<"
 K(x)
{x | K(x)}
x K(x)
{x | K(x)}
Z = {x | x }
Z " Z
R = {x | x x " x} ?
R " R R " R R R " R
R " R R " R
'"
(" Ź "!
"x x "x x
"xA(x)("B C (("xA(x)) (" B) C
A(x) A (" B '" C
 x = y x y
x " y x y x y
Źx " y Źx = y
x " y x = y

x
"x"y(x = y "! "z(z " x "! z " y))
a b
a b b
a
x y
"z(z " x z " y)) x ą" y
x ą" y Źx ą" y
x y x ą" y '" x = y

"x"y(x = y "! x ą" y '" y ą" x)
 ą" "
"x"y(y " x)
x "y(y " x)
x1 x2 x1 = x2
"y(y " x1) "y(y " x2)
"y(y " x1 "! y " x2)
x1 = x2
"
x y z
z
"x"y"z"t(t " z "! (t = x (" t = y))
x, y z
x y {x, y}
{x, y} = {y, x}
 {x, x} {x} x1, . . . , xn
{x1, . . . , xn}
{a, b} = {b, b, a}
" = {"}

"x"y"z(z " y "! "t(z " t '" t " x))
x y
x
y x
x
x

z " x "! "t(z " t '" t " x)

x *" y = {x, y}
(1) z " x *" y "! (z " x (" z " y)
(2) z " x *" y "! (z " x '" z " y)

�! z " x*"y x*"y = {x, y} z " t t " {x, y}
t = x t = y z " x z " y
�! z " x x " {x, y} z "

{x, y} z " y
"x"y"z(z " y "! z ą" x)
 x
x P(x)
z " P(x) "! z ą" x
P(") = {"} P({"}) = {", {"}} P({a, b}) = {", {a}, {b}, {a, b}}
P(x) " x
W (z)
"x"y"z(z " y "! (z " x '" W (z)))
z
W (z) x
x
{z " x | W (z)}, {z " x : W (z)}.
x

x = {z " x | "t(t " x z " t)}
x = " z


z " x "! "t(t " x z " t).

�! �! z " x
x = " t t " x "t(t " x z " t) z " t " x


z " x

x = " " = "

"t(t " " z " t) z

x )" y = {x, y}.
(1) z " x )" y "! (z " x '" z " y)
(2) z " x )" y "! (z " x (" z " y)
x - y = {z " x | z " y}
a
-a

a ą"
-a
"
"x(x = " "y((y " x) '" (y )" x = ")))

v " y " x v " x
x x y " x
x
"z(z " z)
{z} z )"{z} = "
z {z} z " z z )" {z} = "

 "x"a W (x) "x (x " a W (x)) "x (x "
a '" W (x)) "x"a W (x) "x"y . . . "x, y . . .
{t " a | . . . }
{{a, b} | a, b " A} {t " P(A) | "a, b " A (t = {a, b})}
a b
a, b = {{a}, {a, b}}
a, b, x, y
a, b = x, y a = x i b = y.
L = a, b = {{a}, {a, b}} P = x, y = {{x}, {x, y}},
L = P {a} " L {a} " P {a} = {x} {a} = {x, y}
x " {a} a = x
 b = y a = x
P = x, y = {{a}, {a, y}}.
a = b L = {{a}, {a, b}} = {{b}, {b, b}} = {{b}, {b}} = {{b}} {a, y} " P
P = L {a, y} = {b} y " {b} b = y
a = b {a, b} " L {a, b} " P {a, b} = {a}

{a, b} = {a, y} a = b {a, b} = {a} {a, b} = {a, y}

b " {a, y} a = b b = y

a b a � b
t
t " a � b "u"v(t = u, v '" u " a '" v " b).
a � b
a � b = {t " P(P(a *" b)) | "u"v(t = u, v '" u " a '" v " b)}
r a � b
a b a = b r a
x r y x, y " r
R
ą"R= { x, y " R � R | x ą" y}
R R
r a
"x"a (x r x)
"x"a "y"a (x r y y r x)
"x"a "y"a "z"a(x r y '" y r z x r z)
"x"a "y"a (x r y '" y r x x = y)
"x"a "y"a (x r y (" y r x)
(a � b) � c
 r ą" a � b
r-1 = { y, x | x, y " r} ą" b � a
r ą" a � b s ą" b � c r s (r ; s) ą" a � c
x (r ; s) y "z " b (x r z '" z s y).
f ą" a�b a b f : a b
"x"a "y"b ( x, y " f)
"x"a "y"b "z"b ( x, y " f '" x, z " f y = z)
y " b x, y " f f(x) a
f Dom(f) f
Rg(f) = {y " b | "x"a f(x) = y}
f : a b
f(x) x " a f, g a b
f = g "x"a f(x) = g(x)
f = g "x"a f(x) = g(x)

f f
a b

f
1-1
" f : a b f : a - b
"x, y " a (x = y f(x) = f(y))

"x, y " a (f(x) = f(y) x = y)
" f : a b b "y"b "x"a (f(x) = y)
na
b = Rg(f) f : a - b
"
1-1
f : a - b
na
1-1
f : P(A) - P(A � A)
f(z) = z � z z ą" A Ą1 : A � B A
Ą2 : A � B B Ą1( x, y ) = x Ą2( x, y ) = y
f Rg(f)
1-1
f : a - b f-1 f
1-1 1-1
f : a - b f-1 : Rg(f) - a
na
f-1 ą" Rg(f) � a x, y " f-1 y, x " f
y " a x = f(y) " Rg(f)
x " Rg(f) x = f(y) y x, y " f-1
x, y " f-1 x, z " f-1 x = f(y) x = f(z) y = z f
f-1 f-1(x) = f-1(y) = z x = f(z) = y
a y " a y = f-1(f(y))
f : a b g : b c f g
g ć% f : a c (g ć% f)(x) = g(f(x)) x " a
 g ć% f g ć% f = (f ; g)
f : a b g : b c h : c d h ć% (g ć% f) = (h ć% g) ć% f
f-1 f-1 ć% f = idDom(f) f ć% f-1 = idRg(f)
f ć% idDom(f) = f = idRg(f) ć% f
1-1 1-1 1-1
f : a - b g : b - c g ć% f : a - c
na na na
f : a - b g : b - c g ć% f : a - c
f : A B C ą" A f

f (C) = {b " B | "a " C (f(a) = b)}

f (C) = {f(a) | a " C}
D ą" B f

-1
f (D) = {a " A | f(a) " D}
f : N P(N) n " N
n

f ({1, 3, 4, 6, 9}) = {", {2}, {3}, {2, 3}}

-1
P f (P ) = {"} f ({{2}, {1, 2, 27, 36}}) =
{2k | k " N - {0, 1}}
 {At}t"T At
T
{At | t " T }
{At}t"T A Dom(A) = T
A(t) = At t " T
A � B
(A � B) � C
{An}n"N a0, a1, . . .
an " An n " N
N
{At}t"T

