S i i 0, 1, . . . , n Z
∈{
⊆ X x ∈
}
r =k
n r
r k
F n
×
J n
×
1 n J n = F n+1
×
1 1
1 2
J m+n = J m J n + J m−1 J n−1 m + n
J m+n
m
m + 1
J m−1 J n−1 J m J n 3J n = J n+2 + J n−2 n+3 J n
J 3 = 3 3J n
J n−2 KDK .. . KKK .. . D . . . KK .. . KD p
DK . .. n+2
i=0
p J n−i = J n+ p i n + p
i
p n + p 2i ( p 0 p
p i
− 1, n , n + 1
i
− − − i) = n − i
J n2 = J 2n−1 + J n2−2 n
J n+2
2n
n + 2 n
n+1
J n−i
n ...DD... ...KD... ... KK ... 2n 1
−
J n−2 J n−2 . . . DK . . . J 2n−1
·
F n
n F n+1 an a1 = 1, a2 = 2 n
A
⊆ {1, 2, . . . , n} −1
A n
an−1 = F n
{1, 2, . . . , n}
n
−2
n
−1
A an−2 = F n−1
n A an = a n−2 +an−1 = F n−1 +F n = F n+1
cr an+r + cr−1 an+r−1 + . . . + c0 an = 0 r an = x n cr xn+r + cr−1 xn+r−1 + . . . + c0 xn = 0 / : x n cr xr + cr−1 xr−1 + . . . + c0 x0 = 0,
x1 , x2 , . . . , xr an = A1 xn1 + A1 xn1 + . . . + Ar xnr ,
an = 2an−1 + an−2
xn
n−3 ;
a1 = 1, a2 = 2, a3 = 3.
− 2a 1 − a 2 + 2a 3 = 0 x − 2x 1 − x 2 + 2x 3 = 0 / : x 3 , x3 − 2x2 − x + 2 = 0. a = A · (−1) + B · 1 + C · 2 = A · (−1) + B + C · 2 1 1 1 = A · 1 + B + C · 4 =⇒ A = , B = , C = . 6 2 3 = A · (−1) + B + C · 8
an
n−
n
−1, 1, 2 1 2 3
− 2a
n−
n−
n−
n−
n−
n−
n
n
= a1 = a2 = a3
−
x1 , an
··· , x
+
k2 −1 k2 n
Am1 + Am2 n +
n−1 +
15an−2
− 9a
k1 ,
m
k1 −1 k1 n
A11 + A12 n +
+
n
··· + A1 A21 + A22 n + ··· + A2
=
− 7a
n
( 1)n 1 2n an = + + 6 2 3
an
A1 , . . . , Ar
xn1 xn2
km −1 mkm n
··· + A
n−3 =
xnm .
0; a0 = 1, a1 = 2, a2 = 3.
x3 an
1 2 3
= a0 = a1 = a2
− 7x2 + 15x − 9 = 0 =⇒ x1 = 1, x2 = x3 = 3 = A · 1 + B · 3 + C · n · 3 . n
= A+B = A + 3B + 3C = A + 9B + 18C
n
⇒ =
n
A = 0, B = 1, C =
− 13 .
··· , k
m
− · n 3
an = 1
3n
an + an−2 = 0;
a0 = 1, a1 = 1.
cr an+r +
··· + c1a +1 + c0a
n = f (n)
n
Rj. an =
f
1
−i ·i
n
2
1 + i + ( i)n 2
·−
n
r
aH n aP n P an = a H n + an
aP n
f (n)
b n aP n = A b
C bn
·
p (n)
∈ R [x] ,
·
b aP n aP n
p = m
k k
· ·b
= A n
1 = p 1 (n)
n
m
1
k aP n
= n
k
b
· p1 (n)
n aP n = p 1 (n) b
C nm bn
· ·
p1 = m
·
b
k aP n
= n
k
· p1 (n) · b
n
C b A
an+1
− 5a
n =
4n2 + 2n + 6;
a1 = 1.
