23 14 24 14 25 15 26 15 27 15 28 15 29 16 30 N = Q 17 31 R = R 2 18 32 N < R 18 33 19 34 20 35 20 36 21 37 21 38 21 39 22 40 22 2

1 3 2 3 2.1...................................... 4 3 7 4 7 5 7 6 7 7 7 8 8 9 8 10 9 11 10 12 10 13 10 14 10 15 11 16 11 17 11 18 11 19 12 20 12 21 13 22 13 1

23 14 24 14 25 15 26 15 27 15 28 15 29 16 30 N = Q 17 31 R = R 2 18 32 N < R 18 33 19 34 20 35 20 36 21 37 21 38 21 39 22 40 22 2

1 2 P, Q P P P Q P Q P Q P Q P = Q P Q P Q P Q Q P x 2 + y 2 = z 2 x, y, z 3 2 + 4 2 = 5 2 x 3 + y 3 = z 3 x, y, z 4 40 x, y, z x, y, z x 2 + y 2 = z 2 x, y, z x, y, z x 3 + y 3 = z 3 x y z(x, y, z (x 2 + y 2 = z 2 )) x y z(x, y, z = (x 3 + y 3 = z 3 )) x, y, z x, y, z Z + 3

x y z(((x Z + ) (y Z + ) (z Z + )) (x 2 + y 2 = z 2 )) x y z(((x Z + ) (y Z + ) (z Z + )) = (x 3 + y 3 = z 3 )) P Q Q P (1) P Q Q P (2) P (Q R) (P Q) R (3) P (Q R) (P Q) R (4) P (Q R) (P Q) (P R) (5) P (Q R) (P Q) (P R) (6) P ( P ) (7) (P Q) ( P ) ( Q) (8) (P Q) ( P ) ( Q) (9) P = Q ( P ) Q (10) ( xp ) x( P ) (11) ( xp ) x( P ) (12) x yp y xp (13) x yp y xp (14) x(p Q) ( xp ) ( xq) (15) x(p Q) ( xp ) ( xq) (16) P x x(p Q) P ( xq) (17) P x x(p Q) P ( xq) (18) P P = Q Q (19) P Q Q P (P Q) (Q P ) 2.1 4

2.1.1 P Q P, Q P Q (6)(8)(9) (8) x x x = x= x = x x x x x (8) x x x = x= x = x x = (8),,,,,,,, = = = = = = = = = = = (6) = P, Q R 5

P (Q R) ( ) ( ) = = (P Q) (P R) ( ) ( ) ( ) ( ) = = (8)(9) 2.1.2 = (10) P = Q ( P ) Q P Q P Q (P Q) (8) (7) ( P ) Q P P Q P = Q P P Q ( P ) Q 2.1.3 180 180 (11)(12) ( xp ) P x x( P ) x P ( xp ) x P x( P ) P x 6

( xp ) x P x( P ) 3 4 180cm 5 A a A a A A a a A a A a / A A a 6 7 x(x A x B) A B A = B A B A B A B x(x A = x B) A B A B A B B A A B A B A B A B B A A A A x(x = x A) x x 4 (10) x = x A x A 7

A B B A A = B A B B C A C 8 A = {,, } B = {x x 3 2 } N = {0, 1, 2, 3, 4,... } 9 {x x A x B} A B A B {x x A x B} A B A B {x x A x / B} A B A B A B A B = A A B (20) A B = A A B (21) A B = B A (22) A B = B A (23) A (B C) = (A B) C (24) A (B C) = (A B) C (25) A (B C) = (A B) (A C) (26) A (B C) = (A B) (A C) (27) A (B C) = (A B) (A C) (28) A (B C) = (A B) (A C) (29) (26) (20) (26) (28) (20) (= ) x A A = A B B x B A B ( =) A B C A B A A A B x A A B x B x A B A A B 8

(26) A (B C) = (A B) (A C) x A (B C) x (A B) (A C) x A (B C) (x A x (B C)) (x A (x B x C)) 4 (5) (x A (x B x C)) ((x A x B) (x A x C)) ((x A x B) (x A x C)) (x (A B) x (A C)) x (A B) (A C) (28) A (B C) = (A B) (A C) x A (B C) x (A B) (A C) (= ) x A (B C) x A x (B C) x (B C) x (B C) x (B C) x B x C x (B C) (x B x C) 4 (8) x (B C) x B x C x A (x B x C) 4 (5) x (A x B) (A x C) x (A B) (A C) ( =) 10 11 U U U X U X U X def = {x x U X(X U x X)} U (30) X def = {x X(X U x X)} U (31) A, B U U = {A, B} U U X U X U X = A B (32) X = A B (33) 9

