!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

!

N := 0, 1, 2, 3, ...

n n ∈ N

N+: N× N → N · : N× N → N

m, n ∈ Nn m n ≥ mk ∈ N n = m + k k = 0

n m n > m

m n m | n k ∈ Nn = m · k

2 | n n

k ∈ N n = 2k +1 n

n > 1 k ∈ N1 < k < n k | n n

! 3 > 2

! x3 > 10

!

n

! n n3 > 10

nn3 > 10

33 > 1013 > 10

A,B ,C , ...A(x) B(x , y , z)

! A :

! B : 2 > 3

! A(n) : 1 + 2 + 3 + ...+ n = n(n + 1)/2

AB

A A¬A A A

A A : 2 > 3 A¬A : 2 > 3

¬

A ¬A

A ¬A

A BA B A ∧ B

A B A ∧ B

A ∧ B A B AB

A BA B A ∨ B

A B A ∨ B

A ∨ B A B A B

! A : 2 > 0 B : 1+ 1 = 1 A∧B A∨B! A A ∨ (¬A)

A ∧ (¬A)

A ∨ (¬A)A ∨ (¬A)

A

A ¬A A ∨ (¬A)

A ∨ (¬A)A ∨ (¬A)

A ∧ (¬A)A ¬A A ∧ (¬A)

A ∧ (¬A)

A BB A A ⇒ B

A B A ⇒ B

A ⇒ B A B A B A BA B

A(n) : n ⇒ n , n ∈ N,

n

n

! n = 3 n n ⇒! n = 4 n n ⇒! n = 9 n n ⇒

n

! n = 2 n n ⇒n

⇒

A BA B A ⇔ B

A B A ⇔ B

A ⇔ B A B A B

A B

A B

2 > 0 100 = 99 + 1

A BA ⇔ B A ≡ B

¬(A ∨ B) ⇔ (¬A) ∧ (¬B), ¬(A ∧ B) ⇔ (¬A) ∨ (¬B).

¬(A ∨ B)(¬A) ∧ (¬B) ¬(A ∧ B) (¬A) ∨ (¬B)

A ⇒ B(A ⇒ B) ⇔ (¬B ⇒ ¬A).

n n > 0 ⇒ n3 > 0n3 > 0 ⇒ n > 0

A B ¬A ¬B ¬B ⇒ ¬A A ⇒ B (A ⇒ B) ⇔ (¬B ⇒ ¬A)

TT : A ∨ (¬A) FF : A ∧ (¬A)

A ∧ T ≡ A ∧A ∨ F ≡ A ∨A ∧ F ≡ F ∧A ∨ T ≡ T ∨A ∧ A ≡ A ∧A ∨ A ≡ A ∨¬(¬A) ≡ A

A ∧ B ≡ B ∧ A ∧A ∨ B ≡ B ∨ A ∨

(A ∧ B) ∧ C ≡ A ∧ (B ∧ C ) ∧(A ∨ B) ∨ C ≡ A ∨ (B ∨ C ) ∨

A ∨ (B ∧ C ) ≡ (A ∨ B) ∧ (A ∨ C )A ∧ (B ∨ C ) ≡ (A ∧ B) ∨ (A ∧ C )

A ∨ (A ∧ B) ≡ AA ∧ (A ∨ B) ≡ A

∧ ∨

A ⇒ B ≡ ¬A ∨ B ≡ ¬B ⇒ ¬A(A ⇒ B) ∧ (A ⇒ C ) ≡ A ⇒ (B ∧ C )(A ⇒ B) ∨ (A ⇒ C ) ≡ A ⇒ (B ∨ C )

(A ⇒ C ) ∧ (B ⇒ C ) ≡ (A ∨ B) ⇒ C(A ⇒ C ) ∨ (B ⇒ C ) ≡ (A ∧ B) ⇒ C

(A ⇔ B) ≡ ((¬A) ⇔ (¬B))(A ⇔ B) ≡ (A ⇒ B) ∧ (B ⇒ A)(A ⇔ B) ≡ (A ∧ B) ∨ ((¬A) ∧ (¬B))¬(A ⇔ B) ≡ A ⇔ (¬B)

A(x)x x

A(x)

xA(x) A(x)

! N 0

! Z! R! ∅ ∅

M x x ∈ M x M

! ∀! ∃

M A(x)∀

∀x∈M

A(x) ⇔ A(x) x ∈ M

∃

∃x∈M

A(x) ⇔ A(x) x ∈ M

∀x ∈ M : A(x) ∀x∈M

A(x) ∃

x

! ∀x : x > 0 ⇒ x3 > 0

! ∀x : x > 0 ⇔ x2 > 0

! ∃x : x > 0 ⇔ x2 > 0

∃y : y + x2 > 0 ∀x

∃y∀x : y + x2 > 0

∃

∃x∈M

A(x) ⇔ A(x) x ∈ M

⇔ A(x) x ∈ M

⇔ ¬ ∀x∈M

(¬A(x))

∃x ∈ M : A(x) ∀x ∈ M : ¬A(x)

¬(∃x ∈ R : x2 < 0

)⇔ ∀x ∈ R : x2 < 0.

¬ (∀x ∈ M : A(x)) ⇔ ∃x ∈ M : ¬A(x).

∀ M = ∅∀x ∈ M : A(x)

A(x) A(x)

M x x = x + 1

∀x∈M

x > x

x , y

! ∀x∀y : x2 + y2 − 2xy ≥ 0 ∀y∀x : x2 + y2 − 2xy ≥ 0∀x , y : x2 + y2 − 2xy ≥ 0

! ∃x∃y : x + y > 0 ∃y∃x : x + y > 0∃x , y : x + y > 0

! ∀x∃y : x + y > 0

! ∃x∀y : x + y > 0

I R f : I → RI

∀ε>0

∀x∈I

∃δ>0

∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε.

f I

∀ε>0

∃δ>0

∀x∈I

∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε.

II

I I = [a, b]

f I

¬(∀

ε>0∀x∈I

∃δ>0

∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε)

⇔(∃

ε>0¬ ∀

x∈I∃

δ>0∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε)

⇔(∃

ε>0∃x∈I

¬ ∃δ>0

∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε)

⇔(∃

ε>0∃x∈I

∀δ>0

¬ ∀y∈I

|x − y | < δ ⇒ |f (x)− f (y)| < ε)

⇔(∃

ε>0∃x∈I

∀δ>0

∃y∈I

(|x − y | < δ) ∧ ¬(|f (x)− f (y)| < ε))

⇔(∃

ε>0∃x∈I

∀δ>0

∃y∈I

(|x − y | < δ) ∧ (|f (x)− f (y)| ≥ ε))

H : R → R

H(x) :=

1 x ≥ 0,

0 x < 0,

I = Rε > 0 ε = 1/2 x ∈ R x = 0

δ > 0 y ∈ R

|x − y | = |y | < δ |H(x)− H(y)| = |1− H(y)| ≥ ε =1

2.

δ > 0 y = −δ/2 |y | = δ/2 < δ|1− H(y)| = 1 > 1/2 H R

A : ∃ε>0

∃x∈R

∀δ>0

∃y∈R

(|x − y | < δ) ∧ (|H(x)− H(y)| ≥ ε)

B : H .

A

B

(A ∧ (A ⇒ B)

)⇒ B .

P1, ... ,Pn C

(P1 ∧ P2 ∧ · · · ∧ Pn) ⇒ C

P1, ... ,Pn

C

P1

P2

Pn

∴ C

∴

AA ⇒ B

∴ B

(A ∧ (A ⇒ B)

)⇒ B

A B A ⇒ B A ∧ (A ⇒ B)(A ∧ (A ⇒ B)

)⇒ B

⇒

AA ⇒ B

∴ B

¬BA ⇒ B

∴ ¬A

A ⇒ BB ⇒ C

∴ A ⇒ C

∴

∴

∴

∨∧

A ∨ B¬A

∴ B

¬(A ∧ B)A

∴ ¬B

A ∨ B¬A ∨ C

∴ B ∨ C

∴

∴

∴

AB

∴ A ∧ B

A ∧ B

∴ A

A

∴ A ∨ B

∴

BA ⇒ B

∴ A

¬AA ⇒ B

∴ ¬B

A ∨ BA

∴ ¬B

∴

∴

∴

∀x∈M

P(x)

∴ P(x0) x0 ∈ M

P(x) x ∈ M

∴ ∀x∈M

P(x)

∃x∈M

P(x)

∴ P(x0) x0 ∈ M

P(x0) x0 ∈ M

∴ ∃x∈M

P(x)

n ∈ N n2

n

P1 : ∀n∈N

¬(n ∧ n ),

P2 : n ⇒ n2 ,

P3 : n2 ∧ (n ∨ n )

C : n .

P2 n kn = 2k + 1

n2 = (2k + 1)2 = 2(2k2 + 2k) + 1 = 2k ′ + 1

k ′ = 2k2 + 2k n2

P3 : n2 ∧ (n ∨ n )

∴ P4 : n2

P1 : ∀n∈N

¬(n ∧ n )

∴ P5 : ¬(n2 ∧ n2 )

P4 : n2

P5 : ¬(n2 ∧ n2 )

∴ P6 : ¬(n2 )

P6 : ¬(n2 )P2 : n ⇒ n2

∴ P7 : ¬(n )

P3 : n2 ∧ (n ∨ n )

∴ P8 : n ∨ n

P7 : ¬(n )P8 : n ∨ n

∴ C : n

P3

xX x ∈ X

X P

x ∈ X ⇔ P(x).

X = x : P(x)

∅ := x : x = xx = x

X = x1, x2, ... , xnX

X = x : (x = x1) ∨ (x = x2) ∨ · · · ∨ (x = xn).

x ∈ A : P(x) = x : x ∈ A ∧ P(x)

n ∈ N : ∃k∈Z

n = 2k

x ∈ X Y XY X ⊂ Y

X ⊂ Y ⇔ ∀x ∈ X : x ∈ Y .

X = Y X ⊂ Y Y ⊂ X

X Y X ⊂ Y X = YX " Y

⊆ ⊂ ⊂ "

X ∅ ⊂ X ∅∀x ∈ X : x ∈ Y∅ ⊂ X

A = a, b, c a, b, c

B = a, b, a, b, c , c

A A ⊂ B B ⊂ A

x ∈ A ⇔ (x = a) ∨ (x = b) ∨ (x = c) ⇔ x ∈ B .

C = a, b C ⊂ A C " A D = b, cD " A C ⊂ D D ⊂ C

X XX |X | X

P(M) := A : A ⊂ M.

P(M) P(M)M

A ⊂ M A ∈ P(M)

a, b, c

P(a, b, c) =∅, a, b, c, a, b, b, c, a, c, a, b, c

.

a, b, c|P(a, b, c)| = 8

A = x : P1(x) B = x : P2(x)A B

A ∪ B := x : P1(x) ∨ P2(x), A ∩ B := x : P1(x) ∧ P2(x),A \ B := x : P1(x) ∧ (¬P2(x)).

A ⊂ M A

A := M \ A.

A ∩ B = ∅ A B

A− B A \ B AA

A = a, b, c B = c , d

A ∪ B = a, b, c , d, A ∩ B = c, A \ B = a, b.

∧ ∨! A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C )

! A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C )

! (A ∪ B) \ C = (A \ C ) ∪ (B \ C )

! (A ∩ B) \ C = (A \ C ) ∩ (B \ C )

! A \ (B ∪ C ) = (A \ B) ∩ (A \ C )

! A \ (B ∩ C ) = (A \ B) ∪ (A \ C )

n ∈ Nn⋃

k=0

Ak := A0 ∪ A1 ∪ A2 ∪ · · · ∪ An,

n⋂

k=0

Ak := A0 ∩ A1 ∩ A2 ∩ · · · ∩ An.

n = ∞

x ∈∞⋃

k=0

Ak :⇔ ∃k∈N

x ∈ Ak ,

x ∈∞⋂

k=0

Ak :⇔ ∀k∈N

x ∈ Ak .

∞⋂

k=0

Ak ⊂∞⋃

k=0

Ak .

Ak = 0, 1, 2, ... , k k ∈ N

∞⋃

k=0

Ak = N,∞⋂

k=0

Ak = 0.

N ⊂⋃∞

k=0 Ak x ∈ Nx ∈ Ax x ∈

⋃∞k=0 Ak

⋃∞k=0 Ak ⊂ N

x ∈⋃∞

k=0 Ak x ∈ Ak k ∈ N x ∈ N⋂∞

k=0 Ak ⊂ N 0 ∈ Ak

k ∈ N 0 ⊂⋂∞

k=0 Ak x ∈ N \ 0x /∈ Ax−1 x /∈

⋂∞k=0 Ak

a, b = b, a.

(a, b)

(a, b) = (c , d) ⇔ (a = c) ∧ (b = d).

(a, b) := a, a, b.

A,B a ∈ A b ∈ B

A× B := (a, b) : a ∈ A, b ∈ B.

A× B A B

(a, b, c)n (a1, ... , an) n

A1 × · · ·× An Ak k = 1, ... , n

N2 := N× N.

P(x) : x /∈ xA = x : P(x)

A = x : x /∈ xy y ∈ A y /∈ A y = A

A ∈ A P(A) A /∈ A A /∈ A ¬P(A) A ∈ A

A ∈ A A /∈ A A

Ə ჴ

0, 1, 2, 3, ...

N

Nn

n (n) N

N (N, )

NN

Nn N n N

S ⊂ NS S = N

∅

0 := ∅,1 := 0 = ∅,2 := 0, 1 = ∅, ∅,3 := 0, 1, 2 = ∅, ∅, ∅, ∅

(n) := n ∪ n,N := ∅ ∪

n : ∃

m∈Nn = (m)

.

n ∈ (n) n " (n).

<

m < n :⇔ m " n.

0 = ∅ ∈ Nn ∈ N (n) ∈ N

0 = ∅ n (n) = n ∪ n = ∅

S NN

(m) = (n) m = n

n

∀m∈N

m ∈ n ⇒ n ⊂ m,

∀m∈N

n ⊂ m ⇒ m /∈ n

S

S :=n ∈ N : ∀

m∈Nm ∈ n ⇒ n ⊂ m

.

S = Nn ∈ N S = ∅

0 ∈ S 0 = ∅ m ∈ 0 m ∈ N

m ∈ 0 ⇒ 0 ⊂ m

m ∈ N 0 ∈ S

n ∈ S (n) ∈ SS = N

n ∈ S n = nn ⊂ n n /∈ n m = n

(n) = n ∪ n ⊂ n.

(n) ⊂ m m n ⊂ (n) ⊂ mn ∈ S m /∈ n m ∈ n

(n) ⊂ m

(n) ⊂ n (n) ⊂ m m ∈ n(n) n n

m ∈ (n) ⇒ (n) ⊂ m

n ∈ S (n) ∈ S

S 0S

S = N

A

y ∈ x ∧ x ∈ A ⇒ y ∈ A.