aP n
2
n aH n = A 5
·
= Bn + Cn + D
B (n + 1)2 + C (n + 1) + D
− 5Bn2 − 5Cn − 5D = 4n2 + 2n + 6 ⇐⇒ −4Bn2 + (2B − 4C ) n + (B + C − 4D) = 4n2 + 2n + 6. B =
D =
an = a H n
−2
+
aP n
− 1 − 1 − 2 = 5A − 4, a = 5 − n2 − n − 2
−1
C =
−1
1 = a 1 = 5A A = 1
n
an = 6an−1
− 9a
n−2 +
n
n 3n
·
a0 = 2, a1 = 3.
x2 aH n
x1 = x 2 = 3
n
= A 3 + B n 3 2 aP = n (an + b) 3n n
·
·
· ·
·
n
· − 1)2 · [a (n − 1) + b] · 3
− 9 · (n − 2)2 · [a (n − 2) + b] · 3 2 + n · 3 / : 3 ⇐⇒ n2 · (an + b) = 2 · n2 − 2n + 1 (an − a + b) − n2 − 4n + 4 (an − 2a + b) + n ⇐⇒ (1 − 6a) n + (6a − 2b) = 0. −5 1 b = a = a + a A = 2 B =
n2 (an + b) 3n = 6 (n
·
·
n−1
a =
1 6
− 6x + 9 = 0
2 3n−1 an = 2
3
2
n + 3n
− 10n + 12 an
H n
n
·
n−
n
n
P n
3
− 3a
n−1 +
2an−2 = 2n ;
a0 = 3, a1 = 8. [Rj. an = (2n + 1) 2n + 2]
·
an bn
= =
−2a −5a
n−1 +
4bn−1 , n−1 + 7bn−1 ;
bn−1 =
an
− 5a
n−1 +
a1 = 4, b1 = 1.
an + 2an−1 4
6an−2 = 0;
an = 2n+3
−4·3
bn =
a1 = 4, a2 =
−2a1 + 4b1 = −4. bn = 2n+3
n
1
an+1 + 2an 4
×n
1
−5·3
n
×1 1×2
an
1 an−1
1
an−2
×2
×1
an = a n−1 + an−2 . a1 = 1 a2 = 2
an = F n+1
(F n )n∈N
F 1 = F 2 = 1
1 F n = 5
√
F n = F n−1 + F n−2 ,
√ − √ 1+ 5 2
n
−
1
5
2
n
,
∀n ∈ N.
∀n ≥ 3
{1, 2 , . . . , n} n
n
an
n (n
−
an−1
1)
an−2 an = a n−1 + an−2 , a1 = 2 a2 = 3
an = F n+2
n an n
n a1 = 2
an = a n−1 + n;
an
−1
a1 = 2.
= an−1 + n = [an−2 + (n = 2 + 2 + 3 + . . . + n = n (n + 1) = + 1. 2
− 1)] + n = . . .
rn
n
Rj. rn =
n
rn = r n−1 +
n 3
− 6n2 + 17n − 18 , ∀n ≥ 4, r3 = 1, 6
n4
rn
− 6n3 + 23n2 − 42n + 24 24
n!
n k
k
n S (n, k)
k
n
n
Bn
k
20 1, 2 5 x, y z x + 2y + 5z = 20, x , y , z
≥ 0, x , y , z ∈ Z z
1, 2
5
20
f (x) = 1 + x + x2 + x3 +
a20
1 + x2 + x4 +
··· · f (x) = a 0 + a1 x + a2 x2 + ···
··· ·
1 + x5 + x10 +
···
x20
f (x) = a 0 + a1 x + a2 x2 +
··· + a20x20 + ··· [Rj. a20 = 29]
(an )
∞
(an )n∈N
n=0
x ∞
f 1 (x) =
∞
an xn f 2 (x) =
n=0
bn xn
n=0 ∞
(an + bn ) xn
(f 1 + f 2 ) (x) =
n=0
n
∞
(f 1 f 2 ) (x) =
ak bn−k xn
n=0 k=0
∞
d f 1 (x) = dx
nan xn−1
n=0
∞
f 1 (x) dx =
an n+1 x n+1 n=0
|x| < 1 1 + x + x2 + x3 +
··· = 1 −1 x
n
n
(1 + x) =
n k x k
k=0
± ∞
(1
± x)
α
=
k =0
α ( x)k k
α n
∈ R, n ∈ N
α
α α (α = n
− 1) ··· (α − n + 1) n!