At = {f " P(T � At) | (f : T At) '" ("t " T (f(t) " At)}
t"T t"T t"T

f " At �! f Dom(f) = T "t " T (f(t) " At)
t"T

X S ą" X
X S
X
"a " X"t " a (S )" a = {t})

f : X X X f(a) " a a " X
{1, 3, 4} {{1, 2}, {3, 5}, {4, 5}}
{{1}, {2}, {1, 2}}
 X

F : X P(X � X) F (a) =
{a} � a a " X Y = Rg(F ) Y = {{a} � a | a " X} X
Y
Y t " ({a} � a) )" ({b} � b)
t = a, � = b, � � " a " X � " b " X a = b
{a} � a = {b} � b
Y S S X

S ą" X � X S ą" Y t " S t " {a}�a

a " X t = a, � � " a t " X � X
� " a " X
"a " X"� " a ( a, � " S)
a " X {a} � a " Y t " S )" ({a} � a) t
a, �
"a " X"�, � ( a, � " S '" a, � " S � = �)
a, � a, � S )" ({a} � a)

X X
S(a) " a S
R
"a, b " R(a = b a )" b = ")

 {At}t"T

At
t"T

� {At | t " T } f : T {At | t " T }

f(t) = �(At) t " T f " At
t"T
A = "

1-1
f : A - B
na
g : B - A
�! A = " ą " A f

g(b)
b " B

f-1(b), b " Rg(f)
g(b) =
ą, .

�! X = {g -1({a}) | a " A} X
g
X � f(a) = �(g-1({a}) f
g-1({a})
r a
" "x"a (x r x)
" "x"a "y"a (x r y y r x)
" "x"a "y"a "z"a(x r y '" y r z x r z)
r x " a
[x]r = {y " a | x r y}
a ida = { x, x | x " a} a � a
f : a b
ker(f)
x, y " ker(f) �! f(x) = f(y).
r ą" A � A A x " A x " [x]r
r ą" A � A A x, y " A
x r y
x " [y]r
y " [x]r
[x]r = [y]r
[x]r )" [y]r = "

r
�! x r y t " [x]r x r t
y r t [x]r ą" [y]r
�! x " [x]r = [y]r x " [x]r )" [y]r
�! t " [x]r )" [y]r x r t y r t x r y
r a/r
r
 r ą" A � A A
� : A A/r
�(a) = [a]r a " A
ker(�) = r
A P ą" P(A)
" "p"P (p = ")

" "p, q"P (p = q (" p )" q = ")

" P = A "x"A"p"P (x " p)
r A A/r A
P A r A
P = A/r
P A r
r = { x, y " A � A | "p"P (x " p '" y " p)}

r P = A
r x r y y r z
p, q " P x, y " p y, z " q p )" q = " p = q

x " p z " q = p x r z
x " p " P [x]r = p
x " p " P t " [x]r x, y " q
q " P q = p x " p '" q t " p [x]r ą" p
t " p t r x x " p t " [x]r
P = A/r
(ą") p " P p = " x " p p = [x]r p " A/r

(�") x " a p " P x " p [x]r = p
[x]r " A/r P
"
"
"
"
"
W
W
W
W
n n *" {n}
0 = " 1 = {"} 2 = {", {"}} 3 = {", {"}, {", {"}}}
N
"N (" " N '" "z (z " N z *" {z} " N))

R R

" " N N " R " " R z " R
N " R z " N z *" {z} " N z *" {z} " R
N N ą" N N
M
R = {N " P(M) | N }

N = R N
N
N )" M R
N )" M N
N
s : N N s(n) = n *" {n}
A ą" N
 0 " A
"n (n " A s(n) " A)
A = N
A
n " N n ą" N
A = {n " N | n ą" N}
0 " N 0 = " ą" N 0 " A
n " A n ą" A
s(n) = n *" {n} ą" N n ą" A {n} ą" A
m " n " N m ą" n
A = {n " N | "m " N (m " n m ą" n)}
n
m " " m ą" " 0 = " " A
n " A m " n *" {n} m " n m ą" n
m " {n} m = n m ą" n
m, n " N s(m) = s(n) m = n
s(m) = s(n) m *" {m}n *" {n} m " n *" {n}
m " n m = n m ą" n
n ą" m
" 0 = " " N N
" n " N s(n) = n *" {n} " N N
" n *" {n}
"
"
f : A B C ą" A f C
f|C : C B f|C(x) = f(x) x " C
D : N � N N
D(0, m) = m
D(s(k), m) = s(D(k, m))
k, m " N
M : N � N N
M(0, m) = 0
M(s(k), m) = D(M(k, m), m)
k, m " N
D : A�N N A ą" N
m " N k s(k) " A
n " N Dn : s(n)�N N
D0 D0(0, m) = m
Dn : s(n) � N N Ds(n) : s(s(n)) � N N