A x ⊂ A x ∈ A

nn

S0 ∈ S n ∈ S x ∈ (n)

x ∈ n x = n x ∈ n x ⊂ n ⊂ (n)n ∈ S x = n x ⊂ n ∪ n = (n) (n) ∈ S

S = N

m nn

(m) = (n) ⇒ m = n

n∪ n = m∪ m n ∈ m n = m m = n m ∈ nm = n n ∈ m m ∈ n n ∈ n

n ⊂ n

N

a, b ∈ Nc = a + b ∈ N a b

a, b, c ∈ N

a+ (b + c) = (a + b) + c

a+ 0 = 0 + a = a

a+ b = b + a

a · b ∈ Na b a, b, c ∈ N

a · (b · c) = (a · b) · ca · 1 = 1 · a = a

a · b = b · a

a · (b + c) = a · b + a · c .

a1, a2, ... , an

a1 + a2 + · · ·+ an =:n∑

j=1

aj =:∑

1≤j≤n

aj

a1 · a2 · · · an =:n∏

j=1

aj =:∏

1≤j≤n

aj .

n ∈ N

0! := 1 n! := n · (n − 1)! n > 1

ab = a · a · ... · a︸ ︷︷ ︸b

a, b ∈ N

a0 := 1 an := a · an−1

ab+c = ab · ac (ab)c = ab·c .

a, b ∈ N

a | b ⇔ ∃c ∈ N : c · a = b,

a b a | b a b

! n ∈ N n 4→ n2 n n2

! n3 4→ n n3 = p ∈ N n

R

R = (a, b) ∈ N2 : b = a2,R = (a, b) ∈ N2 : a = b3,

R M N

R = (m, n) ∈ M × N : P(m, n)P M = N R

M

R

(a, b) ∈ R a ∼R b a ∼ bR

R

R = a : ∃b : (a, b) ∈ R

R = b : ∃a : (a, b) ∈ R.

R M

(a, a) ∈ R a ∈ M

(a, b) ∈ R (b, a) ∈ R a, b ∈ M

(a, b) ∈ R (b, c) ∈ R (a, c) ∈ Ra, b, c ∈ M

MM

! a ∼ a a ∈ M

! a ∼ b ⇒ b ∼ a

! a ∼ b ∧ b ∼ c ⇒ a ∼ c

! R = (a, b) ∈ N2 : a > b(1, 0), (2, 1), (2, 0) (0, 1)(a, b) ∈ R ⇔ a ∼ b ⇔ a > b ∼a > b b > c a > c a > b ⇒ b > a

a > a! n ∈ N I (n)

I (125) = 1 + 2 + 5 = 8I (78) = 7 + 8 = 15

R = (a, b) ∈ N2 : I (a) = I (b)(22, 4), (14, 5), (3, 30) (4, 1) R

I (a) = I (a) I (a) = I (b)I (b) = I (a) I (a) = I (b) I (b) = I (c)I (a) = I (c) R

A F AA

2N = 0, 2, 4, 6, ... ⊂ N2N+ 1 = 1, 3, 5, 7, ... ⊂ N

2N ∩ (2N+ 1) = ∅ 2N ∪ (2N+ 1) = N

F = 2N, 2N+ 1 N 2N 2N+ 14

2N2N+ 1

2N = [0] 2N+ 1 = [1].

F M∼ M

a ∼ b :⇔ a, b ∈ M

∼

∼ MF = [a] : a ∈ M M

a ∈ [b] :⇔ a ∼ b.

F = M/ ∼

∼ a ∈ [a] a ∈ MM

[a], [b] c ∈ Mc ∈ [a] c ∈ [b] c ∼ b c ∼ a a ∈ [a]

a ∼ c a ∈ [a] a ∼ b a ∈ [b]a ∈ [a] [a] ⊂ [b] a b[b] ⊂ [a] [a] = [b]

n ∈ N −n

n + (−n) = 0

N

N2 = (n,m) : m, n ∈ N.

N N2 n ∈ N(n, 0) ∈ N2

N2

(n,m) ∼ (p, q) :⇔ n + q = m + p.

(5, 0) 5 ∈ N(6, 1) 5 + 1 = 0 + 6

!Z = N2/ ∼ N2

! (n, 0) ∈ N2 n ∈ N

[+n] ∋ (n, 0)

! (0, n) ∈ N2 n ∈ N n ≥ 1[−n] ∋ (0, n)

! [+n] [−n] n ∈ N!

Z = [+n] : n ∈ N ∪ [−n] : n ∈ N \ 0.

N2

(n,m) + (p, q) = (n + p,m + q).

(n,m) ∼ (n, m) (p, q) ∼ (p, q)

(n,m) + (p, q) ∼ (n, m) + (p, q).

[±n], [±m] ∈ Z [±n] + [±m][±n]

[±m]

Z

(n,m) + (0, 0) = (n,m), (n,m) + (p, q) = (p, q) + (n,m)

Z [0] ∈ Z

(n,m) ∈ N2

(n,m) + (m, n) = (n +m, n +m) ∼ (0, 0).

[n] + [−n] = [0]Z

Z n [n]−n [−n] n + (−m) n −m

− Z

!

!n ↔ (n, 0)

! N2

Z = N2/ ∼! Z = [n] ∪ [−n]! N2

N n ↔ (n, 0) m ↔ (m, 0)(n, 0) + (m, 0) = (n +m, 0) ↔ n +m

! N2

Z! Z

Z

Z

Z2

(n,m) ∼ (p, q) :⇔ n · q = m · p.

(n,m), (p, q) ∈ Z2

(n,m) · (p, q) = (n · p,m · q)

Z Z2 n ↔ (n, 1)

Q := Z2/ ∼

[(1, 1)] [(n,m)] ∈ Q[(n,m)−1] = [(m, n)]

Q

(m, n) + (p, q) = (q ·m + p · n, nq).

(n,m) [(n,m)]

(n,m) =:n

m∈ Q.

!

!

< > =

!

∀a,b,c∈Q

a+ (b + c) = (a + b) + c

∃0∈Q

∀a∈Q

a+ 0 = 0 + a = a

∀a∈Q

∃−a∈Q

(−a) + a = a + (−a) = 0

∀a,b∈Q

a + b = b + a.

∀a,b,c∈Q

a · (b · c) = (a · b) · c ,

∃1∈Q1 =0

∀a∈Q

a · 1 = 1 · a = a,

∀a∈Q

∃a−1∈Q

a · a−1 = a−1 · a = 1,

∀a,b∈Q

a · b = b · a.

a = 0 a · b = a · cb = c

0 = 1

∀a,b,c∈Q

a · (b + c) = a · b + a · c .

a− b = b − a a = b

a · b = 0a = 0 b = 0

m < n :⇔ m # n m, n ∈ N,[(0,m)] < [(n, 0)] [(0,m)] ∈ Z \ N, n ∈ N.[(0,m)] < [(0, n)] :⇔ n < m [(0,m)], [(0, n)] ∈ Z \ N.

m ∈ Q

0 < m =p

q:⇔ (p < 0 ∧ q < 0) ∨ (0 < p ∧ 0 < q) p, q ∈ Z,

m < n :⇔ 0 < n −m m, n ∈ Q.

m ≤ n :⇔ (m < n) ∨ (m = n) m, n ∈ Q.

m > n n < m m ≥ n n ≤ m

b ∈ Q a ∈ Q a2 = b

A(n)n ∈ N n ≥ n0 n0 ∈ N

A(n0)

A(n + 1) A(n) n ≥ n0

∀n∈Nn≥n0

(A(n) ⇒ A(n + 1)

)

A(n)A(n + 1) A(n)

A(n)n ≥ n0

n∑

k=1

(2k − 1) = n2 n ∈ N \ 0

A(n) :∑n

k=1(2k − 1) = n2

n > 0

A(1)

1∑

k=1

(2k − 1) = 2 · 1− 1 = 1 12 = 1,

A(1) : 1 = 1

A(n) ⇒ A(n + 1) n ∈ N \ 0∑n+1k=1(2k − 1) = (n + 1)2

∑nk=1(2k − 1) = n2 n

n A(n)

n+1∑

k=1

(2k − 1) =n∑

k=1

(2k − 1) + 2(n + 1)− 1

A(n) nn2

n+1∑

k=1

(2k − 1) = n2 + 2n + 1 = (n + 1)2

A(n + 1) A(n)A(n + 1) A(n) ⇒ A(n + 1)

n0 ∈ N(A(n0) ∧ ∀

n∈Nn≥n0

(A(n) ⇒ A(n + 1)))⇒ ∀

n∈Nn≥n0

A(n).

n0 = 0

S ⊂ NS S = N

S = n ∈ N : A(n) A(0)0 ∈ S A(n) ⇒ A(n + 1) n ∈ N

n ∈ S n + 1 n SS = N

A(n) n ∈ N

n0 > 0

S = n ∈ N : n < n0 ∨ ((n ≥ n0) ∧ A(n)).

S ⊂ N

Sm mSm

m < n :⇔ (n ∈ Sm) ∧ (n = m).

m ≤ n ⇔ m ⊂ n

M m0 ∈ MM ⊂ Sm0 m0 ≤ m m ∈ M

∀M⊂N

∃m∈M

M ⊂ Sm.

S ⊂ N 0 ∈ S n ∈ S(n∈S S = N

n0 ∈ N n0 /∈ S M = n ∈ N : n /∈ SM m0

0 ∈ S m0 = 0 m0 > 0 m0 ∈ N m0 − 1m0 M m0

M m0 − 1 ∈ S Sm0 − 1 S

m0 ∈ S m0 /∈ M

∀n∈N

(1 + 12)

n ≥ 1 + n/2

∀n∈N

∀a,b∈Q

(a + b)n =∑n

k=0n!

(n−k)!k!anbn−k

∀n∈N

∀r∈Q

r=p/qq2>p2

∑nk=0 r

k = rk+1−1r−1

Hk =n∑

k=1

1k n ∈ N \ 0 ∀

n∈NH2n ≥ 1 + n/2

M A1, ... ,An ⊂ M

( n⋂

i=1

Ai

)=

n⋃

i=1

Ai .

M M = n n ∈ NP(M) = 2n

P(n)2n + 1

n = 1

C

P(n) 2n+3

! 2n2n + 1

! 2n + 12n + 1 P(n)

n ≥ 2

n = 2

n nn + 1

n+ 1 n

2, ... , n + 1

2, ... , n

n + 1

A(n0)

A(n + 1) A(n) n ≥ n0

A(n0)

A(n + 1)A(n0),A(n0 + 1), ... ,A(n)

n ≥ 2

n = 22, 3, ... , n n + 1

a, b < n + 1 a b

n + 1 = a · b

(A(n0) ∧ ∀

n∈Nn≥n0

(A(n) ⇒ A(n + 1)))⇒ ∀

n∈Nn≥n0

A(n)

(A(n0) ∧ ∀

n∈Nn≥n0

((A(n0) ∧ · · · ∧ A(n)) ⇒ A(n + 1)

))⇒ ∀

n∈Nn≥n0

A(n).

((A(n0) ∧ · · · ∧ A(n)) ⇒ A(n + 1)

)⇒(A(n) ⇒ A(n + 1)

)

n0 ∈ N B(n) : A(n0) ∧ · · · ∧ A(n) n ≥ n0A(n0) = B(n0)

(B(n0) ∧ ∀

n∈Nn≥n0

(B(n) ⇒ A(n + 1)

))⇒ ∀

n∈Nn≥n0

A(n).

(B(n) ⇒ A(n+ 1)

)≡(B(n) ⇒ (A(n+ 1) ∧ B(n))

)≡(B(n) ⇒ B(n+ 1)

)

(B(n0) ∧ ∀

n∈Nn≥n0

(B(n) ⇒ B(n + 1)

))⇒ ∀

n∈Nn≥n0

A(n).

∀n∈Nn≥n0

A(n) ≡ ∀n∈Nn≥n0

B(n) B

0! := 1

n! :=n∏

k=1

k , n ∈ N \ 0,

0! := 1, n! := n · (n − 1)!, n ∈ N \ 0.

n!n!

t > 0

Γ(t) : =

∫ ∞

0z t−1e−z dz , t > 0.

Γ(1) = 1

Γ(t + 1) = tΓ(t) = tΓ(t + 1− 1) t > 0

Γ(n + 1) = n! n ∈ N

n = 0 n + 1n ∈ N S ⊂ N

Sn ∈ N

n = n0n = 0

f0 := 0, f1 := 1, fn := fn−1 + fn−2, n ∈ N \ 0, 1.

S ⊂ N

3 ∈ S x , y ∈ S ⇒ x + y ∈ S .

3 ∈ S 3 + 3 = 6 ∈ S 3 + 6 = 9 ∈ S 6 + 6 = 12 ∈ S

S = n ∈ N : ∃k ∈ N \ 0 : n = 3k

S S

N

ΣΣ∗ Σ

λ ∈ Σ∗ λ

w ∈ Σ∗ x ∈ Σ wx ∈ Σ∗

Σ = 0, 1 ΣΣ λ λ0 = 0

λ1 = 1 0, 1 ⊂ Σ∗ 01, 10, 11, 00 ∈ Σ∗

000, 001, 010, 011, 100, 101, 110, 111 ∈ Σ∗

Σ

w = λ wx w ∈ Σ∗

x ∈ Σ

w ∈ Σ∗ w · λ = w λw1,w2 ∈ Σ∗ x ∈ Σ

w1 · (w2x) = (w1 · w2)x .

Σ = 0, 1 110, 101 ∈ Σ∗

110w1

· (10w2

1x) = (110 · 10)1 =

((110 · 1)0

)1

=(((110)1

)0)1 = 110101.

l(w) w ∈ Σ∗

l(λ) = 0

l(wx) = l(w) + 1 x ∈ Σ

TF 1 0

¬ ∧∨

b b1 00 1

a b a b1 1 11 0 00 1 00 0 0

a b a b1 1 11 0 10 1 10 0 0

(λ) := λ, (wx) := (w) (x), w ∈ Σ∗, x ∈ Σ.

(011) = (01) (1) = (0) (1) (1) = 100.

r r

T1, ... ,Tn n ∈ N \ 0, 1r1, ... , rn r

r r1, ... , rn r

r1r r r1

r

r1

r1, r2 rr r1 r2

r

r1 r2

n

r

r1 r2 r3 r4 ... rn

T1 T2

r r T1

T2 r

∅∅ · ∅ r

T1 = ∅ T2 = r T1 = rT2 = ∅ T1 = T2 = r

r r

T1 T2

r r T1 T2

r

Σ∗

P(w) w ∈ Σ∗

P(w) w ∈ Σ∗

P(λ) λ

∀w∈Σ∗

∀x∈Σ

P(w) ⇒ P(wx)

l(xy) = l(x) + l(y) x , y ∈ Σ∗

l(w) w ∈ Σ∗

P(y) : ∀x∈Σ∗

l(xy) = l(x) + l(y).

P(λ)

P(λ) : ∀x∈Σ∗

l(xλ) = l(x) + l(λ).

xλ = x l(λ) = 0 P(λ)

P(y) P(ya)a ∈ Σ

P(y) ⇒ ∀a∈Σ

P(ya)

(∀

x∈Σ∗l(xy) = l(x) + l(y)

)⇒(

∀a∈Σ

∀x∈Σ∗

l(xya) = l(x) + l(ya))

l(xya) = l(xy) + 1 l(ya) = l(y) + 1

P(n) : n

P(0) P(n) ⇒ P(n + 1)

T rh(T ) = 0

T1,T2 T = T1 · T2

h(T ) = 1 + (h(T1), h(T2))

n(T )n(T ) = 1 T r

n(T1 · T2) = 1 + n(T1) + n(T2)h(t) n(t)

T n(T ) ≤ 2h(T )+1 − 1

Tn(T ) = 1 h(T ) = 0

n(T ) ≤ 2h(T )+1 + 1 ⇔ 1 ≤ 20+1 − 1,

∀T1,T2

(((n(T1) ≤ 2h(T1)+1 − 1) ∧ (n(T2) ≤ 2h(T2)+1 − 1)

)

⇒ (n(T1 · T2) ≤ 2h(T1·T2)+1 − 1))

n(T1 · T2) = 1 + n(T1) + n(T2)

≤ 1 + 2h(T1)+1 − 1 + 2h(T2)+1 − 1

≤ 2 (2h(T1)+1, 2h(T2)+1)− 1

= 2 · 2 (h(T1),h(T2))+1 − 1

= 2h(T1·T2)+1 − 1,

R X ,Y

R = (x , y) : P(x , y), x ∈ X , y ∈ Y

P

R

∀(x1, y1) ∈ R ∀(x2, y2) ∈ R : (x1 = x2 ⇒ y1 = y2).