an xn
n = k
k
n
{
S = S 1 , S 2 ,
··· , S } n
{
A = S i , S i ,
n
A
1
2
··· , S } ⊆ S k
S
ik
0
0
n
0
·
(1 + x) (1 + x)
n
n
f (x) = (1 + x) =
· · · · · (1 + x) = (1 + x)
n
k=0
n k x k
n
n
··· · − · − · · · · −
f (x) = 1 + x + x2 + 1 + x + x2 + 1 1 1 = 1 x 1n x 1 x 1 = 1 x = a 0 + a1 x + a2 x2 +
−
··· ···
1 + x + x2 +
···
···
∞
∞
∞ n
{S 1 , S 2 , ··· , S }
− n k
− − − 1) ··· (−n − k + 1) k! (−1) (n + k − 1) (n + k) ··· (n) = k! n+k−1 =(−1) =
( n) ( n k
k
k
∞
k
∞
∞ n
{S 1 , S 2 , ··· , S } f ( x)
=
− −− − − −− 1
1
x x)−n n ( x)k k
= (1
∞
=
k =0 ∞
=
( 1)k
k =0 ∞
=
n
k=0
n+k k
n + k k
−
1
1
( 1)k xk
xk
a2 , b , c2 , d
4
a c
0 1
2
b c
f (x) = 1 + x + x2 (1 + x) 1 + x + x2 (1 + x)
x4
f (x) = 1 + 4x + 8x2 + 10x3 + 8x4 + 4x5 + x6 =
⇒
x4 = 8
a10 , b7 , c12
15 a
0
10 b
0
7
c
0
12
··· ··· ··· − · − · − −− · −− · − − · − − − − − − − ··· · ⇒ − − −
f (x) = 1 + x + + x10 1 + x + + x7 1 + x + + x12 1 x11 1 x8 1 x13 = 1 x 1 x 1 x = 1 x11 1 x8 1 x13 (1 x)−3 3 k + 2 k = (1 x)−3 = ( x)k = x = k k k k k + 2 k 8 11 13 19 = 1 x x x +x + x k k
=
x15 =
15 + 2 15
7+2 7
4+2 4
2+2 = 79 2
24 3
4
8 3 4
8 4
··· · ··· · −− · − − · − − − · − − − · − · ·
f (x) = x3 + x4 + + x8 4 = x 12 1 + x + + x5 4 1 x6 12 = x 1 x 4 12 = x 1 x6 (1 x)−4
∞
= x 12
4x6 + 6x12
1
k =0 ∞
= x12
4x18 + 6x24
4x30 + x36
k =0
x24 =
p1 (n)
n
12 + 3 12
4 ( x)k k
4x18 + x24
4
6+3 +6 6
k + 3 k x k
3 0
p2 (n)
n
n = 6
f 1
p1 (n)
0
···
f 1 (x) = (1 + x) ∞
=
·
1 + x2
(1 + xn )
1
···
1 + xk
k=1
f 2
f 2 (x) = 1 + x + x2 + 1 1 = 1 x 1 x3
··· · 1 + x3 + x6 + ··· · 1 · · − − 1 − x5 ···
1 + x5 + x10 +
··· ···
∞
f 1 (x) =
−− 1 + xk
k =1 ∞
x2k = xk k =1 1 x2 1 x4 = 1 x 1 x2 = f 2 (x) 1 1
− · − · 1 − x6 · 1 − x8 ··· − − 1 − x3 1 − x4
1
− x2
k
f ( x) = (1 + x) (1 + x)
n
·
n
(1 + x)
···
k
n
n
x
n x
1 n n = 6
n = 6
∈ N, n ≥ 3
xn 1+2+3
f 1
←→ x1 · x2 · x3
←→ x1 · x5 2 + 4 ←→ x2 · x4 6 ←→ x6 1+5
n
∈N xn xm = x n+m
·
f (x) = (1 + x) (1 + 2x)
·
··· (1 + nx)
n ak = P kn
=
· ⇒ n k
n
n k x k
n
(1 + x) =
k=0
n
e (x) =
k=0
n ak = k k!
k! =
ak k x k!
n (an )n∈N
∞
an n x n! n=0
∞
an n e1 (x) = x e2 (x) = n! n=0
∞
bn n x n! n=0
− ∞
(e1 + e2 ) (x) =
n
∞
(e1 e2 ) (x) =
an + bn n x n! n=0 n ak bn−k n x k n!
n=0 k=0 ∞
d an e1 (x) = xn−1 dx (n 1)! n=0 ∞
e1 (x) dx =
S = an1 , an2 ,
{
1
2
nk k
··· , a }
ai e (x) =
1+
0 x
1!
x2
+
2!
ni
xn
1
+
·
·
an xn+1 (n + 1)! n=0
·
+
n1 !