[3] def = def def X def = {x X(X U x X)} U = X U X U X x U U = X U X U X = U X U U U U U U U U = X = X U 11 A {X X A} A A P(A) A = {,, } P(A) = {, { }, { }, { }, {, }, {, }, {, }, {,, }} P( ) P( ) = { } { } = { } 12 a, b {{a}, {a, b}} (a, b) a b (a, b) (c, d) ((a, b) = (c, d)) (a = c b = d) [2] ((a, b) = (c, d)) (a = c b = d) a b (a, b) a b 13 (a, b, c) = ((a, b), c) (a, b, c) a, b, c (a 1, a 2,, a n 1, a n ) = ((a 1, a 2,, a n 1 ), a n ) n ((a 1, a 2,, a n ) = (b 1, b 2,, b n )) (a i = b i (i = 1, 2,, n)) 14 10

A, B {(x, y) x A y B} A B A B A A A 2 A 2 A A 3 B A n 1 A A n A = A = (A B = B A) ((A = B) (A = ) (B = )) A 15 A A A A R A a, b A (a, b) R arb = 16 A, B G A B (G, A, B) A B G A B (G, A, B) {a a A b B (a, b) G} (G, A, B) {b b B a A (a, b) G} (G, A, B) 17 (G, A, B) G {(b, a) b B a A (a, b) G} G G 1 (G 1, B, A) B A (G, A, B) (G 1 ) 1 = G G G 1 18 (G, A, B) f A B f : A B A f B f A B f = (G, A, B) Domf Imf (G 1, B, A) f 1 (f 1 ) 1 = f f f 1 f : A B G f U A {b b B u U (u, b) G} f(u) f U Imf = f(domf) = f(a) V B f 1 (V ) f 1 V f V a A f({a}) f(a) 11

19 A A R R A a A ara (34) a, b A ((arb bra) = a = b) (35) a, b, c A ((arb brc) = arc) (36) a, b A arb bra (37) R < > (34) 20 A A 19 R A (A, ) (A, ) 11 15 (, ) (34) a A ara a(a A = ara) R = a a R 3 2 P P = Q Q a a R = ara a(a R = ara) (A, ) U A U = (U U) U U (U, U ) (A, ) U (U, U ) U U (A, ) A U A U (U, U ) (U, ) U 12

21 A A R R a A ara (38) a, b A (arb = bra) (39) a, b, c A ((arb brc) = arc) (40) 22 A R A a A {x x A xra} R a C R (a) C(a) C(a) A a A a C(a) C(a) (41) a, b A (C(a) = C(b) arb) (42) a, b A (C(a) C(b) C(a) C(b) = ) (43) C(x) = A C(x) A 28 (44) x A (43) C(a) = C(b) C(a) C(b) = (41) = C(a) C(b) x x C(a) C(b) x x C(a) x C(b) xra xrb (39) (40) arb (42) C(a) = C(b) (43) (44) A C(x)(x A) A {C(x) x A} R A A/R C(b) C(a) C(c) C(d) A 13

23 (G, X, Y ) (G, X, Y ) x X y Y (x, y) G (45) x X (((x, y 1 ) G (x, y 2 ) G) = y 1 = y 2 ) (46) (45) x X y Y (G, X, Y ) X (46) y Y x X y Y y Y x X X f : X Y (45) (46) x X y Y f(x) = {y} f(x) = y X X f f(x) {y} y y = f(x) x X y Y x y Y Y 24 f X Y U X u U u f(u) U Y (45) (46) f U f U G = {(u, f(u)) u U} (G, U, Y ) U Y f U f U f U 14

25 X Y f Y Z g x X x g(f(x)) X Z (45) (46) g f gf f g f : X Y g : Y Z g f : X Z g f X 777777 Z f g 7 Y 26 f X Y f(x) = Y f x 1, x 2 X (x 1 x 2 = f(x 1 ) f(x 2 )) f f f 1 (f 1 ) 1 = f f f 1 U X u U u u u U X U X U X U = X U X X X X id X 27 Λ a A a Λ A Λ a(λ) Λ λ Λ a(λ) a λ a (a λ ) λ Λ a(λ) {a λ } λ Λ {a λ } λ Λ {a λ } λ Λ 28 {X λ } λ Λ λ Λ X λ (X λ ) λ Λ Λ 15