R = x ∈ X : ∃y : (x , y) ∈ R,R = y ∈ Y : ∃x : (x , y) ∈ R.

x ∈ R (x , y) ∈ RR x ∈ R

y ∈ R y = R(x) y = Rx

R = (x ,R(x)) : x ∈ R.

x 4→ R(x)P(x ,R(x)) x ∈ R

R R =: Ω ⊂ X R ⊂ Y RΩ Y

R : Ω → Y , R : x 4→ R(x).

R R( · )x

! Ω Y

! x 4→ R(x)

Γ(f ) = (x , f (x)) ∈ X × Y : x ∈ f

f

Ω ⊂ R Y = R

(x , y) ∈ R2

X ,Y ,Z Σ ⊂ Y f : X → Y g : Σ → Zf ⊂ Σ = g

g f : X → Z , g f : x 4→ g(f (x)).

f , g : X → Y fg

f + g : X → Y , (f + g)(x) := f (x) + g(x),

fg : X → Y , (fg)(x) := f (x) · g(x).

f : X → Y

f (x1) = f (x2) ⇒ x1 = x2 x1, x2 ∈ X

f = Y

f f −1 : Y → Xf −1(f (x)) = x x ∈ X f (f −1(y)) = y y ∈ Y

⌊ · ⌋ : Q → Z, ⌊x⌋ = z ∈ Z : z ≤ x

[ · ] ⌊ · ⌋

⌈ · ⌉ : Q → Z, ⌈x⌉ = z ∈ Z : z ≥ x

|x | =x x ≥ 0,

−x x < 0.

an n

(an)n∈N (an)∞n=0 (an)n = 0, 1, 2, 3, ...

a : n 4→ n2

(an) = (0, 1, 4, 9, 16, 25, ...),

(n2)n∈N an = n2

n ∈ N : n ≥ n0, n0 ∈ N

(an)n ∈ N : n ≥ n0, n0 ∈ N

(an) n → ∞ a

n→∞an = a :⇔ ∀

m∈N∃

N>0∀

n>N|an − a| < 1

m.

an → C n → ∞n→∞

an = a

n (an)

Sn =∑

k≤n

ak .

(Sn) Sn → ∞ (an) (an) S

∞∑

n=n0

an = S ,

(an) n ∈ N n ≥ n0

xan = xn n ∈ N (an) |x | < 1

∞∑

n=0

xn =1

1− x

(an) (bn) (cn)

an = (n/2 + 1)2, bn = n2, cn = n.

(an) (bn)an bn cn

n → ∞(an) (bn)

an =1

n2, bn =

1

n.

an bn

O

(an) (bn)

an = O(bn) n → ∞

(an) (bn) C ≥ 0 N ∈ N

|an| ≤ C |bn| n > N

n + n2 = O(n2) n → ∞1

n2 + n= O

(1

n

)n → ∞

1

n2= O

(1

n

)n → ∞

O

(an)an O(bn)

n2 + 1 = O(n2) n2 + 2 = O(n2) n2 + 1 = n2 + 2.

n2 + 1 = O(n2) n → ∞=

O an = O(bn)n → ∞ an bn

(an)(bn)

O

N > 0 n > Na0, a1, ... , aN−1 N

C n > Nn ∈ N

(an) (bn)

(an) (bn)C ≥ 0

n→∞

|an||bn|

= C ,

an = O(bn) n → ∞

C ≥ 0

n→∞

|an||bn|

= C .

∀m∈N

∃N>0

∀n∈N

n > N ⇒∣∣∣∣|an||bn|

− C

∣∣∣∣ <1

m.

m = 1 N > 0 n > N

∣∣|an|− C |bn|∣∣ < |bn|,

|an| ≤ (C + 1)|bn|.

an = O(bn) n → ∞

an = O(bn) ann bn n

an bn

1

n= O(n100)

anbn

O an = O(bn)an

(an) (bn)n → ∞

an = Ω(bn) :⇔ bn = O(an),

(an) (bn)

an = Θ(bn) :⇔ an = O(bn) an = Ω(bn),

(an) (bn)

an =n∑

k=1

k =n(n + 1)

2

an = Θ(n2) n ∈ N

an =n2

2+

n

2≤ 2

n2

2= n2

an =n2

2+

n

2≥ n2

2.

n

n2

max(a1, a2, ... , an : )max := a1

i = 2 nmax < aimax := ai

max

!

!

!

!

!

!

!

!

!

!

!

!

!

!

x (a1, ... , an)x ak = x

kx

(x : , a1, a2, ... , an : )i := 1

i ≤ n x = aii := i + 1i ≤ nlocation := i

location := 0

(a1, ... , an)

x

(2, 3, 7, 8, 10, 11, 13, 16)(2, 3, 7, 8) (10, 11, 13, 16) 7 > 8

(2, 3) (7, 8) 7 > 3(7) (8) 7 > 7

7

(x : , a1, a2, ... , an : )i := 1 i j := n j

i < j

m := ⌊(i + j)/2⌋x > am i := m + 1

j := m

x = ai location := ilocation := 0

location x x

(3, 2, 1, 4)3 > 2

(2, 3, 1, 4)3 > 1 (2, 1, 3, 4)

3 > 4

n − 1n i

(n − i)n − 1

(a1, a2, ... , an : n ≥ 2)i := 1 n − 1j := 1 n − iaj > aj+1 aj aj+1

(a1, ... , an)

jn

j − 1

(a1, a2, ... , an : n ≥ 2)j := 2 n

i := 1aj > ai

i := i + 1m := aj

k := 0 j − i − 1aj−k := aj−k−1

ai := m(a1, ... , an)

nn

(c1, c2, ... , cr :c1 > c2 > · · · > cr n

i := 1 rn ≥ ci

cin := n − ci

n n

nn

q′ qq = q′

q ≥ q′

q′ < q ≥ 25

!

!

n n − 1

2n − 1

Θ(n)

Θ(n)

Θ(n2)Θ(n2) +Θ(n) = Θ(n2)

i n − i n − 1

n−1∑

i=1

(n − i) = n(n − 1)− n(n − 1)

2=

n(n − 1)

2= Θ(n2)

Θ(1)Θ( n)Θ(n)Θ(n n) n nΘ(nb) b > 0Θ(bn) b > 1Θ(n!)

P = NP

n!

n! n ∈ N \ 0

nfactorial := 1

i := 1 nfactorial := i · factorial

n!

n! n ∈ N \ 0

nn = 1factorial(n) := 1

factorial(n) := n · factorial(n − 1)

100! 101! 200!

L = a1, ... , ann > 1

m := ⌊n/2⌋L1 := a1, ... , amL2 := am+1, ... , anL := merge(mergesort(L1),mergesort(L2))

L

L1, L2 :L :=

L1 L2

L1 L2L

LL

m nn +m − 1

m − 1 + n − 1

O(n2) n

n O(n n)

n = 2m

2m−1 k2k 2m−k m

2m 2m 2m−1

2m−1

kk = m,m − 1, ... , 2, 1 2k 2m−k

2k−1 2k−1

2m−k + 2m−k − 1 = 2m−k+1 − 1

m∑

k=1

2k−1(2m−k+1 − 1) =m∑

k=1

2m −m∑

k=1

2k−1

= m2m − (2m − 1)

= n n − n + 1.

Ω(n n) n

n! factorial(1) = 1 1!factorial(n − 1) = (n − 1)!

factorial(n) = n!

factorial(n) = n · factorial(n − 1) = n · (n − 1)! = n!,

!

!

Sp

q p S Sq S

pSq Sp q

y := 2z := x + y

p : x = 1 q : z = 3

S1 S2S = S1; S2

(pS1q ∧ qS2r

)⇒ pS1; S2r .

conditionS

S

! p condition q S

! p condition q

[((p∧condition)Sq

)∧((p∧¬condition) ⇒ q

)]⇒ p condition Sq

x > yy := x

p :q : y ≥ x

! p x > y S Sy = x y ≥ x

! p x > y y ≥ x

S y := x

conditionS1

S2

[((p ∧ condition)S1q

)∧((p ∧ ¬condition)S2q

)]

⇒ p condition S1 S2q

conditionS

Sp

(p ∧ condition)Spp p

p ¬condition

((p ∧ condition)Sp

)⇒ p condition S(¬condition ∧ p)

n!

i := 1factorial := 1

i < n

i := i + 1factorial := i · factorial

factorial = n! p : (factorial = i ! ∧ i ≤ n)p

pfactorial = i ! i < n

i i = i + 1i ≤ n

i · factorial = (i + 1) · factorial = (i + 1)! = i ! p(p ∧ i < n)Sp p

i = 1 ≤ n factorial = 1 = 1! = i !p p

i < n i ≥ n

(p ∧ (i ≥ n)

)⇔

(factorial = i ! ∧ i ≤ n ∧ i ≥ n

)

⇔(factorial = i ! ∧ i = n

)

⇔ factorial = n!

n − 1i = n

!

!

!

!

!

a | b a b a bc ∈ N a · c = b

a, b ∈ Z

a | b :⇔ ∃c∈Z

a · c = b.

a | b a b b a ab a $ b

a, b, c ∈ Za | b a | c a | (b + c)

a | b a | (bc)a | b b | c a | c

a | b b | c a | (mb + nc)a, b, c,m, n ∈ Z

a ∈ Z d ∈ Z 0 < d ≤ aq, r ∈ Z 0 ≤ r < d

a = dq + r .

r , r q, q

dq + r = a = dq + r .

d(q − q) = r − r d | (r − r) −d < r − r < dr − r = 0 dq = dq d = 0

q = q

S(a, d) := n ∈ N : n = a − dq, q ∈ Zq = −⌈a/d⌉

S(a, d)r = a − dq0 ≥ 0 r < d

a− d(q0 + 1) S(a, d)a = r + dq0 q0 ∈ Z

0 ≤ r < d

da q r

q = a d , r = a d .

a, b ∈ Z m ∈ Z+ a bm m | (a− b)

a ≡ b m.

a, b ∈ Z m ∈ Z+

a ≡ b m ⇔ a m = b m

m m ∈ Z+

Z

a ∼ b ⇔ a ≡ b m

m = 2

a, b ∈ Z m ∈ Z+

a ≡ b m ⇔ ∃k∈Z

a = b + km.

a ≡ b m ⇔ m | (a− b) ⇔ ∃k∈Z

a− b = km

⇔ ∃k∈Z

a = b + km

a, b, c , d ∈ Z m ∈ Z+ a ≡ b mc ≡ d m

a + c ≡ b + d m ac ≡ bd m.

s, t ∈ Z b = a+ sm d = c + tm

b + d = a + c +m(s + t) ⇔ a + c ≡ b + d m

bd = ac +m(at + cs + stm) ⇔ ac ≡ bd m.

a, b ∈ Z m ∈ Z+

(a + b) m =(a m + b m

)m

ab m =((a m)(b m)

)m.

h(k) = k m

k mm = 1000

hk1 k2 h(k1) = h(k2)

j ≥ 1

064212848037149212 107405723 k 111h(064212848) = 14 h(037149212) = 65 h(107405723) = 14

h(107405723) = 15

xn+1 = (axn + c) m,

x0 ac m

0 ≤ x0 < m, 2 ≤ a < m 0 ≤ c < m.

m = 9 a = 7 c = 4 x0 = 3

7, 8, 6, 1, 2, 0, 4, 5, 3, 7, 8, 6, 1, 2, 0, 4, 5, 3, ...

c = 0

m = 231 − 1 a = 75 = 16807 231 − 2

0 1 xn/m

A 4→ 0, B 4→ 1, ... Z 4→ 25.

k

k = 13

k = 3

12 4 4 19 24 14 20 8 13 19 7 4 15 0 17 10

15 7 7 22 1 17 23 11 16 22 10 7 18 3 20 13

k = 13 x + 13 26 = x − 13 26

p ∈ N \ 0, 1p 1 p p

p ∈ N \ 0, 1p

p p

pp

n n√n

n n = ab a, b ∈ N 1 < a ≤ ba >

√n b >

√n a · b > n a b√

nn

√n

101√101 101 101

2, 3, 5 7 101

p1, ... , pnQ = p1 · p2 · · · pn + 1.

QQ

p1, ... , pn Q pkpk | Q pk | p1 · p2 · · · pn

pk | (Q − p1 · p2 · · · pn)︸ ︷︷ ︸=1

,

θ(x)x ∈ N

x→∞

θ(x)

x/ x= 1,

e

ff (n) n ∈ N

p p = n2 + 1n ∈ N

n > 2

f (n) = n2 − n + 41 n ≤ 40 f (41)

n n2 + 1

16, 869, 987, 339, 975 · 2171,960 ± 1

N+ N \ 0

a, b ∈ N+ d ∈ N+ d | ad | b a b

(a, b)

(a, b) = 1 a b

(a, b)

(a, b) = d ∈ N+ : (d | a) ∧ (d | b)

d ≤ a, b

! (24, 36) = 1, 2, 3, 4, 6, 12 = 12

! (17, 22) = 1

a1, ... , an ⊂ N+

(ai , aj) = 1 i = j

a, b ∈ N+ p1, ... , pn

a = pa11 · pa22 · · · pann , b = pb11 · pb22 · · · pbnna1, ... , an, b1, ... , bn ∈ N

(a, b) = p (a1,b1)1 · p (a2,b2)

2 · · · p (an,bn)n .

p (a1,b1)1 · p (a2,b2)

2 · · · p (an,bn)n a b

ab

d | a da

a, b ∈ N+ d ∈ N+ a | db | d a b(a, b)

n ∈ N : (a | n) ∧ (b | n)ab ∈ N (a, b)

a, b ∈ N+ p1, ... , pn

a = pa11 · pa22 · · · pann , b = pb11 · pb22 · · · pbnn

a1, ... , an, b1, ... , bn ∈ N

(a, b) = p (a1,b1)1 · p (a2,b2)

2 · · · p (an,bn)n .

a, b ∈ N+

a · b = (a, b) · (a, b).

124 = 1 · 102 + 2 · 101 + 4 · 100

124 = 1 · 82 + 7 · 81 + 4 · 80.

b ∈ N \ 0, 1 n ∈ N+

a0, ... , ak < b ak = 0

n = akbk + ak−1b

k−1 + · · ·+ a1b + a0.

b nak n b

! b = 2! b = 8! b = 10! b = 16

b = 10 b

124 = (124)10 = (174)8.

aj

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , , , , .

(2 0B)16

2 · 164 + 10 · 163 + 14 · 162 + 0 · 16 + 11 = (175627)10.