∀i = 1, 2, ··· , k
·
1+
x
1!
xn
2
+
·
·
·
+
n2 !
·
·
·
1+
x
1!
xn
k
+
·
·
·
+
nk !
4
S = A3 , B , N 2
4 C
B A N A N A
0 1
2
·
A
·
0
3
B
x 2 x 3 x 2 e (x) = 1 + x + + (1 + x) 1 + x + 2! 3! 2! 19 19 1 1 = 1 + 3x + 4x2 + x3 + x4 + x5 + x6 6 12 2 12 a4 19 = x4 = = 4! 12
⇒ a4 = 38
n
e (x) =
··· ·
x 2 x 4 x3 1+ + + x+ + 2! 4! 3! ex + e−x ex e−x x e 2 2 1 e2x e−2x ex 4 1 e3x e−x 4 ∞ ∞ 1 (3x)k ( x)k 4 k! k!
· − · = · − · − = · − − = · =0 =0 1 3 − (−1) · = (7) = x =
··· ·
x 2 1+x+ + 2!
···
k
k
∞
k=0
k
k
k
4
k!
n
x
1 3n ( 1)n = 4 n!
· − −
⇒a
=
n =
3n
n
− (−1) 4
= 1+x+
e−x
=1
−
−
n
∞
d (x) =
x2 x 3 + + 2!2 3!3 x x x+ + 2! 3!
ex
··· ···
dn
d (x)
dn n x n! n=0
n
S n
n
n
k
k = 0, 1, n
n! =
k=0
n dn−k k
xn n! n
n
x =
k=0
n dn−k n x , k n!
∀n ∈ N
k=0
S n(k)
· ·· , n
S n(k)
∞
n
x
=
n=0
1
1
n
∞
−x
n dn−k n x k n!
n=0 k =0 n ∞
=
n=0 k =0
n dn−k n x k n!
∞
∞
xn = (8) = n! n=0 x = e d (x)
dn n x n! n=0
·
⇒ d (x) = 1 −1 x · e
−x
=
d (x) = =
1
−x
− · e
1∞ x
xn
n=0 n ∞
=
∞
( x)n n! n=0
− −
n=0 k=0
( 1)k n x k!
x =
n n
dn = n!
n
n
( 1)k k!
− k=0
( 1)k k!
· − k=0
A B X Y
| ∪ | ≤ |X | + |Y | |A ∪ B| = |A| + |B| − |A ∩ B|
S
A1 A2 . . . An n
|A1 ∪ A2 ∪ . . . ∪ A | = n
A¯1
|
Ai
i=1
|−
A¯2
∩ ∩ ... ∩
⊆ S
|A ∩ A | + i
1≤i
j
Ai +
i=1
k
· |A1 ∩ A2 ∩ . . . ∩ A | n
n
∩ A | − . . . + (−1) |A1 ∩ . . . ∩ A | j
n
S
{ ∈ S : i|x}
Ai
i
Ai = x
∩ A¯3 ∩ A¯4
j
n
A¯2
n−1
|A ∩ A ∩ A | + . . . + (−1) i
1≤i
| | |− | |
A¯n = S
S = 1, 2, . . . , 10 6
106
|A | = i
A¯2
i
∩ A¯3 ∩ A¯4
| | − | | − |A3| − |A4| + |A2 ∩ A3| + |A2 ∩ A4| + |A3 ∩ A4| − |A2 ∩ A3 ∩ A4|
= S A2 = 166666
1060 2050 3040 1060 = 260
·
∈
∈
∈
A = d N : d 1060 B = d N : d 2050 C = d 560 A 2α 5α 0 α1 , α2 3 B = 101 51 C = 41
|
| |
·
1
·
| |
2
2
2
|
≤
: d 3040 60
|
N
≤
S
¯2 A
|A ∩ A ∩ A | i
A¯1
∩ A¯2 ∩ . . . ∩ A¯9
j
k
Ai =
∩ A¯3 ∩ . . . ∩ A¯9 i j = 6!, . . . , A1
|A| = 612
|A ∩ B| = 61 · 51
2
|B ∩ C | = 41 , |A ∩ C | = 41 , |A ∩ B ∩ C | = 41 |A ∪ B ∪ C | = 73001
i = 2, . . . , 9
(i
|S | = 9! |A | = 8! A ∩ A = 7! k
i, j
i
j
− 1). 2 ≤ k ≤ 9
| ∩ A2 ∩ . . . ∩ A9 | = 1
n
=
| ∩ | − | |− | | − · · − · − · · − S
Ai +
i=1
= 9!