Dn(k, m), k " s(n)
Ds(n)(k, m) =
s(Dn(n, m)), k = s(n).
Dn
D : s(s(n))�N N D|s(n)�N : s(n)�N N
Dn D|s(n)�N Ds(n)(k, m) = Dn(k, m) = D|s(n)�N(k, m)
 k " s(n) Ds(n)(s(n), m) = s(Dn(n, m)) = s(D|s(n)�N(n, m)) = D(s(n), m))
Ds(n) D
D : N � N N
D(k, m) = Dk(k, m).
D(0, m) = D0(0, m) = m D(s(k), m) =
Ds(k)(s(k), m) = s(Dk(k, m)) = s(D(k, m)) D : N � N N
D |s(n)�N : s(n) � N N Dn n
D (n) = (D |s(n)�N)(n) = Dn(n) = D(n)
D M
D(k, m) k + m
M(k, m) k � m km
2 � 2 = 1 � 2 + 2 = (0 � 2 + 2) + 2 = (0 + 2) + 2 = 2 + 2 = s(1 + 2) =
s(s(0 + 2)) = s(s(2)) = s(s(s(s(0)))) = 4
f(0, n1, . . . , nk) = g(n1, . . . , nk);
f(s(m), n1, . . . , nk) = h(m, n1, . . . , nk, f(m, n1, . . . , nk)).
f
g h
f(m, n) = h(m, f|m�N)
h : N � P((N � N) � N)) N f(m, n)
f(k, r) k " m r " N
m d" n "k(m + k = n).
d"
m < n m d" n m = n

d"
 m, k, l " N
m + (k + l) = (m + k) + l
m + k = m k = 0
k + l = 0 k = 0
m + 0 = m
s(m) + k = m + s(k)
m + k = k + m
m 0 + (k + l) = (k + l) = ((0 + k) + l)
m + (k + l) = (m + k) + l s(m) + (k + l) = s(m + (k + l)) =
s((m + k) + l) = s(m + k) + l = s((m + k) + l)
m 0 + k = k 0 + k = 0
k = 0 s(m) + k = s(m) s(m + k) = s(m)
s(m) + k = s(m + k) m + k = m k = 0
k + l = 0 k = 0 k = s(k ) k 0 = k + l = s(k ) + l =

s(k + l) = 0

m 0 + 0 = 0 s(m) + 0 =
s(m + 0) = m
m s(0)+k = s(0+k) = s(k) = 0+s(k)
s(m)+k = m+s(k) s(s(m))+k = s(s(m)+k) = s(m+s(k)) = s(m)+s(k)
m m = 0
s(m) + k = s(m + k) = s(k + m) = s(k) + m = k + s(m)
m, n " N
m < n
"k(m + k = n '" k = 0)

s(m) d" n
m " n
 �! m + k = n m = n k = 0

�! m + k = n k = 0 n = m + s(k ) = s(m) + k k

�! k m n
n = s(m) + k m " n k = 0 n = s(m) = m *" {m} m " n
n = s(m) + s(k) n = s(s(m) + k) = (s(m) + k) *" {s(m) + k}
m " s(m) + k ą" n
�! n n = 0 m " n
m < 0
m < n m " n m " s(n) = n *" {n} m " n
m = n m < s(n) s(n) = s(0 + n) =
s(0) + n = n + s(0) = n + 1
d"
m d" n n d" p m + k = n m + l = p
k, l m + (k + l) = (m + k) + l = n + l = p
m d" p
m d" n n d" m m + k = n m + l = m
k, l m + (k + l) = (m + k) + l = n + l = m k + l = 0 k = 0
m = m + 0 = n
n " N
"m " N(m d" n (" n d" m)
n = 0 n d" m m = 0+m "m " N(m d" n("n d" m)
"m " N(m d" s(n) (" s(n) d" m) m " N m d" n
m + k = n k m + s(k) = s(m) + k = s(m + k) = s(n)
n d" m n < m n = m
s(n) d" m
A N
a " A "b (b " A a d" b)
 A ą" N
B = {n " N | "k(k " A n < k)}.
B B = N A = "
0 " A 0
0 d" m 0 + m = m 0 " B "k(k " A 0 < k)
n " B "k(k " A n < k) "k(k " A s(n) d" k)
s(n) " A s(n) A
s(n) " A "k(k " A s(n) < k)
B ą" N "n " N(n ą" B n " B)
B = N
A = N - B B = N A = " n

n ą" B n " B
B
n B n
B
N � N
m, n <" m , n m + n = m + n
<"
Z = (N � N)/<"
[ m, n ]<" + [ m1, n1 ]<" = [ m + m1, n + n1 ]<"
[ m, n ]<" � [ m1, n1 ]<" = [ mm1 + nn1, mn1 + nm1 ]<";
-[ m, n ]<" = [ n, m ]<"
m, n <" m , n m1, n1 <" m , n
1 1
" m + m1, n + n1 <" m + m , n + n
1 1
" mm1 + nn1, mn1 + nm1 <" m m + n n , m n + n m
1 1 1 1
" n, m <" n , m
1-1
i : N - Z
i(n) = [ n, 0 ]<"
n
i(n) i(m + n) = i(m) + i(n) i(m � n) = i(m) � i(n)
Rg(i) ą" Z
A B A <" B
1-1
f : A - B
na
1-1
" (a, b) (c, d) f : (a, b) - (c, d)
na
d-c bc-ad
f(x) = � x +
b-a b-a
Ą
" (-Ą , )
2 2
R
1-1
" (0, 1] (0, 1) f : (0, 1] - (0, 1)
na

1 1
, x = n " N
n+1 n
f(x) =
x, .
" R
 A B C
" A <" A
" A <" B B <" A
" A <" B B <" C A <" C
A A <" n n
A
a " A b " B
" A *" {a} <" B *" {b} A <" B
1-1
" f : A *" {a} - B *" {b}
1-1
f : A - B
1-1
f : A - B g = f *" { a, b } A *" {a} B *" {b}
f B g (�!)
1-1
f : A *" {a} - B *" {b}
h : A B

f(a), f(x) = b
h(x) =
f(x), .
h f h
n, m " N
1-1
f : s(n) - n
na
f : n - s(n)
m <" n m = n
n = 0
n
"m " N(m <" n m = n)
n = 0
m <" s(n) m <" n *" {n}
m = 0 m = s(m ) = m *" {m } m m <" n

m = n m = s(n)
n " N A = n A <" n
A n
A f : A A f
A
(�!) f
a " A - Rg(f) A = " A

A = "

s(n) n A A = (A - {a}) *" {a}
A - {a} <" n
1-1 1-1 1-1
h : n - A - {a} g : s(n) - A h-1 ć% f ć% g : s(n) - n
na na
na
(�!) f : A - A a, b " A
f(a) = f(b) A
1-1 1-1
h : n - A - {a} g : s(n) - A
na na
na
g-1 ć% (f|A-{a}) ć% h : n - s(n)
 A A *" {a}
A B A - B = "

na
A f : A - B B
A = n A *" {a} = s(n)
n
B ą" s(n) = n *" {n} B ą" n n " B
B = (B - {n}) *" {n} B - {n}
B
1-1
B ą" A A f : A - n n " N
na