28 − 1

b

! n = bq0 + a0 0 ≤ a0 < b

q0 = n b, a0 = n b.

a0 b n!

q1 = q0 b, a1 = q0 b

a1qk b = 0 k ∈ N

(12345)1012345 = 8 · 1543 + 1

1543 = 12345 8, 1 = 12345 8,

192 = 1543 8, 7 = 1543 8,

24 = 192 8, 0 = 192 8,

3 = 24 8, 0 = 24 8,

0 = 3 8, 3 = 3 8.

(12345)10 = (30071)8

b

b (n : )q := nk := 0

q = 0

ak := q bq := ⌊q/b⌋k := k + 1

b n (ak−1 ... a1a0)b

(1101 1001 1100 0000)2 = ( 9 0)16.

a+ b

a = (an−1an−2 ... a0)2 b = (bn−1bn−2 ... b0)2

a0 + b0 = c0 · 2 + s0

s0 a + b s0 = (a + b)0

a1 + b1 + c0 = c1 · 2 + s1,

(a + b)1 = s1an−1 + bn−1 + cn−2 = cn−1s + sn−1 sn = cn−1

a+ b

(a, bs : )a = (an−1an−2 ... a0)2 b = (bn−1bn−2 ... b0)2c := 0

j := 0 n − 1

d := ⌊(aj + bj + c)/2⌋sj := aj + bj + c − 2dc := d

sn := c (snsn−1 ... s1s0)2

O(n)

(a, b : )a = (an−1an−2 ... a0)2 b = (bn−1bn−2 ... b0)2

j := 0 n − 1

bj = 1 cj = a jcj = 0

c0, ... , cn−1 p0 := 1

j := 0 n − 1p := p + cj

p ab

O(n2)

O(n 3) = O(n1.584)

(a : , d : )q := 0r := |a|

r ≥ d

r := r − dq := q + 1

a < 0 r > 0

r := d − rq := −(q + 1)

q = a d r = a d

O(q a)O( a · d)

bn mn ∈ N b ∈ Z m ∈ N \ 0, 1 bn

bn

n bn

bn = bak−12k−1+···+a1·2+a0 =k−1∏

j=0

baj2j, a0, ... , ak−1 ∈ 0, 1.

b2j

b2j+1

= (b2j)2

m

(b : , n = (ak−1 ... a0)2,m : )

x := 1power := b m

i := 0 k − 1

ai = 1 x := (x · power) mpower := (power · power) m

x bn m

O(( m)2 n)bn m

(a, b)a b

a, b, q, r ∈ Z a = bq + r(a, b) = (b, r)

a bb r

d | a d | b d | (a − bq)d | r

d | b d | r d | (r + bq) d | a

(a, b : )x := ay := b

y = 0

r := x yx := yy := r

(a, b) x

a, b ∈ Z+ a ≥ b(a, b)

b

a = r0 b = r1

r0 = r1q1 + r2, 0 ≤ r2 < r1,

r1 = r2q2 + r3, 0 ≤ r3 < r2,

rn−2 = rn−1qn−1 + rn 0 ≤ rn < rn−1

rn−1 = rnqn,

rn = (a, b) q1, ... qn−1 ≥ 1 qn ≥ 2rn−1 > rn

(fn)n∈N(1, 1, 2, 3, 5, ...)

rn ≥ 1, rn ≥ f1,

rn−1 ≥ 2rn ≥ 2, rn−1 ≥ f2,

rn−2 ≥ rn−1 + rn, rn−2 ≥ f2 + f1 = f3,

r2 ≥ r3 + r4 r2 ≥ fn−1 + fn−2 = fn,

b = r1 ≥ r2 + r3, b ≥ fn + fn−1 = fn+1.

nb ≥ fn+1 fn ≥ αn−1 n > 2α = (1 +

√5)/2

10 α > 1/5

10 b > (n − 1) 10 α >n − 1

5.

n − 1 < 5 10 b b kb < 10k n < 5k + 1 n n ≤ 5k

O( b)(a, b) a ≥ b

(a, b) ab

a, b ∈ Z+ s, t ∈ Z(a, b) = sa+ tb

a, b, c ∈ Z+ (a, b) = 1 a | bc a | c

s, t ∈ Z sa+ tb = 1

sac = c − tbc .

a | c

m ∈ Z+ a, b, c ∈ Z ac ≡ bc m(c ,m) = 1 a ≡ b m

ac ≡ bc m ⇒ m | c(a− b) ⇒(c,m)=1

m | a − b

⇒ a ≡ b m.

p ∈ N \ 0, 1 a1, ... , an ∈ Zp | a1a2 ... an p | ai ai

p1, ... , ps q1, ... , qt

n = p1p2 ... ps = q1q2 ... qt .

pi1 ... piu = qj1 ... qjv

u, v ∈ Z+

pi1 | qjk k

a, b ∈ Z m ∈ Z+

ax ≡ b m

x ∈ Z xa aa ≡ 1 m.

a m

a ∈ Z+ m ∈ N \ 0, 1 (a,m) = 1a m m

s, t ∈ Z sa+ tm = 1sa+ tm ≡ 1 m

sa ≡ 1 m

s a m

a m a ma a m

3x ≡ 4 7(3, 7) = 1

7 + (−2) · 3 = 1 −23x ≡ 4 7 −2

−6x ≡ −8 7 ⇔ x ≡ 6 7,

−6 ≡ 1 7 −8 ≡ 6 7

孫子 / 孙子孫子算經

今有物不知其数，三三数之剩二，五五数之剩三，七七数之剩二，问物几何？

x ≡ 2 3,

x ≡ 3 5,

x ≡ 2 7.

m1, ... ,mn ∈ Z+

a1, ... , an ∈ Zx ≡ a1 m1,

x ≡ a2 m2,

x ≡ an mn.

m = m1m2 ...mn

Mk :=m

mk=∏

i =k

mi .

(mk ,Mk) = 1 ykMk mk

Mkyk ≡ 1 mk .

x =n∑

k=1

akMkyk

Mj ≡ 0 mi i = j

m = m1m2m3 = 3 · 5 · 7 = 105

M1 = m/3 = 35, M2 = m/5 = 21, M3 = m/7 = 15.

Mk mk

y1 = 2, y2 = 1, y3 = 1.

x = 2 · 35 · 2 + 3 · 21 · 1 + 2 · 15 · 1 = 233 = 23 105.

m1, ... ,mn ∈ N \ 0, 1 m =∏n

i=1mi

a ∈ Z 0 ≤ a < ma n

(a m1, ... , a mn).

m1 = 3 m2 = 43 · 4 = 12

n(n 3, n 4)

n(n 3, n 4)

99 · 98 · 97 · 96 = 89 403 930

123684 + 413456(n 99, n 98, n 97, n 95)

(33, 8, 9, 89) + (32, 92, 42, 16)

= (65 99, 100 98, 51 97, 105 95)

= (65, 2, 51, 10)

x ≡ 65 99, x ≡ 2 98, x ≡ 52 97, x ≡ 10 95

x = 537 140

n

2n−1 ≡ 1 n.

p a ∈ Zp

ap−1 ≡ 1 p.

a ∈ Z p

ap ≡ a p.

p qM

A 4→ 01,B 4→ 02, ... ,Z 4→ 26

e ∈ Z+

(p − 1)(q − 1) ee $ (p − 1)(q − 1)

p q Mp q

M pq

M C

C = Me n,

n = pq p q

C Md ∈ Z+ e (p − 1)(q − 1) d

∃k∈Z

de = 1 + k(p − 1)(q − 1)

Cd ≡ (Me)d = Mde = M1+k(p−1)(q−1) n

Cd ≡ M · (Mp−1)k(q−1) p

Cd ≡ M · (Mq−1)k(p−1) q.

Mp−1 ≡ 1 p Mq−1 ≡ 1 q

Cd ≡ M p Cd ≡ M q.

Cd ≡ M n.

n = pq en d

n e n e(n, e)

(n, d)(n, e)

d n e dp q d de ≡ 1 (p − 1)(q − 1)

p q nn

p qd e n

M

MS ⊂ N

M Mn ϕ : M → 1, ... , n

M n M = n

∅ ∅ := 0

M M MM → N M

M = ℵ0 M

M M

M = 00, 01, 10, 11M = 4

ϕ : M → 1, 2, 3, 4

00 4→ 1, 01 4→ 2, 10 4→ 3, 11 4→ 4.

MM ϕ

2N = n ∈ N : n = 2k , k ∈ Nϕ : n 4→ n/2 + 1

0 1 M = x ∈ R : 0 ≤ x ≤ 1

11

12

13

14

15

16 · · ·

21

22

23

24

25

26

31

32

33

34

35

36

41

42

43

44

45

46

51

52

53

54

55

56

61

62

63

64

65

66

· · ·

· · ·

· · ·

· · ·

· · ·

M M = nM P(M) 2n

M M = a1, ... , anP(M) n

S ∈ P(M) S ⊂ M Sn i ai ∈ S ai /∈ S

P(M) n

n2n − 1

P(M) → 1, ... , 2n

M,N M = m N = nM N M × N m · n

M = a1, ... , am N = b1, ... , bn

ϕ : M × N → 1, ... ,m · n, (ai , bj) 4→ (i − 1)n + j .

ϕ−1 : 1, ... ,m · n → M × N, x 4→ (a1+((x−1) n), b1+((x−1) n)),

ϕ

M1, ... ,Mn

Mk = mk

(M1 ×M2 × · · ·×Mn) =n∏

i=1

mi .

( , ).

24 · 32 = 768

M,N M = mN = n M ∩ N = ∅ M N M ∪ N

m + n

M = a1, ... , am N = b1, ... , bn

ϕ : M ∪ N → 1, ... ,m + n, c 4→i c = ai ,

m + j c = bj .

M1, ... ,Mn

Mk = mk

( n⋃

i=1

Mi

)=

n∑

i=1

mi .

M,N M = m N = n(M ∩ N) = k M N M ∪ N

m + n − k

M = a1, ... , am N = a1, ... , ak , b1, ... , bn−kk ≥ 1

M ∪ N = a1, ... , am, b1, ... , bn−k

(M ∪ N) = m + n − k

(M ∪ N) = M + N − (M ∩ N).

φ : (1, x1, x2, ... , x7) 4→ (x1, x2, ... , x7).

0, 1727

26

25

27 + 26 − 25 = 160

!

!

!

!

11001010011110000010111010111001.

!

!

27 − 1 = 255 224 − 2 = 16 777 214255 · 16 777 214 = 2 130 706 178

1 073 709056532 676 608 3 737 091 842

M f : M → Mf

nn1, ... , n

1, ... , n

f : 1, ... , n → 1, ... , n.

ff

M,N f : M → Nf M = N

M = a1, ... , am M = m N = n f

n = N = f = f (a1), ... , f (am) ≤ m

f f (a1), ... , f (am)f (a1), ... , f (am) = m m = n

m = n f (a1), ... , f (am) = m f (a1), ... , f (am)f

M f : M → M f : M → f( f ) = M

f ⊂ M f = M f : M → M

f : M → M f

M,N M > Nf : M → N f

f : N → N n 4→ n+1

n ∈ Z+ n

n ∈ Z+ n + 1

M =1, 11, 111, ... , 11 ... 1︸ ︷︷ ︸

n

, 11 ... 11︸ ︷︷ ︸n + 1

.

f : M → 0, ... , n − 1 f (x) = x nf M

n

n n

m kM M = m

N N = n

X ,Y f : X → YV ⊂ Y

f −1(V ) := x ∈ X : f (x) ∈ V .

V V = vf −1(v) f −1(v)f : R2 → R f (x , y) = x2 + y2

f −1(r2) = (x , y) ∈ R2 : f (x , y) = r2

r

M,NM = m N = n

f : M → N n0 ∈ N(f −1(n0)) ≥ ⌈m/n⌉

y ∈ f

(f −1(y)) < ⌈m/n⌉.

(f −1(y)) < m/n (f −1(y)) ∈ Nf −1(y) ∩ f −1(y ′) = ∅ y = y ′

M =( ⋃

y∈ f

f −1(y))=∑

y∈ f

(f −1(y)) < nm

n= m,

AA

AB ,C ,D A

A B ,C ,D

B ,C ,D A

m, n ∈ N \ 0, 1 R(m, n)m

n R(3, 3) ≤ 6R(3, 3) > 5 R(3, 3) = 6

R(m, n) = R(n,m) R(2, n) = n3 ≤ m ≤ n

R(4, 4) = 1843 ≤ R(5, 5) ≤ 49

x1, ... , xn nxk = xj j = k

π : x1, ... , xn → x1, ... , xn

nx1, ... , xn 1, ... , n

f (x , f (x))x

π (1,π(1)), ... , (n,π(n))

π =

(1 2 ... n

π(1) π(2) ... π(n)

)

1, 2, 3

π0 =

(1 2 31 2 3

)

(1 2 33 1 2

),

(1 2 32 3 1

),

(1 2 33 2 1

),

(1 2 32 1 3

),

(1 2 31 3 2

).

n (π(1), ... ,π(n))(1 2 33 1 2

)(3, 1, 2).

(3, 1, 2)

1, 2, 3

n! π 1, ... , n

π(k) k = 1, ... , nπ(1) n 1, ... , n

1, ... , n \ π(1) nπ(1) n− 1 π(2)

n − k + 1 π(k)

n · (n − 1) · · · (n − n + 1) = n!

(π(1), ... ,π(n))

(π(1), ... ,π(n))(1, ... , n)

r ≤ n 1, ... , n

n, r ∈ N \ 0 r ≤ n

π : 1, ... , r → 1, ... , n

r r 1, ... , n

π(

1 2 ... rπ(1) π(2) ... π(r)

)(π(1), ... ,π(r)),

π(k) ∈ 1, ... , n k = 1, ... , r

n = 3 r = 21, 2, 3

(1, 2), (2, 1), (1, 3), (3, 1), (2, 3), (3, 2).

n · (n − 1) · · · (n − r + 1) =n!

(n − r)!

r 1, ... , n

r nr

n = 3 r = 21, 2, 3

1, 2, 1, 3, 2, 3.

r 1, ... , nA ⊂ 1, ... , n A = r

(n

r

)=

n!

r !(n − r)!

r 1, ... , n

r 1, ... , n

(π(1), ... ,π(r)) −→ π(1), ... ,π(r).

(π(1), ... ,π(r)) r1, ... , n (π(1), ... ,π(r))

(π(1), ... ,π(r))r ! (π(1), ... ,π(r))

r 1, ... , n r !

r 1, ... , n

r1, ... , n

1, ... , nr

1, ... , n A1

A2

N := 1, ... , n

N !

A1! A2!=

n!

n1!n2!=

n!

r !(n − r)!=

(n

r

)

N A1 A2

! A1 ⊂ N A1 = n1 = r

! A2 = N \ A1 A2 = n2 = n − r

N := 1, ... , n

n!

n1!n2! ... nk !

N k A1, ... ,Ak

! N =k⋃

i=1Ai Ai ∩ Aj = ∅ i = j

! Ai = ni i = 1, ... , k

n1 A1

(n

n1

)=

n!

n1!(n − n1)!.

n2 n − n1A2

(n − n1n2

)=

(n − n1)!

n2!(n − n1 − n2)!.

n1 A1 n2A2

(n

n1

)(n − n1n2

)=

n!

n1!(n − n1)!

(n − n1)!

n2!(n − n1 − n2)!=

n!

n1!n2!(n − n1 − n2)!

(n

n1

) k−1∏

i=1

(n −

∑ij=1 nj

ni+1

)=

n!

n1!(n − n1)!

k−1∏

i=1

(n −

∑ij=1 nj

)!

ni+1!(n −

∑i+1j=1 nj

)!