8 8! +
8
=
( 1)k
k =0
Ai
Aj
. . . + ( 1)9
i
8 2
8 k
7!
(9
8 3
6! + . . .
− · |A1 ∩ A2 ∩ . . . ∩ A9|
−1
k)!
n
3n
n!2 n! n
A¯1
A¯2
∩ ∩ ... ∩
A¯n
Ai Ai = (n 1)!2 Ai Aj = (n 2)!2 , . . . , A1
| | | ∩ |
−1
− −
i
| ∩ A2 ∩ . . . ∩ A | = 1 n
n
| − | |
2
= n!
Ai +
i=1
2
i
− n · (n −
= n!
n
=
Ai
− · · n k
( 1)k
k=0
j
·
n 1)! + 2 2
(n
n
∩ A | − . . . + (−1) · |A1 ∩ A2 ∩ . . . ∩ A |
− · n 3
2
− 2)!
(n
n
(n
2
− 3)!
−
n
+ . . . ( 1)
·
n 0!2 n
− k)!2
n S n π(i) = i, i
Ai = π
∀
{ ∈
n S n : π(i) = i
|S | |A | |A ∩ A | n
i
i
j
| A1 ∩ A2 ∩ . . . ∩ A | n
n
− · · n k
( 1)n
k =0
∈ S
π
} = n! = (n = (n
n
− 1)! − 2)!
= 1
(n
− k)!
n ϕ(n)
n
n
n Ai
n = p α pα . . . pα 1 2 k A¯2 . . . A¯k = n A1 1
pi 1
¯1 ∩ ≤i≤k ∩ ∩ ϕ(n) = A |A | = pn |A ∩ A | = p n· p i
·
2
· · − | ∪ A2 ∪ . . . ∪ A | k
k
i
i
j
i
|A1 ∩ A2 ∩ . . . ∩ A | k
j
= 1
− ( p n1 + pn2 + . . . + pn − p1 n· p2 − p1 n· p3 − . . . − p 1n · p 1 1 1 n(1 − )(1 − ) · . . . · (1 − ) p1 p2 p
ϕ(n) = n
k
=
k−
k
− · p1 · p2 n· . . . · p
+ . . . + ( 1)k
)
k
k
(7, 5)
− · − · · · Rj.