B f(B) n B
A ą" B A
1-1
g : B - A B
Rg(g) ą" A
n m n + m
A A
A
"AB (A <" B �! A = B)
A " N
 A
A
A
A
1-1
n " N f : N - n
N
1-1
f|s(n) : s(n) - n
N 5!0
A
5!0 A
5!0
A 5!0
(�!) Ń P(A)-{"}
f : N A

f(n) = Ń(A- f (n))
n

f (n) A- f (n)
f m = n


m " n f(m) "f (n) f(n) f (n)
1-1
f(m) f : N - Rg(f) Rg(f) 5!0
na
A
1-1 1-1
(�!) N <" B ą" A A = n " N f : N - B g : A - n
na na
1-1
g ć% f : N - n
(�!)
(�!) A
1-1
B 5!0 f : N - B g : A A
na

f(f-1(x) + 1), x " B
g(x) =
x, .
g A
f(0) Rg(g) A <" Rg(g) A
B ą" N min B B
B
C B C
C B
1-1
B g : B - N C
na

g
(C) N
N
A f : N A

f(n) = min(A- f (n))
f
f A
m " A - Rg(f) n

m " A- f (n) m = f(n) m > f(n) Rg(f) ą" m Rg(f)

f Rg(f) <" N
 A
na
f : N - A
(�!) A = 5!0
1-1
A = n " N n = 0 g : n - A

na

g(m), m " n
h(m) =
g(0), .
na 1-1
(�!) f : N - A g : A - N g(a) =
min{i " N | f(i) = a} A Rg(g) N
na
A f : A - B B
na na
g : N - A f ć% g : N - B
B
A B A *" B A � B
A B
na na
A B f : N - A g : N - B
na
� : N - A *" B

f(k), n = 2k k
�(n) =
g(k), n = 2k + 1 k
A *" B
na
� : N - A � B
 n = 0

n = 2i3jq,
q

0, 0 , n = 0
�(n) =
f(i), g(j) , n = 2i3jq q
� a " A b " B i, j f(i) = a
f(j) = b a, b = �(2j3j)
t : N�N N f(n, m) = 2n3m
u, v : N � N N
u(m, n) = 2m(2n + 1) - 1
(m + n)(m + n + 1)
v(m, n) = + m
2
v(m, n) m + n
" N � N
" Z Z = (N � N)/<"
na
� : N�N - Z �(m, n) = [ m, n ]<"
�(m, n) = m - n
" Q Z
H" Z � (Z - {0})
x, y H" u, v x � v = u � y
Q = (Z � (Z - {0}))/H" x, y H" x , y
u, v H" u , v
xv + yu, yv H" x v + y u , y v xu, yv H" x u , y v
N � N N
[ x, y ]H" + [ u, v ]H" = [ xv + yu, yv ]H"
[ x, y ]H" � [ u, v ]H" = [ xu, yv ]H"
j(z) = [ z, 1 ]H".
na
x
[ x, y ]H" � : Z � (Z - {0})) - Q
y
x
�(x, y) =
y
"
Q � Q
A

" A = " A = "


" " " A A = (A - {"}) A A - {"}
na
F : N - A
A m " N
na
fm : N - F (m)

na
G : N � N - A G(m, n) = fm(n) G

F a " A F (m)

a fm(n) fm A
na
m fm : N - F (m)
A
A A
w : n A n
 w n = |w| baba
w : 4 {a, b}

b, i
w(i) =
a, i
n A n An
n A A A"
0
�
A A"
An
A0 = {�}
An+1 <" An � A A" An n " N
1-1
A <" B C <" D f : A - C
1-1
g : B - D
1-1 1-1 1-1
� : B - A � : C - D � ć% f ć% � : B - D
na na na
f

A C

�
�


B D
A B
1-1
A d" B f : A - B A d" B
A B A < B A
B
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" m, n m d" n A d" B A = m B = n
1-1
" f : A - B f A < B
N N N < N

" A ą" B A d" B
" n n < 5!0
1-1
" A A d" P(A) ś : A - P(A)
ś(a) = {a} a " A
" A 5!0 d" A
A, B
A d" B
na
g : B - A
A B
1-1
" f : A - B A <" Rg(f)
1-1 1-1
" f : A - C ą" B f :- B
na
A, B, C
" A d" A
" A d" B B d" C A d" C
 A d" B B d" A A = B
1-1
� : A - C ą" A C <" A
Xn n " N
X0 = A - C;

�
Xn+1 = (Xn).