=n!

n1!(n − n1)!

∏k−1i=1

(n −

∑ij=1 nj

)!

∏k−1i=1 ni+1!

∏k−1i=1

(n −

∑i+1j=1 nj

)!

=n!

∏ki=1 ni !

∏k−1i=2

(n −

∑ij=1 nj

)!

∏k−2i=1

(n −

∑i+1j=1 nj

)!

=n!

∏ki=1 ni !

A1, ... ,Ak Ai = ni

r rn ≥ r r r

rr

r

r1, ... , n

r 1, ... , nrnr nr r

a, ba, b, b = a, b

b

2a, 4b, ca, b, c 2, 4, 1

rr r

r r nr ≤ n

S = a, b, c

(b, a, b, c , a, b, b)

S

2a, 4b, c

S

nr = 37 = 2187

2a, 4b, c = a, a, b, b, b, b, c.

S = a, b, c

x , x , |, x , x , x , x , |, x

2a, 4b, c xa | x b

|xxx |xxxx 3b, 4c.

xa, b, c x

x xS = a, b, c(92

)= 36

r , n ∈ Z+

(r + n − 1

n − 1

)=

(r + n − 1

r

)

r n rn

x1 + x2 + x3 = 11, x1, x2, x3 ∈ N.

x1 x2 x3

(11 + 3− 1

11

)= 78

3! = 6 1, 2, 3

a = (a1, a2, ... , an) b = (b1, b2, ... , bn)n a b a ≺ b a1 < b1

k > 1 ai = bi 1 ≤ i ≤ k − 1ak < bk

(2, 3, 4, 1, 5) (2, 3, 5, 1, 4)

a = (a1, ... , an−2, an−1, an)b a ≺ b

! an−1 < an (a1, ... , an−2, an, an−1) a

! an−1 > an an−2

! an−2 < an−1

(a1, ... , (an−1, an), (an−2, x), (an−2, x))

x = (an−1, an)! an−2 > an−1 an−3

! j

aj < aj+1 aj+1 > aj+2 > · · · > an.

! aj

aj+1, ... , an : ak > aj , k = j + 1, ... , n.

j + 1n

S = 1, 2, 3, ... , n

(1, 2, ... , n − 1, n)

(n, n − 1, ... , 2, 1).

(1, 2, 3)

(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1).

r n ≥ rn

r S = (a1, a2, ... , an)S C

n r 1ak ∈ C k

S = (1, 2, 3, 4)S

1, 2, 3 1110,

2, 3, 4 0111,

1, 2, 4 1101,

1, 3, 4 1011.

2, 3, 41, 2, 3

1, 2, ... , n

r a1, ... , ar iak = n − r + k k ≤ i ai ai + 1

j = i + 1, ... , r aj ai + j − i + 1

1, 2, 3, 4, 5, 61, 2, 5, 6

a1 = 1 = n − r + 1 = 3, a2 = 2 = n − r + 2 = 4, a3 = 5 = n − r + 3 = 5

i = 2 a2 3 a3 a4 51, 3, 4, 5 1, 2, 5, 6 110011 1, 3, 4, 5

101110

16

A

AP [A]

P [A] =A

P [ ] =1

2= 0.5

S = 2, 3, ... , 12S = 3

S = 3

6× 6 = 36

P [3] =2

36=

1

18= 0.056

17

Wang Yifei

S

S

S

S = N

A S

A1,A2 A1 ∩ A2 = ∅

S = N10

(1, 2, 3, 2, 3, 3, 1, 1, 4, 4) ∈ N10

A ⊂ S A

(1, 2, 3, 2, 3, 3, 1, 1, 4, 4) ∈ A(1, 2, 3, 2, 3, 3, 1, 1, 3, 4) /∈ A

410 = 1048576S = N10

A10

10!

5!3!1!1!= 5040

A

5040

1048576≈ 0.00481 ≈ 0.5 .

20

S P(S)S P : P(S) → R A 4→ P [A]

S

P ≥ 0

P [S ] = 1

Ak ⊂ P(A) Ai ∩ Ak = ∅ i = k

P[⋃

Ak

]=∑

P [Ak ].

A ∈ P(S)

A = SA = S \ AA

P [S ] = 1, P [∅] = 0, P [A ] = 1− P [A],

A1,A2 ∈ P(S) A1 ∩ A2

A1 A2

A1 ∪ A2 A1 A2

P [A1 ∪ A2] = P [A1] + P [A2]− P [A1 ∩ A2].

P [A1 ∪ A2] ≤ P [A1] + P [A2],

A1,A2 ∈ P(S)

A1 ⊂ A2 P [A1] ≤ P [A2]

S = (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), ... , (4, 3), (4, 4),

P [(i , j)] = 1/16 i , j = 1, 2, 3, 4S

A ⊂ S

A1

A2

A1 = (1, 1), (1, 2), (2, 2), A2 = (1, 1), (2, 2), (3, 3), (4, 4).

P [A1] = P [(1, 1)] + P [(1, 2)] + P [(2, 1)] = 3

16,

P [A2] =4

16=

1

4.

P [A1 ∩ A2] = P [1, 1] = 1

16,

P [A2] = 1− P [A2] =3

4.

! A

! A

! A B

! A B

B A

P [B | A] B A

P [A1 | A2]

A2

A1 ∩ A2 = (1, 1)

A2

A1 ∩ A2 A1 A2

P [A1 | A2] =(A1 ∩ A2)

A2=

P [A1 ∩ A2]

P [A2].

A,B ⊂ S P [A] = 0

P [B | A] := P [A ∩ B ]

P [A].

A,B A B

P [A ∩ B ] = P [A]P [B ].

P [A | B ] = P [A] P [B ] = 0

P [B | A] = P [B ] P [A] = 0

m1/n

n

npn n

m ≥ n p1 = 1

P [ 2] =m − 1

m,

m − 1 m

k

P [ k ] =m − k

m,

pn = P [ 1, ... , n]

=n∏

k=1

P [ k ]

=m

m· m − 1

m· m − 2

m· · · m − n + 1

m=

m!

(m − n)!mn.

1− pn1− pn > 1/2 n ≥ 1.177

√m

A1 A2

B1 B2

A1 A2

(B1,A1) (B2,A1) (B1,A2) (B2,A2)P[B1 ∩ A1] P[B2 ∩ A1] P[B1 ∩ A2] P[B2 ∩ A2]

P[A1] P[A2]

P[B1 | A1] P[B2 | A1] P[B1 | A2] P[B2 | A2]

P[ ] = 0.36 P[ ] = 0.24 P[ ] = 0.24 P[ ] = 0.16

!

!

!

P[ ∩ ] = 0.17

P[ ] = 0.43

P[ | ]

P [ | ] =P [ ∩ ]

P [ ]=

0.17

0.43≈ 0.40

O(2n)

n ∈ N s ∈ N tn − 1 = 2st n b

bt ≡ 1 n b2j t ≡ −1 n

j 0 ≤ j ≤ s − 1

! n 1 < b < n n b

! n n/4 b1 < b < n n b

nk b

nn n

nn

k

pk =1

4k.

k = 30 pk < 10−18

SS

S

S

k ∈ N \ 0, 1 R(k , k) ≥ 2k/2 R

k = 2 k = 3R(2, 2) = 2 = 22/2 R(3, 3) = 6 > 23/2

n < 2k/2

k

n

(nk

)k

S1, S2, ... , S(nk)Ei k Si

kS

P [ k ] = P[(

nk)⋃

i=1

Ei

].

(k2

)= k(k − 1)/2 Si

1/2 Si2−k(k−1)/2

2−k(k−1)/2

P [Ei ] = 2 · 2−k(k−1)/2.(nk

)≤ nk/2k−1

P[(

nk)⋃

i=1

Ei

]≤

(nk)∑

i=1

P [Ei ] ≤ 21−k(k−1)/2 nk

2k−1= 22−k/2 nk

2k2/2.

n < 2k/2 k ≥ 4

P[(

nk)⋃

i=1

Ei

]< 1.

P [ k n ] < 1.

P [ k n ]

=k

n

1 n < 2k/2 k ≥ 4n k

R(k , k) ≥ 2k/2

(an)

an = f (a0, ... , an−1), n ≥ k

k ∈ N (an)a0, ... , ak−1

f0 = 1 f1 = 1f3 = 2

f4 = 3

fn = fn−1 + fn−2, n ≥ 2, f0 = 0, f1 = 1.

n

!!

Hn

nHn−1 n − 1

Hn−1

Hn = 2Hn−1 + 1, n ≥ 2, H1 = 1.

(Hn)

1, 3, 7, 15, 31, ...

Hn = 2n − 1H1 = 21 − 1 = 1 Hn−1 = 2n−1 − 1

Hn = 2Hn−1 + 1 = 2 · (2n−1 − 1) + 1 = 2n − 1.

!

!

!

n n

n n n

↑ ↓

↑↑↓↓↑↓

↑↓↓↑↓↑

w

↑, ↓∅ w

(↓) = (↑) = n

k w (↑) ≥ (↓)k = 1, ... , 2n

2n nCn

C1 = 1 Cn

! ∅! w1 w2 ↑w1 ↓w2

↑w1 ↓w2

w1,w2

w l(w) = 2n↑w1 ↓w2 l(w1) + l(w2) = 2(n− 1)

2nw1 w2 2(n − 1)

Cn =n−1∑

k=0

CkCn−1−k , n ≥ 2.

! n + 1

! 2n

! n + 2

! n

! n

a0, ... , ak−1 ∈ R(an) an

(an)n a0, ... , ak−1

n ≥ k ak , ... , an−1

n k , ... , n − 1

an := f (a0, ... , an−1)

an

k ∈ N

an = c1an−1 + · · ·+ ckan−k + F (n) n ≥ k

c1, ... , ck ∈ R ck = 0 F : N → R F (n) = 0n

an = c1an−1 + · · ·+ ckan−k .

an = rn

r ∈ R

rk − c1rk−1 − · · ·− ck−1r − ck = 0,

≥ 3

c1, c2 ∈ R r2 − c1r − c2 = 0r1 r2 (an)

an = c1an−1 + c2an−2

an = α1 · rn1 + α2 · rn2 , α1,α2 ∈ R, n ∈ N.

an = c1an−1 + c2an−2

an = c1an−1 + c2an−2

an = α1rn1 + α2rn2 r1, r2 r2 − c1r − c2 = 0

c1an−1 + c2an−2 = c1(α1rn−11 + α2r

n−12 ) + c2(α1r

n−21 + α2r

n−22 )

= α1rn−21 (c1r1 + c2) + α2r

n−22 (c1r2 + c2)

= α1rn−21 r21 + α2r

n−22 r22

= an

(an)a0 a1

α1 α2

a0 = α1 + α2, a1 = α1r1 + α2r2.

an = α1rn1 + α2rn2 n ∈ N

α1 =a1 − a0r2r1 − r2

, α2 =a0r1 − a1r1 − r2

an =a1 − a0r2r1 − r2

rn1 +a0r1 − a1r1 − r2

rn2 .

f0 = 0 f1 = 1fn = fn−1 + fn−2

r1,2 =1±

√5

2,

fn =1√5

(1 +

√5

2

)n

− 1√5

(1−

√5

2

)n

c1, c2 ∈ R c2 = 0 r2 − c1r − c2 = 0r0 (an)

an = c1an−1 + c2an−2

an = α1 · rn0 + α2 · nrn0 , α1,α2 ∈ R, n ∈ N.

an = 6an−1 − 9an−2

a0 = 1 a1 = 6r = 3

an = α13n + α2n3

n.

α1,α2 a0 = α1 = 1 a1 = 3α1 + 3α2 = 6

an = (n + 1)3n.

k

k

c1, c2, ... , ck ∈ R

rk − c1rk−1 − · · ·− ck−1r − ck = 0

t r1, ... , rt m1, ... ,mt

(an)an = c1an−1 + c2an−2 + · · ·+ ckan−k

an = (α1,0 + α1,1n + · · ·+ αn,m1−1nm1−1) · rn1

+ (α2,0 + α2,1n + · · ·+ α2,m2−1nm2−1) · rn2

+ · · ·+ (αt,0 + αt,1n + · · ·+ αt,mt−1nmt−1) · rnt , n ∈ N.

αi ,j ∈ R 1 ≤ i ≤ t 0 ≤ j ≤ mi − 1

k

an = −3an−1 − 3an−2 − an−3, a0 = 1, a1 = −2, a2 = −1.

r3 + 3r2 + 3r + 1 = (r + 1)3 = 0

r = −1 3

an = α1,0(−1)n + α1,1n(−1)n + α1,2n2(−1)n.

a0 = α1,0 = 1,

a1 = −α1,0 − α1,1 − α1,2 = −2,

a2 = α1,0 + 2α1,1 + 4α1,2 = −1

an = (1 + 3n − 2n2)(−1)n.

an = c1an−1 + · · ·+ ckan−k + F (n) n ≥ k

c1, ... , ck ∈ R

(an )(an) = (an + an ) (an )

F (n) = 0 n

an = 3an−1 + 2na1 = 3 an = 3an−1

an = α · 3nF (n) = 2n

an = cn + d c , d ∈ R

cn + d = 3(c(n − 1) + d) + 2n ⇒ (2 + 2c)n + (2d − 3c) = 0,

c = −1 d = −3/2

an = α · 3n − n − 3/2

a1 = 3α = 11/6

an =11

23n−1 − n − 3

2.

F (n)

F (n) = snt∑

j=0

bjnj , s, b0, ... , bt ∈ R.

! sp0, ... , pt ∈ R

an = snt∑

j=0

pjnj .

! s mp0, ... , pt ∈ R

an = nmsnt∑

j=0

pjnj .

an = 6an−1 − 9an−2 + F (n),

F (n) = 3n

F (n) = n3n

F (n) = n22n

F (n) = (n2 + 1)3n

an = 6an−1 − 9an−2

r = 3 2

an = p0n23n

an = (p0 + p1n)n23n

an = (p0 + p1n + p2n2)2n

an = (p0 + p1n + p2n2)n23n

nf (n) = f (n/2) + 2.

n/2

n = 2k

f (n) = 2f (n/2) + 2.

a b2n

a = (a2n−1 ... a1a0)2, b = (b2n−1 ... b1b0)2.

A0 = (an−1 ... a0)2, A1 = (a2n−1 ... an)2,

B0 = (bn−1 ... b0)2, B1 = (b2n−1 ... bn)2

a = 2nA1 + A0, b = 2nB1 + B0.

a · b = (22n + 2n)A1B1 + 2n(A1 − A0)(B0 − B1) + (2n + 1)A0B0,

2nn

2n

f (2n) = 3f (n) + Cn

C ∈ N

n n/2n

M(n)

M(n) = 2M(n/2) + n.

f

f (n) = af (n/b) + c , a ≥ 1, b ∈ N \ 0, 1, c > 0

n b

f (n) =

O(n b a) a > 1,

O( n) a = 1

a > 1 n = bk k ∈ Z+

f (n) = C1n b a + C2

C1 = f (1) +c

a − 1, C2 = − c

a − 1.

f (n) = af (n/b) + g(n) g

f (n) = af(nb

)+ g(n) = a

(af( n

b2

)+ g

(nb

))+ g(n)

= a2f( n

b2

)+ ag

(nb

)+ g(n)

= ak f( n

bk

)+

k−1∑

j=0

ajg( n

bj

)

k ∈ N n = bk k ∈ N

f (n) = ak f (1) +k−1∑

j=0

ajg( n

bj

).

a = 1 n = bk k ∈ N

f (n) = f (1) + c · k = f (1) + c · b n = O( n).

bk < n < bk+1 k ∈ N f

f (n) ≤ f (bk+1) = f (1) + c(k + 1) = f (1) + c + c · b n = O( n).

a = 1

a > 1 n = bk

f (n) = ak f (1) + ck−1∑

j=0

aj = ak f (1) + cak − 1

a − 1

= ak(f (1) +

c

a− 1

)− c

a− 1

= C1n b a + C2,

ak = a b n = n b a

bk < n < bk+1 k ∈ N f

f (n) ≤ f (bk+1) = ak(af (1) +

ac

a − 1

)− c

a − 1= O(n b a).

f (n) = f (n/2) + 2

f (n) = O( n).

f (n) = 2f (n/2) + 2

f (n) = O(n 2 2) = O(n).