7+5 7
2+2 2
4+3 4
4+2 2
3+2 2+2 + 2 2
1
3+2 2
e
∈ E
A
∈
(V, E ) V e
V B
{
E
∈ V
e = A, B
{
}
V = a,b,c,d,e,f,g
E =
E A V
{A, B} ∈ E
}
∈
{{a, b}, {a, d}, {b, f }, ··· , {f, g}} e
c d
f
b a
g
v(d(v))
v
G = (V, E )
d(v)
v ∈V
G = (V, E )
d(v) = 3, v
∀ ∈ V | |
2 E =
·
d(v) = 7 3 = 21
v ∈V
G = (V, E )
(v, 0, e1 , v1 , e2 , . . . , ek , vk )
ek =
i = 1, 2, . . . , k
V 1
E 1 =
}
ϕ : E 1
−→ E 2
{{
G1 = (V 1 , E 1 ) vi , vj E : v i , vj
}∈
G = (V, E )
∈ V 1}
G1 = (V 1 , E 1 ) v
G2 = (V 2 , E 2 ) e G1
⊆ V
V 1 G1
E 1
{v
k−1 , vk
⊆ {{v , v } ∈ E : v , v ∈ i
j
θ : V 1 θ(v)
}
i
j
−→ V 2
ϕ(e)
G2 G1 G2
|V 1| = |V 2| |E 1| = |E 2| ∀ ∈ V 1 (v0 , v1 , ··· , v0 )
d(v) = d(θ(v)), v
V
n
(θ (v0 ), θ(v1 ),
⊆ V 1
··· , θ(v0)) θ(V )
4
3
n
⊆ V 2
d
c
8
7
h
g
5
6
e
f
1
2
a
b
G = ( a,e,g,c ,
{
} {{ae}, {gc}})
|E | = i
|E | = i
|E | = 6 − i
|E | = 6 − i
|E | = 0
|E | = 1
|E | = 3
|E | = 4
|E | = 5
|E | = 6
|E | = 2
g
7
f
a
1
5
2
e
b
3
d
c
6
4
a
{b,c,f,g} −→ {2, 4, 5, 7} {f,g,b,c} 4, c → 7, g → 5, f → 2 b → g e → 6
a
{7, 4, 5, 2} {d, e}−→{3, 6}
d
→ 3
d
n
{ ··· , v }
V = v1 ,
m
{
n
n 2
G = (V, E )
u
(u, e1 , v1 , . . . , en , v) i,j, i < j vi = v j
} {
e = vi , vj , i = j e = vi , vj , i = j n 1 2 +n+m m
n
u
v
u
} −
v
v (u, e1 , . . . , vi , ej +1 , vj +1 , . . . , v)
u
v
G = (V, E )
∼ u
u
∈ V
V
∼ v ⇐⇒
u
u
u
(u)
v
u
∼ u, ∀u ∈ V.
V
∈ V
u, v
V
∼v
u
(u, e1 , v1 , . . . , en , v)
u
v
u
z
(v, en , . . . , v1 , e1 , u) v
u
v
∼ u
∈
u,v,z V V v (v, f 1 , w1 , . . . , fm , z) v z (u, e1 , v1 , . . . , en , v , f1 , w1 , . . . , fm , z) u z
u
u
∼ v v ∼ z
u
(u, e1 , v1 , . . . , e n , v)
z
∼
− − n
n
2
n
G
+
n
x
2
= x 2
i=2
1
n
2
Gi
x
G1
n−x )
| −
m
G1 G1=
G1, G2, . . . , G
m
G1 |E G1 | ≤ |E (K |E | ≤ x2
1
|E (G1) | ≤ |E (K ) | x
2 − nx + n 2− n =: f (x)
x
x2
∈ {1, 2, . . . , n − 1}
f (x)
− − − − − ⇒ | | ≤ − ⇒⇐ 1 n 1 + 2 2
f (1) =
f (n =
⇒
− 1)
max
x∈[1,n−1]
n
=
1
2
f (x) =
n
1 2
+
1
=
2
n
=
1
2
n
=
1
2
n
E
1
2
1
G = (V, E )
n
|E | = n − 1 d
|E | = n − 1 2 (n
− 1) =
v
| |
2 E =
d
v
v ∈V
v ∈V
d + (d
1) 1 + (n
1
2
2
n
d
d (v)
≥ − · − · d (v)
d
d) 2 = 2n
− 1 ⇒⇐
3
−d
H
G
H
G
G
G
H
G
V ( ) = V ( )
nn−2
K
n
A
∈
v1
(v0 , e1 , v1 , . . . , ek , vk ) B, v2 A, v3 B , . . . , vk
∈
∈
B
k 2N 1 v0 = vk