X = {Xn | n " N} Y = A-X C = A-X0 = (X *"Y )-X0 =
1-1
(X - X0) *" Y Y )" X0 = " � : A - C
na

x, x " Y
�(x) =
�(x), x " X
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Y Xn
�
1-1
� idY : Y -
1-1
Y �|X : X - X X Y �
C c " C c " Y c = �(c)
c " X - X0 c " Xn+1 n c = �(x) = �(x)
x " Xn
A d" B B d" A A = B
1-1 1-1
f : A - B g : B - A C = Rg(g)
1-1
B � = g ć% f � : A - C
A C B
A
B B A A B
K d" A d" L K ą" A ą" L
K = L
A A < P(A)
A d" P(A)
1-1
F : A - P(A) B = {x " A | x " F (x)}
na
F b " A F (b) = B b " B b " B
B b " F (b) = B b " B
b " F (b) b " B
1-1
F : A - P(A)
na
x " F (y) x = y
"x " A(x " F (b) �! x " F (x))
"x(b goli x �! x nie goli x)
&! "x(x " &!)
P(N)
&! &!
P(&!) ą" &! P(&!) d" &!
N < P(N) < P(P(N)) < P(P(P(N))) < � � �
 f : N Q
"� " Q (� > 0 "n " N "k e" n(f(n) - � < f(k) < f(n) + �))
C a"
f a" g �! "� " Q (� > 0 "n " N "k e" n(f(n) - � < g(n) < f(n) + �)).
R C/a"
[f]a" + [g]a"
h h(n) = f(n) + g(n) Q ą" R
q " Q q
C
2N N 2 = {0, 1}
C = P(N) = 2N

1-1
-1
F : 2N - P(N) F (f) =f (1)
na
f = g n f(n) = 1 g(n) = 0 n " F (f)

n " F (g) F P(N)
B ą" N B = F (�B) �B B

1, n " B
�B(n) =
0, .
P(N) 2N
2N d" C C d" P(N)
2N d" C H : 2N R
(0, 1)
H(01100011100 . . .) = 0, 01100011100 . . .
f " 2N
"

f(i)
H(f) =
10i+1
i=0
 H f = g n =


f(i) g(i)
min{i | f(i) = g(i)} = b

i10i+1 10i+1
f(n) = 0 g(n) = 1
"

f(i) 1
H(f) = b + < b + d" H(g)
10i+1 10n+1
i=n+1
1-1
C d" P(N) ą : N - Q
na
G(x) = {n " N | ą(n) < x},
x " R G : R P(N)
x = y x < y q

x < q < y q " G(x) - G(y)
A d" B C d" D
A )" C = " B )" D = " A *" C d" B *" D
A � C d" B � D
C = " AC d" BD

1-1 1-1
f : A - B g : C - D
f g f *" g
A *" C B *" D
1-1
F : A � C - B � D F (a, c) = f(a), f(c)
F f g
na
C = " h : D - C G : AC BD

G(ą) = f ć% ą ć% h
ą

C A

f
h


D B
G(ą)
 G ą, � " AC ą = � ą(c) =

�(c) c " C h d " D h(d) = c
f G(ą)(d) = f(ą(h(d))) = f(ą(c)) =

f(�(c)) = f(�(h(d))) = G(�)(d) G(ą) = G(�)

A = B C = D
" A � C = B � D
" AC = BD
A )" C = " B )" D = "
" A *" C = B *" D
A" = 1
A
m + n m n A *" C
A = m C = n A )" C = "
m � n m n A � C
A = m C = n
mn m n AC
A = m C = n
" 5!0 + 5!0 = 5!0 Z <" N
" 5!0 � 5!0 = 5!0 N � N <" N
0
" 25! = C
n
" 0 = 5!0 n+1 = 2
P(N) = R = 1 P(P(N)) = 2
m e" 5!0 m + 5!0 = m
A = m C = 5!0 A )" C = "
B ą" A 5!0 A*"C = (A-B)*"(B *"C) <" (A-B)*"B = A
B *" C 5!0 m + 5!0 = A *" C = A = m
m n p
" m + 0 = m A *" " = A
" m + n = n + m A *" B = B *" A
" (m + n) + p = m + (n + p) (A *" B) *" C = A *" (B *" C)
" m � 1 = m A � 1 <" A
" m � 0 = 0 A � " = "
" m � n = n � m A � B <" B � A
" (m � n) � p = m � (n � p) (A � B) � C <" A � (B � C)
" m � (n + p) = m � n + m � p A � (B *" C) <" (A � B) *" (A � C)
" m0 = 1 " A"
" m1 = m A1
" 1m = 1 {0}A
" 0m = 0 m = 0

 m n p
mn � mp = m(n+p)
mn � pn = (m � p)n
(mn)p = mn�p
AB �AC <" AB*"C B)"C = "
1-1
F : AB � AC - AB*"C F (f, g) = f *" g f : B A
na
g : C A
1-1
G : AB�CB - (A�C)B
na
G(f, g)(b) = f(b), g(b) f : B A g : B C b " B
1-1
H : (AB)C - AB�C
na
H(�)(b, c) = �(c)(b) � : C AB c " C b " B
m d" n p d" q
" m + p d" n + q
" m � p d" n � q
" mp d" nq p = 0

5!0 � C = C � C = C
0
0
5!5! = C5! = C
0
2C = 5!C = CC
0
0 0 0 0
C d" 5!0 � C d" C � C = 25! � 25! = 25! +5!0 = 25! = C
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0 0 0 0 0 0
C = 25! d" 5!5! d" C5! = (25! )5! = 25! �5!0 = 25! = C
0
0 0
2C d" 5!C d" CC = (25! )C = 25! �C = 2C
0
d" <
5 < 5!0
" 5 + 5!0 = 5!0 + 5!0 = 5!0
" 5 � 5!0 = 5!0 � 5!0 = 5!0
0
0
" 25! = 5!5! = C
0
0
" C5 = C5! = C
m + p = m + q = n p = q
n - m = p 5!0 + 5 = 5!0 + 5!0 = 5!0 5!0 - 5!0
"
m 5!0 < m < C
r A
"x"A (x r x)
"x"A "y"A "z"A(x r y '" y r z x r z)
"x"A "y"A (x r y '" y r x x = y)
A, r A
 r
"x"A "y"A (x r y (" y r x)
" d"
" N - {0}
m|n "k " N - {0} (k � m = n)
"
A A, r a, b " A
a r b a ą" b
A, r A, ą"
d"
d" A <
x < y x d" y x = y

z"
A, r B ą" A
B, r )" (B � B)
B, r
A, d"
a, b " A a d" b b d" a a, b
B ą" A B B, d"
B A
B ą" A B
B A
x d" y x > y
x, y
 A A n
w : n A �
w : n A v : m A w � v v
w w � v : n + m A i < n + m