Θ

f

f (n) = af (n/b) + cnd , a ≥ 1, b ∈ N \ 0, 1, c > 0, d ≥ 0

n = bk

f (n) =

⎧⎪⎨

⎪⎩

O(nd) a < bd ,

O(nd n) a = bd

O(n b a) a > bd

M(n) M(n) = 2M(n/2) + n

M(n) = O(n n).

f (n) = 3f (n/2) + Cn

f (n) = O(n 2 3) = O(n1.585).

O(n2)

O(n n)O(n2)

(an)

G (x) =∞∑

n=0

anxn = a0 + a1x + a2x

2 + ...

ϱ > 0 G(an) (a1, ... , ak)

an = 0 n > k

1

1− x=

∞∑

n=0

xn,

G (x) = 1/(1− x) (an)an = 1 n ∈ N

a = 0 G (x) = 1/(1− ax)(1, a, a2, a3, ...)

(1 + x)n =n∑

k=0

(n

k

)xk

G (x) = (1 + x)n

n + 1 (n

0

),

(n

1

), ... ,

(n

n

).

∑ak

∑bk∑

ck

ck :=∑

i+j=k

aibj

∑ck =

(∑ak)(∑

bk)

(ak) ∗ (bk) := (ck), ck :=∑

i+j=k

aibj ,

(ak) (bk)

(x) =∞∑

n=0

xn

n!.

( x)( y) =( ∞∑

n=0

xn

n!

)( ∞∑

m=0

ym

m!

)=

∞∑

n=0

((xk

k !

)∗(yk

k !

))

n

=∞∑

n=0

∑

l+m=n

1

l !m!x lym =

∞∑

n=0

1

n!

∑

l+m=n

n!

l !m!x lym

=∞∑

n=0

1

n!(x + y)n

= (x + y)

( x)( y) = (x + y).

x = ex

(an) an = 1/n!∑

akxk∑

bkxk

|x | < ϱ

( ∞∑

l=0

alxl)( ∞∑

m=0

bmxm)=

∞∑

n=0

cnxn

cn =∑

i+j=n

aibj

n ∈ Z+ k = 0, ... , n(n

0

)(n

k

)+

(n

1

)(n

k − 1

)+ · · ·+

(n

k

)(n

0

)=

(2n

k

).

k∑

i=0

(n

i

)(n

k − i

)=∑

i+j=k

(n

i

)(n

j

)=

(2n

k

)

(2nk

)k = 0, ... , 2n

G (x) = (1 + x)2n =2n∑

k=0

(2n

k

)xk .

k = 0, ... , nxk G (x)

(1 + x)2n =((1 + x)n

)2

=( n∑

l=0

(n

l

)x l)( n∑

m=0

(n

m

)xm)

=2n∑

k=0

( ∑

i+j=k

(n

i

)(n

j

))xk .

朱世杰k = n

n∑

k=0

(n

k

)2

=

(2n

n

),

(α

0

):= 1,

(α

j

):=

α(α− 1) ... (α− j + 1)

j !, j ∈ N, α ∈ R.

−1 < x < 1 α ∈ R

(1 + x)α =∞∑

n=0

(α

n

)xn

α = 1/2

√1 + x =

∞∑

n=0

(1/2

n

)xn.

n = 0(1/2

0

)x0 = 1.

n ≥ 1(1/2

n

)=

1/2(1/2− 1)(1/2− 2) ... (1/2− (n − 1))

n!

= (−1)n−1 1/2(1− 1/2)(2− 1/2) ... (n − 1− 1/2)

n!

=(−1)n−1

2n1(2− 1)(4− 1) ... (2(n − 1)− 1)

n!

(1/2

n

)=

(−1)n−1

2n1(2− 1)(4− 1) ... (2(n − 1)− 1)

n!

=(−1)n−1

2n1 · 1 · 3 · 5 · · · (2n − 3)

n!

=(−1)n−1

2n1

2n−1(n − 1)!

1 · 2 · 3 · 4 · 5 · · · (2n − 2)

n!

= −2(−1)n

4n1

n

(2n − 2)!

(n − 1)!(n − 1)!= −2

(−1)n

4n1

n

(2n − 2

n − 1

).

√1 + x = 1− 2

∞∑

n=1

(−1)n

4n

(2n − 2

n − 1

)xn

n

|x | < 1

Cn

Cn+1 =n∑

k=0

CkCn−k =∑

j+k=n

CkCj , n ∈ N.

C0 = 1

c(x) =∞∑

n=0

Cnxn.

c(x)2 =( ∞∑

j=0

Cjxj)( ∞∑

k=0

Ckxk)=

∞∑

n=0

( ∑

j+k=n

CkCj

)xn

=∞∑

n=0

Cn+1xn =

1

x(c(x)− 1)

c(x) = 1 + xc(x)2.

c(x) =1−

√1− 4x

2x=

2

1 +√1− 4x

c(x)x = 0

c(x) =1

2x(1−

√1− 4x) =

1

2x· 2

∞∑

n=1

(−1)n

4n

(2n − 2

n − 1

)(−4x)n

n

=∞∑

n=1

(2n − 2

n − 1

)xn−1

n

=∞∑

n=0

(2n

n

)xn

n + 1=

∞∑

n=0

Cnxn.

Cn =1

n + 1

(2n

n

).

e1 + e2 + e3 = 17

2 ≤ e1 ≤ 5, 3 ≤ e2 ≤ 6, 4 ≤ e3 ≤ 7.

x17

(x2 + x3 + x4 + x5)(x3 + x4 + x5 + x6)(x4 + x5 + x6 + x7),

xe1xe2xe3 = xe1+e2+e3

A,B

(A ∪ B) = A+ B − (A ∩ B).

|A| A

|A ∪ B ∪ C | = |A|+ |B |+ |C |− |A ∩ B |− |B ∩ C |− |A ∩ C |+ |A ∩ B ∩ C |.

n

A1, ... ,An n ∈ N

|A1 ∪ A2 ∪ ... ∪ An| =∑

1≤i≤n

|Ai |−∑

1≤i<j≤n

|Ai ∩ Aj |

+∑

1≤i<j<k≤n

|Ai ∩ Aj ∩ Ak |

−+ ...+ (−1)n+1|A1 ∩ A2 ∩ ... ∩ An|

⋃Ai a

r A1, ... ,An 1 ≤ r ≤ n ar =

(r1

)a(r2

)(r2

)a

m a( rm

)

a

(r

1

)−(r

2

)+− · · ·+ (−1)r+1

(r

r

)= 1−

r∑

m=0

(−1)m(r

m

)

= 1− (1− 1)r = 1

SA1, ... ,An ⊂ S P(Ai ) ∈ [0, 1] i = 1, ... , n

P(A1 ∪ A2 ∪ ... ∪ An) =∑

1≤i≤n

P(Ai )−∑

1≤i<j≤n

P(Ai ∩ Aj)

+∑

1≤i<j<k≤n

P(Ai ∩ Aj ∩ Ak)

−+ ...+ (−1)n+1P(A1 ∩ A2 ∩ ... ∩ An)

S

n n

n

Ai iP(A1 ∪ A2 ∪ · · · ∪ An)

P(Ai ∩ Aj) i j

P(Ai ∩ Aj) =: p2

∑

1≤i<j≤n

P(Ai ∩ Aj) =

(n

2

)p2.

P(A1 ∪ A2 ∪ ... ∪ An)

=∑

1≤i≤n

P(Ai )−∑

1≤i<j≤n

P(Ai ∩ Aj) +∑

1≤i<j<k≤n

P(Ai ∩ Aj ∩ Ak)

−+ ...+ (−1)n+1P(A1 ∩ A2 ∩ ... ∩ An)

=

(n

1

)p1 −

(n

2

)p2 +

(n

3

)p3 −+ · · ·+ (−1)n+1

(n

n

)pn.

p1 = P(Ai )i 1/n p2 = P(Ai ∩ Aj) = P(Ai )P(Aj)

Ai Aj

pr P(Ai1 ∩ · · · ∩ Air ) n!r

(n − r)!

pr =(n − r)!

n!.

P(A1 ∪ A2 ∪ ... ∪ An) =

(n

1

)p1 −

(n

2

)p2 +− · · ·+ (−1)n+1

(n

n

)pn

= −n∑

r=1

(−1)r(n

r

)(n − r)!

n!= −

n∑

r=1

(−1)r

r !.

e−1 =∑∞

r=0(−1)r/r !

P(A1 ∪ A2 ∪ ... ∪ An) ≈ 1− 1

e≈ 0.63212.

n n = 70.63214

S A1, ... ,An ⊂ S

A1 ∩ A2 ∩ · · · ∩ An = S \ (A1 ∪ A2 ∪ · · · ∪ An)

|A1 ∩ A2 ∩ ... ∩ An| = |S |− |A1 ∪ A2 ∪ ... ∪ An|

= |S |−∑

1≤i≤n

|Ai |+∑

1≤i<j≤n

|Ai ∩ Aj |

−+ ...+ (−1)n|A1 ∩ A2 ∩ ... ∩ An|.

n n ∈ N \ 0, 1p √

p

n = 120S = 2, 3, ... , 120

⌊√120⌋ = 10

2, 3, 5, 7

A1 := q ∈ S : 2 | q, A2 := q ∈ S : 3 | q,A3 := q ∈ S : 5 | q, A4 := q ∈ S : 7 | q.

S 2 3 5 7A1 ∩ A2 ∩ A3 ∩ A4

|A1 ∩ A2 ∩ A3 ∩ A4| = |S |−4∑

i=1

|Ai |+∑

1≤i<j≤4

|Ai ∩ Aj |

−∑

1≤i<j<k≤4

|Ai ∩ Aj ∩ Ak |+ |A1 ∩ A2 ∩ A3 ∩ A4|

|S | = 119, |A1| = 60, |A2| = 40, |A3| = 24, |A4| = 17.

A1 ∩ A2 = q ∈ S : 6 | q, |A1 ∩ A2| = 20,

A1 ∩ A3 = q ∈ S : 10 | q, |A1 ∩ A3| = 12,

A1 ∩ A4 = q ∈ S : 14 | q, |A1 ∩ A4| = 8,

A2 ∩ A3 = q ∈ S : 15 | q, |A2 ∩ A3| = 8,

A2 ∩ A4 = q ∈ S : 21 | q, |A2 ∩ A4| = 5,

A3 ∩ A4 = q ∈ S : 35 | q, |A3 ∩ A4| = 3,

A1 ∩ A2 ∩ A3 = q ∈ S : 30 | q, |A1 ∩ A2 ∩ A3| = 4,

A1 ∩ A2 ∩ A4 = q ∈ S : 42 | q, |A1 ∩ A2 ∩ A4| = 2,

A1 ∩ A3 ∩ A4 = q ∈ S : 70 | q, |A1 ∩ A3 ∩ A4| = 1,

A2 ∩ A3 ∩ A4 = q ∈ S : 105 | q, |A2 ∩ A3 ∩ A4| = 1

A1 ∩ A2 ∩ A3 ∩ A4 = q ∈ S : 210 | q, |A1 ∩ A2 ∩ A3 ∩ A4| = 0.

|A1 ∩ A2 ∩ A3 ∩ A4| = 119− 60− 40− 24− 17

+ 20 + 12 + 8 + 8 + 5 + 3− 4− 2− 1− 1 + 0

= 26

4 + 26 = 30 120

n√n

2, 3, ... , 1202, ... , ⌊

√120⌋ = 2, ... , 10

m n m × n

A = (aij)1≤i≤m1≤j≤n

=

⎛

⎜⎜⎜⎝

a11 a12 · · · a1na21 a22 · · · a2n

am1 am2 · · · amn

⎞

⎟⎟⎟⎠, aij ∈ R.

m × n(m × n,R) (m × n) A = (aij)

i j

A,B ∈ (m × n) λ ∈ R

A+ B = (aij) + (bij) = (aij + bij) ∈ (m × n),

λA = λ · (aij) = (λ · aij) ∈ (m × n).

A ∈ (m × n) B ∈ (n × p)

A · B = (aij) · (bjk) = (cik) ∈ (m × p)

cik :=n∑

j=1

aijbjk .

A ∈ (n × n) A A · A ∈ (n × n)

A ∈ (n × n)

Ak := A · A · · ·A︸ ︷︷ ︸k

k ∈ Z+

A0 := , := (δij).

δij :=

1 i = j ,

0 i = j

A ∈ (m × n) A

AT = (aij)T := (aji ) ∈ (n ×m).

A ∈ (n × n) A = AT A A = −AT

A A ∈ (n× n)

A = (aij) :=n∑

i=1

aii .

(m × n,Z2) (m × n, 0, 1)b1, b2 ∈ 0, 1

b1 ∧ b2 := (b1, b2) =

1 b1 = b2 = 1,

0 ,

b1 ∨ b2 := (b1, b2) =

0 b1 = b2 = 0,

1 .

A,B ∈ (m × n,Z2)

A ∧ B = (aij) ∧ (bij) := (aij ∧ bij),

A ∨ B = (aij) ∨ (bij) := (aij ∨ bij).

A ∧ B A B ∨ A B

A ∈ (m × n,Z2) B ∈ (n × p,Z2)

A⊙ B = (aij)⊙ (bjk) = (cik) ∈ (m × p)

cik := (ai1 ∧ b1k) ∨ (ai2 ∧ b2k) ∨ · · · ∨ (ain ∧ bnk)

=1≤j≤n

(aij , bjk).

A ∈ (n × n,Z2)

A[k] := A⊙ A⊙ · · ·⊙ A︸ ︷︷ ︸k

k ∈ Z+

A[0] :=

B =

⎛

⎝0 0 11 0 01 1 0

⎞

⎠ .

B [2] =

⎛

⎝1 1 00 0 11 0 1

⎞

⎠ , B [3] =

⎛

⎝1 0 11 1 01 1 1

⎞

⎠ ,

B [4] =

⎛

⎝1 1 11 0 11 1 1

⎞

⎠ , B [5] =

⎛

⎝1 1 11 1 11 1 1

⎞

⎠ .

k > 5 B [k] = B [5]

M,N R M NM × N

! M = N R M

! (a, a) ∈ R a ∈ M R

! (a, b) ∈ R (b, a) ∈ R a, b ∈ M R

! (a, b) ∈ R (b, c) ∈ R (a, c) ∈ R a, b, c ∈ MR

! (a, b) ∈ R (b, a) ∈ R a = b a, b ∈ M R

R Z+ (a, b) ∈ R ⇔ a | b

! R a | a a ∈ Z! R 1 | 2 2 $ 1! R a | b b | c a | c! R a | b b | a a = b

Z2 | (−2) (−2) | 2

M N M × NR1 R2 M N

R1 ∪ R2, R1 ∩ R2 R1 \ R2.