v0
∈ A
∈ − ⇒⇐
∈ B
A B
∈ V, A ∪ B = V ∀a ∈ A, ∀e = {a, b} ∈ E B = B ∪ {b} ∀b ∈ B, ∀e = {b, a} ∈ E A = A ∪ {a} v
{}
A = v , B =
∅
w
∈ A w ∈ B
w
∈ V
w w w
B
w
A
w
⇒⇐ K , |A| = n, |B| = m
A
n,m
A = {a,b,c} , B = {1, 2}
K3 2 ,
a
1
b
2
c
K2
,m
K2 5 ,
v
B
|V | = |A| + |B| = 2 + m
m + 1
B
2
B
2
m
m
−1 # = m · 2
B
2
A
m−1
1 B
A m−1
#=
m−1
A m−1
−− − k=1
m k = k
k=1
m m k k k
1 = m 1
k =0
m
1
k
B = m 2m−1
·
m
−1
G
G = (V, E )
( , ω) ω (e)
→ [0, ∞)
ω : E
e∈E
G = (V, E ) S = ∅
ω
E
(V, S )
e
∪{e}
S = S
∈ E \ S
∪ {e}
S
9
A
B
6 8 3
5 5
D
S =
7
C
E
∅ { }
S = CD DE
CE
◦
DE S = CD,DE
◦
CE
◦
{ } S = {CD,CE } CE
CB S = CD,DE,CB
{
}
◦
CB S = CD,CE,CB
◦
AD
◦
AD
{ } S = {CD,DE,CB,AD} S = {CD,CE,CB,AD }
A
B
A
B
6 8
6 8
C
C
3
D
5
3 5
E
D
E
ω (v) = 22
v ∈S
G = (V, E ) n ω v0 ∈ V T = {v0 } , S = V \{v0 } , F = ∅ |F | < n − 1 e = {v, w} ∈ E v ∈ T, w ∈ S T = T ∪ {w} , F = F ∪ {e} , S = S \ {w}
E
A
B
9
T F S
{A} ∅ {B,C,D,E }
T F S
A
8
D
T
S
{A, D} {AD} {B,C,E }
B
A
9
9 A
B
{A,C,D} {AD,DC } {B, E }
T F S
C
3
T F S
6 D
5
D
{A,C,D,E } {AD,DC,CE } {B}
E
5 5 E
T
C
S
S
T
A
9 C
{A,B,C,D,E } {AD,DC,CE,CB} ∅
T F S
6
D
B
7
E
S
2 R
R2
G
p
q p
r
− q + r = 2
f d(f )
f ∈F
·| |
d(f ) = 2 E
n d(f )
·| |
2 E =
d(f )
f ∈F
3n
−6
≥ 3, ∀f ∈ F
≥ 3 · |F | ⇒ |F | ≤ 32 |E |
| | | | | | ≤ n + 23 |E |
2 + E = V + F 1 E 3
| | ≤ n − 2 ⇒ |E | ≤ 3n − 6
K K5 K5
n
K3 K4
K , n ≥ 5 n
K5 |V | = 5, |E | =
5 = 10 2
|E |
· −6 = 9
3 5
− 4
n
2n
d(f ) 2 E =
·| |
d(f )
f ∈F
≥ 4, ∀f ∈ F
≥ 4 · |F | ⇒ |F | ≤ |E 2 |
| | | | | | ≤ n + |E 2 | ⇒ |E | ≤ 2n − 4
2 + E = V + F
K 3,3
· −4 =8
2 6
{A, C } {C, B}
{A, B} G H
C
K5
{A, B} K3 3 ,
a
b
c
j
d
e
i
g
h
f
K3 3 K5 K3 3 ,
a d i f
,
a
b
d
f
c
e
g
h
K 3,3
n
m 4 108 > 360 o
·
o
|V | = 5 · n = 6 · m2
v ∈V
·| |
·| |
= 3d(v) V = 2 E =
f ∈F
d(f ) = 5n + 6m,
|F | = n + m 2 + E = F + V
| | | | | |
2+
5n + 6m 5n + 6m = n + m + 2 3 n = 12, m = 20
⇒
C 60
v 1 , v2 , vi
··· v
n , v1
vi
n n
n
n 2
a
b
c
d
e
f
g
h
i
a
− b − c − f − i − h − g − d − e − f − h − e − b − d − a a
a i
b h
c
f g e
a
|A| = 4, |B| = 6
V
A
B
|A| = |B| a1 , a2
··· a
∈ | | | | n−1 +2
n , a1
A
ai A i A = B
B
ai
∈ B
i
⇒n
n
2
A B
G
A B
G
− n
1
2
+2
≤ |E | = |E | + d(a) + d(b) ≤ |K 2| + d(a) + d(b) n−2 = + d(a) + (b)
n−
2