w(i), i < n
(w � v)(i) =
v(i - n), .
ein � (und � zwanzig) = (ein � und) � zwanzig = einundzwanzig
� � w = w � � = w
w
w, v " A"
w ą" v �! "u " A"(v = w � u)
(�!) w ą" v Dom(w) ą" Dom(v) |w| d" |v|
k = |v| - |w| i < k u(i) = v(|w| + i) v = w � u
(�!) v = w � u w = v||w| w ą" v
w ą" v w v
A
A d"
A w v
" w ą" v
" u ua ą" w ub ą" v a, b " A a < b
a < b � ab aba baba bba
wu = vu w ą" v v ą" w
x = wu = vu w = x||w| v = x||v|
|w| d" |v| w ą" v v ą" w
 A"
ą"
w v v x
w ą" v v ą" x w ą" x
w ą" v = uav x = ubx a < b
w ą" u u w w ą" x ua ą" w
w = uaw w x
w = uaw v = ubv ą" x a < b x = ubv x w x
w = uaw v = ubv v = u a v x = u b x a < b
a < b v = ubv = u a v u ą" u u ą" u u ą" u x = ubx
w x u ą" u w = u a w w x
w v v w
x = w
w = uaw = ubw a < b w ą" v
v ą" w w = v
w, v " A"
|w| d" |v| w ą" v i i < |w|
w(i) = v(i) i w|i = v|i u

w(i) < v(i) w = uw(i)w uv(i)v = v v(i) < w(i) v w
A, d" a " A
a A
"x " A (x d" a)
"x " A (a d" x a = x)
"x " A (a d" x)
"x " A (x d" a a = x)
a A, d"
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a d" x a x d" a a = x b " A
a b d" a b = a b
a A
" N - {0, 1}, | |
" Z d"
" Z *" {�}, � " Z
x y �! [(x, y " Z) '" (x d" y)] (" [x = y = �]
�
r
r
r-1
A, d" a " A
a
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n
n
A, d" n + 1
A A = B *" {a} B
n b
b d" a b A
a a d" c c = a c " B
a = b = c b B
A, d" a " A b " A
b d" a a d" b a = b
A, d" B ą" A a " A
a B a e" B b d" a
b " B
a B a = sup B
B
" a e" B
" c e" B c e" a c " A
a d" B
a = inf B
" A

X ą" P(A) X
" X

X X
" Q {q " Q | q2 < 2}
" {a, b, c, d} {c, d}
a
b













c
d
A, d"
A
A
A V
k1v1 + � � � + knvn = 0 v1, . . . , vn " A k1 = � � � = kn = 0 A
V
A
A V
A
Z = {A ą" V | A }
Z

Z Z B =
B
k1v1+� � �+knvn = 0 v1, . . . , vn " B v1, . . . , vn
 v1 " A1, . . . , vn " An A1, . . . , An " {A1, . . . , An}
i v1, . . . , vn " Ai A
k1v1 + � � � + knvn = 0
k1 = � � � = kn = 0
B B " Z B
A, d"
" B A a, b " B
c " B a, b d" c
" A
" A A
a0 d" a1 d" a2 d" . . .
sup " Ą"
A, d" B ą" A C
B
C = {x " A | x d" B}
b " B b e" C c = sup C b e" c c
B c x d" B
c d" x
A, d" B, d"
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x, y " A x d" y f(x) d" f(y)
" A, d" B, d" f : A B
f

X ą" A sup f (X)

f(sup X) = sup f (X)
" f : A A f(a) = a a f
x d" y {x, y} y
f(y) {f(x), f(y)} f(x) d" f(y)
A, d"
f : A A
B = {x " A | f(x) d" x} a = inf B a
f
x " B a d" x f(a) d" f(x) d" x f(a)
B f(a) d" a a
f(a) d" a f(f(a)) d" f(a) f(a) " B a d" f(a)
f B a
f : A A fn
n f f0 = idA fn+1 = f ć% fn
A, d"
f : A A sup{fn(Ą") | n " N}
 Ą" d" f(Ą") f
fn(Ą") fn(Ą") d" fm(Ą") n d" m
{fn(Ą") | n " N}
f(sup{fn(Ą") | n " N}) = sup{fn+1(Ą") | n " N} = sup{fn(Ą") | n " N}
a = sup{fn(Ą") | n " N}
b fn(Ą") d" b
n " N Ą" d" b
fn+1(Ą") d" f(b) = b b e" {fn(Ą") | n " N}
b e" a
r A (s; s )
s s P(A � A)
f : P(A � A) P(A � A)
f(s) = r *" s *" (s; s)
f r+
r
f(s) = s s (s; s) ą" s
r ą" s r+ r
1
� �� � +�
� � list
1+(int�list) list
F ą F (ą) = 1+(int�ą) list
2 n
Ą", F (Ą"), F (Ą"), . . . Ą" F (Ą")
n - 1
A B f : A B
" "
A ą" A f : A B f : A B Dom(f) = A f
A B
" "
[A B] [A B]
A B A
BĄ" = B *" {Ą"} Ą" " B
BĄ"
b d" b b = Ą" b = b
"
[A B]
f ą" g "a " A (f(a) d" g(a))
"
f : Z � Z Z
f(m, n) = if m = n then 0 else f(m + 3, n) + 3 fi
f
" "
Ś : [Z � Z Z] [Z � Z Z]
Ś(f)(m, n) = if m = n then 0 else f(m + 3, n) + 3 fi
f
fk = Śk(Ą") Ą" fk
m, n k - 1
 ą = int � ą
stream
stream
, G( ), G2( ), . . .
G(ą) = int � ą
A P
A P
a, b " ą ą " P a ą b
<" A
a1, a2 " A a1 <" a2 ą " P
" a1 ą b1 b1 b2 a2 ą b2 b1 <" b2
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a1 <" a2
a1 a2
H"
a1 H" a2
F : P(A � A) P(A � A)
F(r) = { a1, a2 | "ą"b1 (a1 ą b1 "b2 (b1 r b2 '" a2 ą b2))}
)" { a1, a2 | "ą"b2 (a2 ą b2 "b1 (b1 r b2 '" a1 ą b1))}
F
F
H"
, F( ), F2( ), . . .
A � A P(A � A)
k Fk( ) H"
k
F
A, d" B, d"
1-1
f : A - B
na
a d" a �! f(a) d" f(a )
a, a " A A, d" H" B, d" A H" B
f
"
"
"
"
" N
1
A = {1 - | n " N - {0}}
n
N R
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" A A *" {1} A *" {1, 2} B = {m - | m, n " N - {0}}
n
A H" A *" {1} A
A a, b " A a < b
a < c < b c
"
Q
"
Q
A, d"
A A, d"
A, d"
A
"
" Z Q R
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d" A" {anb | n " N}
A, d"
{ai | i " N} ai+1 < ai i
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(�!) B ą" A B
B b0
b1 " B b1 < b0
b0 > b1 > b2 > � � �
B A
A
"x, y " A (x " B '" y d" x y " B).
x " A
OA(x) = {y " A | y < x}
OA(x) x " OA(x)
OA(x) O(x)
A
T A" A
A", ą"
"w, u " A" (w � u " T w " T )
{a, b}
{�, a, b, aa, ab, ba, bb, aaa, aab, abb, bab, bba, bbb, aaba, aabb, baba, bbab}
a < b c
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a
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b
a