R1,R2 2, 3, 4, 5, 6

(a, b) ∈ R1 ⇔ a | b, (a, b) ∈ R2 ⇔ (2 | a) ∧ (2 | b).

R1 = (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6),R2 = (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6).

R1 ∪ R2 = (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 2), (4, 4), (4, 6),(5, 5), (6, 2), (6, 4), (6, 6),

R1 ∩ R2 = (2, 2), (2, 4), (2, 6), (4, 4), (6, 6),R1 \ R2 = (3, 3), (3, 6), (5, 5),R2 \ R1 = (4, 2), (4, 6), (6, 2), (6, 4).

R1 M N R2 NP R2 R1

R2 R1 =(m, p) ∈ M × P : ∃

n∈N(m, n) ∈ R1 ∧ (n, p) ∈ R2

.

R1 = (x , y) ∈ R2 : x2 = yR2 = (x , y) : y2 = x

R2 R1 =(x , y) ∈ R2 : ∃

z∈R(x , z) ∈ R1 ∧ (z , y) ∈ R2

,

=(x , y) ∈ R2 : ∃

z∈Rx2 = z ∧ z = y2

=(x , y) ∈ R2 : x2 = y2

R M

R M

R1 := R , Rn+1 := Rn R , n ∈ Z+.

R = (x , y) ∈ R2 : x2 = y

R2 = R R =(x , y) ∈ R2 : ∃

z∈R(x , z) ∈ R ∧ (z , y) ∈ R

,

=(x , y) ∈ R2 : ∃

z∈Rx2 = z ∧ z2 = y

=(x , y) ∈ R2 : x4 = y

R = (a, b) ∈ N2 : a > b

R2 =(a, b) ∈ N2 : ∃

c∈N(a, c) ∈ R ∧ (c, b) ∈ R

,

=(a, b) ∈ N2 : ∃

c∈Na > c ∧ c > b

=(a, b) ∈ N2 : a > b + 2

" R

R = (a, b) ∈ N2 : a ≥ b

R2 =(a, b) ∈ N2 : ∃

c∈N(a, c) ∈ R ∧ (c, b) ∈ R

,

=(a, b) ∈ N2 : ∃

c∈Na ≥ c ∧ c ≥ b

=(a, b) ∈ N2 : a ≥ b

= R

R M RRn ⊂ R n ∈ Z+

(⇒) R Rn ⊂ Rn ∈ Z+ n = 1Rn ⊂ R (a, b) ∈ Rn+1 c ∈ M(a, c) ∈ Rn (c , b) ∈ R Rn ⊂ R(a, c) ∈ R R (a, b) ∈ RRn ⊂ R

(⇐) R2 ⊂ R (a, b), (b, c) ∈ R (a, c) ∈ R2 ⊂ RR

M N M = p N = q RM N M N

M = m1, ... ,mp, N = n1, ... , nq.R ⊂ M × N

rij =

1 (mi , nj) ∈ R ,

0

R = (rij)

M = 1, 2, 3, 4, 5 R M(a, b) ∈ R ⇔ a | b

R =

⎛

⎜⎜⎜⎜⎝

1 1 1 1 10 1 0 1 00 0 1 0 00 0 0 1 00 0 0 0 1

⎞

⎟⎟⎟⎟⎠

R MR = RT R R

(rij , rji ) = 0 i = j

R M rii = 1 i R = M

R1,R2 ⊂ M × N

R1 ∪ R2 = R1 ∨ R2, R1 ∩ R2 = R1 ∧ R2,

R1 ⊂ R2 ⇔ R1 ∪ R2 = R2 ⇔ R1 ∨ R2 = R2.

R1 M N R2 NP

R2 R1 = R1 ⊙ R2,

M = 1, 2, 3 R1 = (1, 2), (1, 3), (2, 3)R2 = (2, 1), (2, 2), (3, 1)

R2 R1 = (1, 1), (1, 2), (2, 1).

R1 =

⎛

⎝0 1 10 0 10 0 0

⎞

⎠ , R2 =

⎛

⎝0 0 01 1 01 0 0

⎞

⎠ .

R1 ⊙ R2 =

⎛

⎝1 1 01 0 00 0 0

⎞

⎠

R2 R1

R M R

R [n] ∨ R = R

n ∈ N

G = (V ,E )V E

V E ⊂ V 2 (a, b)a b

V = a, b, c , d

E =(a, b), (a, d), (b, b), (b, d),

(c , a), (c , b), (d , b).

R MV = M E = R

M = 1, 2, 3, 4, 5

(a, b) ∈ R ⇔ a | b.

R

V = M E = R

R

R ⊃ R RR

M = 1, 2, 3 R(a, b) ∈ R ⇔ a < b

R = (1, 2), (1, 3), (2, 3).

R = (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3)

R R

˜R = (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2),

R

R R˜R R

M R M RM

R

R ⊂ R

S RR ⊂ S R

R R

M

∆ :=(a, a) ∈ M2 : a ∈ M

.

R ∈ M2

R−1 :=(a, b) ∈ M2 : (b, a) ∈ R

.

M R M! R ∪∆ R! R ∪ R−1 R

RR R R

M = 1, 2, 3, 4

R = (1, 3), (1, 4), (2, 1), (3, 2).

(2, 1) ∈ R (1, 3) ∈ R(2, 3) /∈ R

R = (1, 3), (1, 4), (2, 1), (3, 2), (1, 2), (2, 3), (2, 4), (3, 1).

R (3, 1) (1, 4)(3, 4)

G = (V ,E )(e1, ... , en) ej ∈ E j = 1, ... , n n ∈ Z+ na b

! ej = (xj−1, xj) j = 1, ... , n! x0 = a xn = b

(x0, ... , xn) a ∈ V0 a a

a b(a, d , c , b, d , b)((a, d), (d , c), (c , b), (b, d), (d , b)

)

(a, b, c)(b, c)

M R M G = (M,R)(x0, ... , xn)

R G

M R M a, b ∈ Mn a b R (a, b) ∈ Rn

n = 1 n = 1a b (a, b) (a, b) ∈ R1 = R

n = 1

nn+ 1 a b c ∈ R

n a c 1 c b(a, c) ∈ Rn

(c , b) ∈ R c(a, b) ∈ Rn R = Rn+1

M R MR R∗

(a, b) ∈ R∗ ⇔ R a b

R∗ =∞⋃

k=1

Rk .

M R M R∗

R

R∗

R R∗ (a, b), (b, c) ∈ R∗

a b b c Ra c (a, c) ∈ R∗ R∗

S R ⊂ SSn ⊂ S

S∗ =∞⋃

k=1

Sk ⊂ S .

R ⊂ S R∗ ⊂ S∗ R SR∗ ⊂ S∗ ⊂ S

R∗ R

M M = n a, b ∈ MR M R

a b na b

a = ba b n − 1

M M = n RM

R∗ =n⋃

k=1

Rk .

a bm

(x0, x1, ... , xm) x0 = a xm = b

a = b m ≥ n + 1 M = nxi = xj

0 ≤ i < j ≤ m − 1xi(x0, ... , xi , xj+1, ... , xm)

a b m ≤ n

a = b

M M = n R MR

R∗ = R ∨ R [2] ∨ · · · ∨ R [n].

M = 1, 2, 3R = (1, 1), (1, 3), (2, 2), (3, 1), (3, 2) R

R∗ = R ∨ R [2] ∨ R [3] =

⎛

⎝1 0 10 1 01 1 0

⎞

⎠ ∨

⎛

⎝1 1 10 1 01 1 1

⎞

⎠ ∨

⎛

⎝1 1 10 1 01 1 1

⎞

⎠

=

⎛

⎝1 1 10 1 01 1 1

⎞

⎠

R∗ = R ∪ (1, 2), (3, 3) R

M = 1, 2, 3 R = (1, 3), (2, 1), (3, 1), (3, 2)R

R∗ = R ∨ R [2] ∨ R [3] =

⎛

⎝0 0 11 0 01 1 0

⎞

⎠ ∨

⎛

⎝1 1 00 0 11 0 1

⎞

⎠ ∨

⎛

⎝1 0 11 1 01 1 1

⎞

⎠

=

⎛

⎝1 1 11 1 11 1 1

⎞

⎠

R∗ = R ∪ (1, 2), (1, 1), (2, 2), (2, 3), (3, 3)

O(n4)

O(n3)

M R MR M

(M,R)

≤ Z ≤(a, b) ∈ Z2 a ≤ b

M ⊂P(M)

| Z+

M #

(M,#)

(M,#)

a, b ∈ M a # b b # a a b

M (M,#)#

≤ ZM ⊂P(M)

| Z+

(M,#) (M,#)# M

(Z,≤)

(N,≤)

(N2,#)

(a1, a2) # (b1, b2) ⇔ (a1 < b1) ∨((a1 = b1) ∧ (a2 ≤ b2)

)

(M,#) a, b ∈ M

a ≺ b :⇔ (a # b) ∧ (a = b).

(M,#)P(x) x ∈ M

∀y∈M

(∀

x≺yP(x)

)⇒ P(y)

z ∈ MP(z) A = x ∈ M : ¬P(x)(M,#) a A a

A P(x) x ≺ aP(a)

P(x)M a M

∀x≺a

P(x) x ≺ a

P(a)

n,m ∈ N

am,n :=

⎧⎪⎨

⎪⎩

0 n = m = 0,

am−1,n + 1 n = 0 m > 0,

am,n−1 + n n > 0.

am,n = m +n(n + 1)

2m, n ∈ N.

N2

(m, n) ≺ (m0, n0) ⇔ (m < m0) ∨((m = m0) ∧ (n < n0)

).

(m0, n0) ∈ N2

(m, n) ≺ (m0, n0)

(m0, n0) = (0, 0)

n0 = 0 m0 > 0 am0,n0 = am0−1,n0 + 1(m0 − 1, n0) ≺ (m0, n0)

am0,n0 = am0−1,n0 + 1 = m0 − 1 +n0(n0 + 1)

2+ 1 = m0 +

n0(n0 + 1)

2

(m, n) = (m0, n0)

n0 > 0 am0,n0 = am0,n0−1 + n0(m0, n0 − 1) ≺ (m0, n0)

am0,n0 = am0,n0−1 + n0 = m0 +n0(n0 − 1)

2+ n0 = m0 +

n0(n0 + 1)

2

(m, n) = (m0, n0)

n

(A1,#1), ... , (An,#n) nA1 × · · ·× An

(a1, ... , an) # (b1, ... , bn) :⇔(

∀1≤i≤n

ai = bi)∨(a1 ≺1 b1

)∨

(∃

j∈2,...,naj ≺j bj ∧ ∀

i<jai = bi

).

A1 × · · ·× An

A1 = A2 = · · · = An #Ak #

A1 × · · ·× An

A #a = a1a2 ... am b = b1b2 ... bn n

m A t = (m, n)

a ≺ b :⇔((a1, ... , at) = (b1, ... , bt) ∧m < n

)

∨((a1, ... , at) ≺ (b1, ... , bt)

).

A = 0, 1 #0 ≺ 1 a = 110011 b = 11001 c = 1100001d = 1110 c ≺ b ≺ a ≺ d

(1, 2, 3, 4,≤)

! ≤

! ≤

!

(1, 2, 3, 4, 6, 8, 12, |)

(M,#) a ∈ M

a b ∈ M a ≺ b

a b ∈ M b ≺ a

a M b # a b ∈ M

a M a # b b ∈ M

(1, 2, 3, 4, 6, 8, 12, |)

! 1! 8 12! 1!

(M,#) A ⊂ M

u ∈ M a # u a ∈ AA

l ∈ M l # a a ∈ AA

x A x # u u AA x = A

x A l # x l AA x = A

(1, 2, 3, 4, 6, 8, 12, |)A = 1, 2, 3

! 6 12 A

! 1 A

! A = 1 A = 6

A = a, b, ec f

A

(M,#)M M

b, c

(M,#)#t a #t b a # b

#(M,#) (M,#)

(M,#)M

a ∈ M aa1 ≺ a a1 M

(M,#)

a1 (M,#)

(M ′,#′) M ′ = M \ a1 #′

# M M ′

M ′ = ∅ (M,#) (M ′,#′)a2 M ′

M = ∅a1, ... , an n = M #t

a1 ≺t a2 ≺t a3 ≺t · · · ≺t an.

#t # a # b ab a #t b

(1, 2, 3, 4, 6, 8, 12, |)

1 ≺t 2 ≺t 3 ≺t 4 ≺t 6 ≺t 8 ≺t 12

1 ≺t 3 ≺t 2 ≺t 6 ≺t 4 ≺t 12 ≺t 8

G = (V ,E )V E

V a, ba b

V < ∞ G

E a, b a, b ∈ Va, a E P(V )

G

GE G

G a ⊂ VE

G = (V ,E )V E

V (u, v)u v

E (a, b) a, b ∈ V(a, a) E ⊂ V 2 G

G

E

u v GG u v G

e u, v e u ve u v

v (v)

G = (V ,E ) |E |

2|E | =∑

v∈Vv .

V1 V2

G = (V ,E )

2|E |︸︷︷︸ =∑

v∈V1

v︸ ︷︷ ︸+∑

v∈V2

v︸ ︷︷ ︸ .

(u, v)u v u v

v u

v −(v)v v +(v)

v

(V ,E )

|E | =∑

v∈V

−(v) =∑

v∈V

+(v).

−( ) = 0, +( ) = 4.

Kj j ∈ Z+

j

Kj j = 1, 2, 3, 4, 5, 6

Cn n ≥ 3 G = (V ,E )V = v1, ... , vn vj , v(j+1) n j = 1, ... , n

Cj j = 3, 4, 5, 6

Wn n ≥ 3 G = (V ,E )Cn

Wj j = 3, 4, 5, 6

Qn n ≥ 1 G = (V ,E )V n

Q1 Q2 Q3

Qn Qn−1

G = (V ,E )V1 V2

E V1 V2

(V1,V2) G

C6 C3

(⇒) G = (V ,E ) V = V1 ∪ V2

V1 V2

V1

V2

(⇐) G = (V ,E ) V = V1 ∪ V2 V1

V2

V1 V2 (V1,V2)G

Km,n m, n ∈ Z+

(V1,V2) |V1| = m |V2| = nV1 V2

K3,3

K1,n

Cn Wn

Kn(n2

)

i(i − 1) (i + 1)

P1, ... ,Pn

P1 P2 P3 P4 Pn

n = m2

Pi ,j 0 ≤ i , j ≤ m − 1 Pi ,j

Pi±1,j±1

P1,0

P2,0

P3,0

P0,0 P0,1 P0,2 P0,3

P1,1 P1,2

P2,1 P2,2

P3,1

P1,3

P3,2

P2,3

P3,3

O(√n) =

O(m)

n = 2m

m = 3

P0 P1 P2 P3 P4 P5 P6 P7

G = (V ,E ) H = (W ,F )G W ⊂ V F ⊂ E H = G H

G

C6 W6

K7

G1 = (V1,E1) G2 = (V2,E2)G1 G2 G1 ∪ G2

V1 ∪ V2 E1 ∪ E2

G1 G2 G1 ∪ G2

G = (V ,E ) V = (v1, ... , vn)AG ∈ (n × n,Z2)

aij =

1 vi , vj ∈ E ,

0 .

v1

v2v3

v4v5

AG =

⎛

⎜⎜⎜⎜⎝

v1 v2 v3 v4 v5

v1 0 1 1 0 1v2 1 0 0 0 0v3 1 0 0 1 1v4 0 0 1 0 1v5 1 0 1 1 0

⎞

⎟⎟⎟⎟⎠

aijvi , vj

AG ∈ (n × n,Z2)G = (V ,E )

aij =

1 (vi , vj) ∈ E ,

0 .

|V | = n n2

c · n c =1≤i≤n

(vi )

c ≪ n

G = (V ,E )V = v1, ... , vn E = e1, ... , em

MG ∈ (m × n,Z2)

mij =

1 ej vi0

v4 v5 v6

v1 v2 v3

e1 e2

e3

e4e5MG =

⎛

⎜⎜⎜⎜⎜⎜⎝

e1 e2 e3 e4 e5

v1 1 0 0 0 0v2 0 1 1 0 1v3 0 0 1 1 0v4 1 1 0 0 0v5 0 0 0 1 1v6 0 0 0 0 0

⎞

⎟⎟⎟⎟⎟⎟⎠

G1 = (V1,E1) G2 = (V2,E2)

ϕ : V1 → V2,

V1 V2

ϕ∗ : E1 → V2 × V2, (a, b) 4→ (ϕ(a),ϕ(b)).