λ Λ λ Λ λ Λ X λ def = X λ def = X λ def = X (X λ ) λ Λ (47) X {X λ } λ Λ X (X λ ) λ Λ (48) X {X λ } λ Λ { f f Λ f } X λ λ Λ f(λ) X λ (X λ ) λ Λ (49) λ Λ Λ = {1, 2,, n} X λ λ Λ X λ λ Λ n X λ X λ λ=1 Λ = {1, 2} λ=1 λ Λ n X λ λ=1 n X λ λ=1 2 2 X λ = X 1 X 2 X λ = X 1 X 2 λ=1 2 X λ X 1 X 2 12 ((x 1, x 2 ) = (y 1, y 2 )) λ=1 (x 1 = y 1 x 2 = y 2 ) {1, 2} X 1 X 2 ; (1 x 1, 2 x 2 ) 2 X λ X 1 X 2 n X λ n n λ=1 λ=1 29 0 *1 1, 2, 3, A A {,, } = 3 0 1, 2, 3, A, B A B A B A = B A B A = B A B B A A B B A *1 0 16

A B A B (50) B A A B (51) A B (50) ( A B B A ) ( A = B ) (52) ( A B B C ) = ( A C ) (53) A, B A B B A (54) ( A B ) (B A ) (55) A f B B f 1 A (52) = A f B B g C A g f C (53) (52) = (54) (55) [3] (52) (54) (55) (50) (51) A B B A A B A = B A < B B > A 30 N = Q N Q N = Q N Q N Q N Q Q N 29 (52) N = Q Q ± n m n, m 0 +0 1 + n m 2n 3 m n (56) m 3n 5 m Q N Q N 17

N ℵ 0 a N Q N = N 2 Q = Q 2 31 R = R 2 R R = R 2 a, b R a < b {x x R a < x < b} (a, b) a, b R (0, 1) = R (0, 1) R x (0, 1) x tan(π(x 1/2)) (0, 1) = R (0, 1) 2 = R 2 R = R 2 (0, 1) = (0, 1) 2 (0, 1) x (x, x) (0, 1) 2 (0, 1) (0, 1) 2 x (0, 1) x = 0.x 1 x 2 x 3 (0, 1) 2 (0.x 1 x 2 x 3, 0.y 1 y 2 y 3 ) 0.x 1 y 1 x 2 y 2 x 3 y 3 (0, 1) (57) (0, 1) (0, 1) 2 (0, 1) = (0, 1) 2 R = R 2 c 0.123 9 = 0.124 0 9 0 = 0.000 / (0, 1) 32 N < R 30 31 N < R A A < P(A) N < R N R N R 31 (0, 1) = R N (0, 1) 31 (0, 1) 31 (0, 1) 0. 0 9 0.123 0.123000000000 9 18

0.1 9 = 0.2 0 0. 9 = 1 / (0, 1) 0. 0 = 0 / (0, 1) (0, 1) {x x = 0. 0 9 0 9 } (0, 1) N (0, 1) 1 0.a 11 a 12 a 13 a 14 a 15 a 16 a 17 2 0.a 21 a 22 a 23 a 24 a 25 a 26 a 27 3 0.a 31 a 32 a 33 a 34 a 35 a 36 a 37 4 0.a 41 a 42 a 43 a 44 a 45 a 46 a 47 5 0.a 51 a 52 a 53 a 54 a 55 a 56 a 57.. 1 a nn 1 a n = a = 0.a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 1 2 a nn = 1 2 a (0, 1) a n a nn a N (0, 1) A < P(A) A x {x} P(A) A P(A) A P(A) A P(A) N < R A P(A) A f P(A) B = {x x A x / f(x)} B A B P(A) f a A f(a) = B B a a B B a / f(a) = B a / B = f(a) a a A a / f(a) a B 33 (A, ) a A x A x a a A a A x A a x = a = x a A a A x A a x a A a A x A x a = a = x a A A maxa mina 19

(A, ) (U, ) a A x U x a a U U A U A a A x U a x a U U A U A A supa infa U U U U U U U A U A U A U A U A U A a A a a x(x = x a) a {1, 2} 2 1 1 2 34 (A, ) a A {x x A x a x a} A a A a a A B = A a B A A a B A A a B A A a = a A 35 (A, ) A A A a A a b A a a b x A a x b x = a x = b U = {x x A a x a x} a A A U A U minu a N N n N n + 1 Z Z Z 20

(, ) 36 0 ( 1) (+1) ( 2) (+2) ( 3) (+3) (Z, ) R A (A, ) A(A ) (A, ) A(A ) (A, ) 38 [3] 37 N N N N N ω Ω = N {ω} n N n ω ω ω Ω (Ω, ) (Ω, ) ω Ω A 12 20 A A A 38 21

[3] 39 X X X P(X) P(X) X P(X) X P(X) X X < P(X) X X = {x x x / x} X X X X X X X / X X X X X / X X / X X X X 40 [1] I [2] [3] [4] I.N.Bronstein K.A.Semenjaef G.Groche V.Ziegler D.Ziegler 22