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b
a
a
b






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b
a b b
a
b



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b
a a
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X
a " X Sa a
A A e" Sa a " X
1-1
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X X = {Xn | n " N}
Xn = {a " X | O(a) d" n}.
X0 = {Ą"} a0 X
1-1
fn : Xn - A" a, b " Xn
a d" b �! fn(a) d" fn(b)
 Rg(f)n
f = {fn | n " N}
f0(Ą") = � fn

fn(b), b " Xn
fn+1(b) =
fn(a) � �a(b), b " Xn+1 - Xn a b
T A
T � = a0, a1, a2, . . .
ai+1 ai
T T
T
T
a " T Ta = {b " T | a d" b}
� = a0, a1, a2, . . . i Ta
i
T� = T Ta
n
an b1, . . . , bk
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n 1 k j k
Tb an+1 bj
j
A
a0 a1 a2 � � �
A a " A
Sa = {b " A | a b} a " A
n a = a0 a1 a2 � � � ak k d" n
O(x)
 a " A T = { b, i | a i b} i i
0 i+1= (i; )
T
b, i d" c, j �! (i d" j) '" (b j-i c)
K
K r
s M ą" Z�Z f : M K M
f(x, y), f(x + 1, y) " r f(x, y), f(x, y + 1) " s
x, y M
M ą" R � R M )" (Z � Z)
Wn = {p " Z | - n < p < n} n " N
2
T = {f | f Wn n " N}.
T
Wn 2n - 1 (K)8n
Wn+1 T
" ą" f1 ą" f2 ą" f3 ą" . . . fn
2
Wn fn
A, d" P ą" A
a " A
OA(a) ą" P �! a " P
P = A
 P = A A - P

a OA(a) ą" P a " P
A


�! �!
�!
a �! b c a c �! b
�!
a, b, c " A
b �! a c b �! c
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a, b, c " A
b �! a c b �! c
" �!- " �! " - "
A
�!
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A, �!
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b, c b �! a c b �! c
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a = b a = c
b �! b1 �! a c1 c
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d b1 d �! c1
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b1 c1 b e �! d f �! c1 e, f
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d g e g �! f
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b g �! c
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g
k Nk k
k
N
k k = 0, 1
Nk B ą" Nk+1
" b = min{n | "w " B (w(0) = b)}
" B = {w " Nk | bw " B}
B Nk w bw
B
{1, 2, 2, 3, 4, 4, 4}
M M(1) = M(3) = 1 M(2) = 2 M(4) = 3 x = 1, 2, 3

M(x) = 0
A M : A N
 a " M
M(a) > 0 M ą" N M(a) d" N(a) a " A
(M *" N)(a) = M(a) + N(a) (M - N)(a) = max{0, M(a) - N(a)}
a " A
{a " A | a " M}
M, N N M N
a, N N = (M - {a}) *" N a > b
b " N
M N
M0 M1 M2 � � �
k = 1 + max{n | n " M0} Mi
k i wi " Nk
wi(j) = Mi(k - j - 1) j = 0, . . . , k - 1 wi
Mi(l) l = 0, . . . , k - 1
k = 4 Mi = {0, 0, 2, 3, 3, 3} wi = 3, 1, 0, 2
w0 > w1 > w2 > � � �
M N
�!
�!
A
A OA(x)
 B A x
A - B B = OA(x)
" b " B b < x x d" b x " B b " OA(x)
" b " OA(x) b < x b " A - B b " B
A A
x = min{y " A | A H" OA(x)}
f : A OA(x) f OA(x)
OA(x) OA(f(x)) A H" OA(f(x))
f(x) < x x
A B
"x " A"y " B (OA(x) H" OB(y))
A B
Ś = { x, y | A � B | OA(x) H" OB(y)}
x, y , x , y " Ś
x < x �! y < y
(�!) x < x y e" y f : OA(x) OB(y)
OB(y ) ą" OB(y) OB(y ) OA(f-1(y ))
OA(x) H" OA(f-1(y )) f-1(y ) < x f-1(y ) " OA(x)
OA(x)
(�!)