ϕ∗ = E2 G2 G1

(ϕ,ϕ∗) ϕ G1 G2 ϕ∗

G1 = (V1,E1) G2 = (V2,E2)G1 G2

ϕ : V1 → V2

ϕ∗ : E1 → E2, (a, b) 4→ (ϕ(a),ϕ(b))

ϕ

G = (U,E ) H = (V ,F )

u1 u2

u3 u4

v1 v2

v3 v4

G H ϕ : U → V

ϕ : u1 4→ v1, u2 4→ v4, u3 4→ v3, u4 4→ v2.

E =u1, u2, u1, u3, u2, u4, u3, u4

ϕ∗E =ϕ(u1),ϕ(u3), ϕ(u1),ϕ(u2), ϕ(u3),ϕ(u4), ϕ(u2),ϕ(u4)

=v1, v3, v1, v4, v2, v3, v2, v4

= F

ϕ G H

ϕ(U,E ) (ϕ(U),F )

U = (u1, u2, u3, u4),

ϕ(U) = (v1, v4, v3, v2),

E =u1, u2, u1, u3, u2, u4, u3, u4

,

F =v1, v3, v1, v4, v2, v3, v2, v4

.

ϕ

A(U,E) =

⎛

⎜⎜⎝

u1 u2 u3 u4

u1 0 1 1 0u2 1 0 0 1u3 1 0 0 1u4 0 1 1 0

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

v1 v4 v3 v2

v1 0 1 1 0v4 1 0 0 1v3 1 0 0 1v2 0 1 1 0

⎞

⎟⎟⎠ = A(ϕ(U),F )

!!

! k ∈ N k

G = (U,E ) H = (V ,F )

u1

u2

u3

u4 u5

v1

v2

v3

v4 v5

Hv1 G

G = (U,E ) H = (V ,F )

u1 u2

u3u4

u5 u6

u7u8

v1 v2

v3v4

v5 v6

v7v8

1 2 3

ϕ : U → V ϕ∗ : E → FU ′ ⊂ U G ′ = (U ′,E ′) (U,E ) ϕU ′ ϕ|U′ V

(V ,F )

(U ′,E ′)k ∈ N ϕ

U ′ Vk

k ∈ N

u1

u3

u5

u7

v2

v3

v6

v7

u2

u4

u6

u8

v1

v4

v5

v8

GH

G = (U,E ) H = (V ,F )

u1

u2

u3

u4

u5

u6

v1

v2

v3

v4

v5

v6

2 3G H

G = (U,E ) u, v ∈ U! n = 1 u v

e ∈ E u, v! n ≥ 2 u v (e1, e2, ... , en)

! e1 u, x1! ek xk−1, xk k = 2, ... , n − 1! en xn−1, v

x1, ... , xn−1 ∈ U e1, ... , en ∈ E! u, x1, ... , xn−1, v

e1, ... , en! G

(u, x1, ... , xn−1, v)! u = v!

G AG

V = (v1, ... , vn)k ∈ N \ 0 vi vj (i , j)

AkG

k = 1

(i , j) Ak

vi vj Ak+1 = Ak · A (i , j) Ak+1

bi1a1j + bi2a2j + · · ·+ binanj

Ak = (bij)1≤i ,j≤n

k + 1 vi vj k vivm 1 vm vj

vm ∈ V vjk vi vm

k ∈ Z+

k k ∈ Z+

A kAk = 0

u1

u2

u3

u4

u5

u6

v1

v2

v3

v4

v5

v6

A =

⎛

⎜⎜⎜⎜⎜⎜⎝

u1 u2 u3 u4 u5 u6

u1 0 1 0 1 0 1u2 1 0 1 0 0 0u3 0 1 0 1 0 1u4 1 0 1 0 1 0u5 0 0 0 1 0 1u6 1 0 1 0 1 0

⎞

⎟⎟⎟⎟⎟⎟⎠

A3 =

⎛

⎜⎜⎜⎜⎜⎜⎝

u1 u2 u3 u4 u5 u6

u1 0 6 0 8 0 8u2 6 0 6 0 4 0u3 0 6 0 8 0 8u4 8 0 8 0 6 0u5 0 4 0 6 0 6u6 8 0 8 0 6 0

⎞

⎟⎟⎟⎟⎟⎟⎠

G = (U,E )

u1

u2

u3

u4

u5

u6

u1 u5 E

U ′ = u5, u6 E ′ = u5, u6U ′′ = u1, u2, u3, u4

E ′′ = u1, u2, u2, u3, u3, u4, u1, u4

U ′′′ = u1, u2, u3 E ′′′ = u1, u2, u2, u3

(U ′′,E ′′)

v GG

G v

u1

u2u5

u6

u3u4

G

G G

G G

(⇒)v1

v1 v1

v1

(⇐) Gv0

v0, v1v1

v2vn

G

vn = v0

v0

GG

H GH \ G w0 H

G \ H GH

w0 H \ GG

GG

(⇒)v1 vn v1

vnv1 vn

(⇐) G v1 vn

v1, vn G Gv1 vn

v1, vn

G

GG

GG

G n ≥ 3n/2 G

G n ≥ 3n

G

a za z

kSk

Lk(v) v

k = 0

S0 := ∅, L0(v) =

0 v = a,

∞ v = a.

Sk+1 Sk k ∈ N u SkSk+1

Lk+1(v) =Lk(v), Lk(u) + w(u, v)

w(u, v) u, v

Lk(v) = a v

Sk .

z Skz

a z

S0 = ∅

0

∞

∞

∞

∞

∞

S1 = a

0

4 (a)

2 (a)

∞

∞

∞

S2 = a, c

0

3 (a, c)

2 (a)

10 (a, c)

12 (a, c)

∞

S3 = a, b, c

0

3 (a, c)

2 (a)

8 (a, c , b)

12 (a, c)

∞

S4 = a, b, c , d

0

3 (a, c)

2 (a)

8 (a, c , b)

10 (a, c , b, d)

14 (a, c , b, d)

S5 = a, b, c , d , e

0

3 (a, c)

2 (a)

8 (a, c , b)

10 (a, c , b, d)

13 (a, c , b, d , e)

S5 = a, b, c , d , e, z

0

3 (a, c)

2 (a)

8 (a, c , b)

10 (a, c , b, d)

13 (a, c , b, d , e)

az

Ga G k

Lk(v) v ∈ Ska v

Lk(w) w /∈ Ska w w Sk

k ∈ N k = 0 S0 = ∅a

a S0 ∞k = 0

k ∈ N Ska u ∈ Sk+1

u /∈ Sk u k uSk

uSk u

Lk(u) a u Sk vSk Lk(v) < Lk(u) v

Sk uu a u kk + 1

v ∈ Sk+1 a vSk+1 u uLk+1(v) = Lk(v) a v

k ua u u v

Lk+1(v) kk + 1

O(n2)

n

n(n − 1)!/2 = O(n!)

n

wc · w c = 3/2

K4

K3,3

v4 v5 v6

v1 v2 v3

v1 v2 v4 v5R1 R2

v1 v5

v4 v2

R2 R1

v3 R2 R2

R21 R22

v1 v5

v4 v2

v3

R21

R22

R1

v6v6 ∈ R1 v3, v6

(v1, v5, v2, v4, v1) v6 ∈ R22 v1, v6v6 ∈ R21 v2, v6

v3 ∈ R1

Ge v r

G

r = e − v + 2.

G = (V ,E )G1,G2, ... ,Ge = G G G1

GGk+1 = (Vk+1,Ek+1) Gk = (Vk ,Ek)

E \ Ek Vk

Ek Vk Vk

G

rk ek vkGk

rk = ek − vk + 2

k ∈ Z+ k = 1rk = ek − vk + 2

ak+1, bk+1 Gk

ak+1, bk+1 ∈ Vk ak+1

bk+1 Rak+1, bk+1

ak+1, bk+1 R

rk+1 = rk + 1, ek+1 = ek + 1, vk+1 = vk

k k + 1

ak+1 ∈ Vk , , bk+1 /∈ Vk bk+1

ak+1 ak+1, bk+1

rk+1 = rk , ek+1 = ek + 1, vk+1 = vk + 1.

k k + 1

G ev ≥ 3 e ≤ 3v − 6

GG r ∈ Z+

2e =∑

Ri

(Ri ) ≥ 3r .

e − v + 2 = r ≤ 2e

3

e ≤ 3v − 6

G G

G G

2e ≤ 6v − 12.

2e =∑

vi

(vi ) ≥ 6v ,

G ev ≥ 3 G e ≤ 2v − 4

K5 v = 5e = 10 3v − 6 = 9 ≥ 10 = e

K3,3

v = 6 e = 9 e ≤ 3v − 6

K3,3

e ≤ 2v −4e = 9 ≤ 8 = 2v − 4

G = (V ,E ) v1, v2 ∈ Vv1, v2 ∈ E G = (V , E )

V = V ∪ w, E = (E \ v1, v2) ∪v1,w, w , v2

GG

G1,G2 GG1 G2 G

G1 G2 G1

G2

G1 G2 GG1 G2

G1 G

G G2 G2

G GG

G

K3,3 K5

v = 10 e = 15e ≤ 2v − 4

v1

v2v3

v4v5

u1

u2

u3

u4

u5

u1

v1

v2v3

v4v5

u1

u2

u3

u4

u5

v1

v2v3

v4v5

u1

u2

u3

u4

u5

u2 u5 v1

K3,3 (u3, v4, v5, v2, v3, u4)

G χ(G )

χ(Kn) = n

χ(Cn) = 2 n ≥ 2 n

χ(Cn) = 3 n ≥ 3 n

χ(Wn) = χ(Cn) + 1

χ(Km,n) = 2

n n

n nr

1 ≤ r ≤ n r

n n

rr

r

nn = 1

1 ≤ r < nr r

n n

r < n rn − 1

n − 1

2n

1 < r0 < n r02r0

n − r0

s n − r0r0 sr0 + s r0

r0 ss s n − r0

s n− r0

n − r0

G = (V ,E ) GG E

G G G

a1 a2 a3

a4 a5 a6

G2n (V1,V2)V1 = V2 = n

G r × sr + s > 2n

G2n

G

後宮

n m · nm

m r1 ≤ r ≤ n m · r

m

G1 G2 G4

G1 G2 G3 G4

G = (V ,E ) u = vG u v

(u, x1, ... , xn−1, v)

xi = xj i < j(xi+1, ... , xj)

(⇒) T Tx , y T

x y

(⇐) TT T

Tx

y x y

x y

T r! v = r T v u

(u, v) T! u v v u! v1, v2 u! v = r v

r v v r! u u

!

! a T aa

a

m

mm m

m mm = 2

uu u

n 2n+2

4 2 6 3 8

n 4 10 4 10

T

n n − 1

n = 1

n Tn + 1 T

nn − 1

T n

m in = mi + 1

mi

m · i n = mi + 1

m

n i = (n − 1)/ml = [(m − 1)n + 1]/mi n = mi +1 l = (m− 1)i +1l n = (ml − 1)/(m − 1) i = (l − 1)/(m − 1)

n = mi +1 n = i + l

m

T r

! v = r r vr

! T T

! m hh h − 1

m h mh

hh = 0

m0 = 1

m

T hm

h − 1 mh−1

m ·mh−1 = mh

m h l h ≥ ⌈ m l⌉m h = ⌈ m l⌉

l ≤ mh ⇔ h ≥ m l

insertion(T : , x : )v := T T null

v = null label(v) = x

x < label(v)v = null v := v

v v := null

v = null v := vv v := null

T = null v xv = null label(v) = x x

vv x

T n

UU

U

U h U TU U n

U n + 1

h ≥ ⌈ 2(n + 1)⌉.

T⌈ (n+ 1)⌉ = O( n)

!

!

!

n

⌈ 3 8⌉ = 2

n n!n!

⌈ 2(n!)⌉

2(n!)

Sn = (n!) =n∑

k=1

k .

∫ n+1

1x dx > Sn >

∫ n+1

2(x − 1) dx =

∫ n

1x dx .

1 2 3 4 5 6 7x

lnH2LlnH3LlnH4LlnH5LlnH6LlnH7L

y

y á lnHxL

y á lnHx - 1L

∫(x) dx = x (x)− x

n (n)− n − e + 1 < (n!) < (n + 1) (n + 1)− n − e.

n! = Θ(n n)Ω(n n)

O(n n)

e 4→ 0, a 4→ 1, t 4→ 01,

0101

s

e

a

t

n

e 4→ 0, a 4→ 10, t 4→ 110, n 4→ 1110, s 4→ 1111.

0.08, 0.10, 0.12, 0.15, 0.20, 0.35.

A 4→ 000, B 4→ 001, C 4→ 100, D 4→ 101, E 4→ 01, F 4→ 11.

0 k1, 2, 3, ... , k

v n A kvA.l ,A.2, ... ,A.kv

T r

T r r T

T1,T2, ... ,Tn r Tr T1,T2, ... ,Tn

T r

T r r T

T1,T2, ... ,Tn r TT1 r

T2,T3, ... ,Tn

T r

T r r T

T1,T2, ... ,Tn r TT1, ... ,Tn

r

!

!

!

+,−, ·,÷, ↑(x + y)2 + (x − 4)/3

(x + y) ↑ 2 + (x − 4)÷ 3

−

÷

+

↑

+

x y

2

x 4

3

(x + y)2 + (x − 4)/3x + y ↑ 2 + x − 4 ÷ 3

(x + y)2 + (x − 4)/3+ ↑ + x y 2 ÷ − x 4 3

!

!

!

!

(x + y)2 + (x − 4)/3x y + 2 ↑ x 4 − 3 ÷ +

G GG G

(⇒) G G GG

GG

(⇐) G T GT T G

G G

vv

v

v ww v v

(G : v1, ... , vn)T := v1

(v1)

(v : G )w v T

w v ,w T(w)

Tn n(n − 1)/2

O(n2)

TG

T0 G Tn+1

Tn Tn

Tn Tn

n Tn

Tn+1 Tn Tn

G G Tn

T

O(n2)

(G : )T :=

i := 1 n − 2

e := TT T

T := T eT G

(G : )T :=

i := 1 n − 1

e := GT T

T := T eT G

TT T

e vO(e (v))

O(e (e))