1-1
Ś : A - B A H" (A) (A)
Ś Ś

B y " (A) y d" y OB(y )
Ś

(OB(y )) A
Ś-1

y " (A)
Ś
 A B
B
A A
"x " A"y " B (OA(x) H" OB(y))
x " A OA(x ) x < x
OB(y ) OA(x)
B B B
A OA(x) H" OB(y) y
A
B B
a
a
A Ś
P(A) - {"} D, d" D ą" A
"x " A (x = Ś(A - OD(x))).
D1, d"1 D2, d"2
 D2 D1
d"1 x " D1
x " D2
OD (x) = OD (x)
1 2
1
OD (x) x
OD (x) = {OD (y) | y < x}
1 1
D2 OD (x) D2
1
x OD (x) = OD (x )*"{x } = OD (x )*"{x }
1 1 2
x x OD (x ) *" {x }
2
D2
OD (x) D2
1
D2 = OD (x) D1 OD (x) = OD (y) y
1 1 2
x = Ś(A - OD (x)) = Ś(A - OD (y)) = y
1 2
D1 D1 ą" D2
D1 D2 x " D1
y < x y " D2 y " OD (x) = OD (x) y " D1
2 1
D1, d"D D2, d"D a d"D b b d"D a
1 2 1 2
a, b " D1 )" D2
F d"F F
" a " F a " D D, d"D a d"D a
a d"F a
" a d"F b b d"F a a d"D b b d"D a
1 2
D1, d"D D2, d"D
1 2
a d"D b a = b
2
" a d"F b b d"F c D1, d"D
1
D2, d"D a d"D b b d"D c D1
2 1 2
a d"D b b d"D a
2 2
a d"D c a d"F c
2
" a, b " F a " D1 b " D2 D1 D2
D1 ą" D2 a b D2 F
" B ą" F a
D a " D B )" D
D b
B d"F c " F
c e"F a e"F b c d"F a c " D c e"F b
F, d"F
F A F = A

a = Ś(A - F ) F
F1 = F *"{a} a
F1
A, d"F
A B A d" B B d" A
A B
A, d"
A
A
A
A,
a " A
a
ą" {x " A | x z" a}
a
d"
a

b z" a = { | b z" a}
b a b
a a
b

*" {b}, *" {b}
b b
=
a
, .
b

= { | a " A}
a
d" c
c d"
c d" a a
a " b
a b
a
a " a d" c

F : P(N)N P(N) F (x) = {x(i) | i " N}
F
F P(N)

-1
A ą" N ({A})
F

-1
A ą" N ({A})
F
k " N - {0} rk, r ą" Z � Z
x, y " rk x y x - y k
x, y " r x y x � y > 0
�k = rk *" r
x " Z [x]� k
k
Z/�
k
k = 4
k = 3
T ą" P(N)� N a, x
" a, x " T b, x " T a ą" b a = b
Ś : {T ą" P(N) � N | T } P(N)P(N)
Ś(T )(a) = {x " N | "b (b ą" a '" b, x " T )}
Ś P(N)P(N)
T
Ś(T ) = idP(N)
Ś(T )
Ś
F P (N) A ą" N F (x)
x A F (x) = F (y)
x(i) = N

N, i
y(i) =
", .
A = " x F (x) = " x(i) = "

-1 -1
({"}) A = " ({A})

F F
yk

A, i = k
yk(i) =
", i = k


-1
k A ą" N ({A})
F
x x - x = 0 k x
x � x d" 0 x = 0

x, y , y, z " �k x z y
x - z = (x - y) + (y - z) k
1
k x, z " �k x � z = (x � y) � (y � z) � > 0
y2
a
[a]� = {b " Z | b > 0} [a]� = {b " Z | b < 0} a
k k
[a]� = {a + nk | n " Z i nk } k
k
k = 4
"
"
"
"
k = 3
"
"
"
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"
f : P(N) P(N) Ś(T )
T

N, A = "
f(A) =
", .
T = { {x}, x | x " N} Ś(T ) = idP(N) Ś(T )(a) = {x " N | {x} ą" a} =
{x " N | x " a} = a T = { ", x | x " A}
A Ś(T )(a) = {x " N | x " A} = A
Ś(T ) = Ś(S) T = S a, x " T - S

x " Ś(T )(a) = Ś(S)(a) b, x " S b ą" a x " Ś(S)(b) =
Ś(T )(b) c ą" b c, x " T c ą" a c = a a, x =
c, x " S
a f a
d a b, c " a [b]d )" [c]d = "
A = C B AB = C
A AA = C
N
{0, 1}N
f d" g �! "x (f(x) d" g(x))
 NN
f d" g �! "x (f(x) d" g(x))
A
B ą" A
" B
" A - B
B
d
R
 Z N
Z d" C Z ą" P(N � N) C d" Z
1-1
F : P(N - {0}) - Z A ą" N - {0}
F (A) = { a, 0 | a " A} *" { n, n | n " N}.
F (A) N
A = B a " A - B a, 0 " F (A) - F (B)

Z d" C C d" Z
g0 g1
fn

0, m d" n
fn(m) =
1, .
P(N), ą"
{gn | n " N}

1, m = n
gn(m) =
0, .
P(N), ą"
ą : N 0, 1
fą : N N fą(0) = 1

2fą(n), ą(n) = 0
fą(n + 1) =
2fą(n) + 1, ą(n) = 1
Aą = Rg(fą) Aą ą = �

m = min{k " N | ą(k) = �(k)} fą(m + 1) " Aą - A�

f�(m + 1) " A� - Aą
Aą
fą NN
1









2 3









4 5 6 7








8 9 10 11 12 13 14 15
. . . . . . . . . . . . . . . . . . . . . . . .
Z = {B ą" A | }
Z
Z a, b, c "
a, b, c

" Z
Z
a h : b c
s a b
{aT }T "t
a b, d"
 F : NN � NN NN
F (f, g)(n) = min(f(n), g(n))
f, g " NN n " N
F
F
K, d" S
f : K K P ą" S a
P K
{b " S | b e" a}
a
S, d"
F f = F (f, f) f " NN
n
F ( , 1) = F ( , 2)
NN = 2C
-1
F ({f}) NN
-1
C F ({f})
f, f + g g N N f + g
(f + g)(n) = f(n) + g(n)
f, f + g
C
f : a b ker(f) = { x, y " a � a | f(x) = f(y)}
 S
A, d" S
L
L
S

a, b " L a " A b " B A, B " L
L A ą" B

a, b " B c " B a, b d" c c " L a

b L L
f f(a) e" f(p) = p
p " P f(a) P a d" f(a)
fn(a) n " N
b K
f(b) = f(sup{fn(a) | n " N}) = sup{fn+1(a) | n " N} = sup{fn(a) | n " N} = b
b b
a c e" a
c e" fn(a) n c e" b
f : P(N) P(N)

X, X d" 1
f(X) =
X *" {7}, .
f
P = {{2}, {3}} P P(N) {2, 3}
P S
S