応用数学特論.dvi

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1 ( ). P,Q,R, = ( ). P T (true) F (false) T F P P T P. T 2 F ( ). 2 P Q P Q P Q P Q P or Q P Q P Q P Q T T T T F T F T T F F F. P = 5 4 Q = 3 2 P Q = P F Q T P Q T ( ). (=) (1) P P = P ( ) (2) P Q = Q P ( ) (3) (P Q) R = P (Q R) ( ) ( ). 2 P Q P Q P Q P Q P and Q P Q P Q P Q T T T T F F F T F F F F. P = 5 4 Q = 3 2 P Q = P F Q T P Q F ( ). 1

2 (4) P P = P ( ) (5) P Q = Q P ( ) (6) (P Q) R = P (Q R) ( ) ( ). P P P P not P P P P T F F T. P = 3 P = 3 P F P T ( ). (7) P = P ( ) ( ). (8) P (Q R) =(P Q) (P R) ( ) (9) P (Q R) =(P Q) (P R) ( ) ( ). (10) (P Q) = P Q ( ) (11) (P Q) = P Q ( ) ( ). 2 P Q P Q P Q if P, then Q P Q P Q P Q T T T T F F F T T F F T. P = 3 > 2 Q = 3 < 2 R = 9 > 4 S = 9 < 4 P R = 3 > 2 9 > 4 T P S = 3 > 2 9 < 4 F Q R = 3 < 2 9 > 4 T Q S = 3 < 2 9 < 4 T P R T P S F Q R Q S T 2 0 = ( ). (12) P Q = P Q 2

3 (13) (P Q) = P Q ( ). 2 P Q P Q P Q Q P P Q P if and only if Q P Q P Q P Q T T T T F F F T F F F T. P = 3 > 2 Q = 3 < 2 R = 9 > 4 S = 9 < 4 P R = 3 > 2 9 > 4 9 > 4 3 > 2 T P S = 3 > 2 9 < 4 9 < 4 3 > 2 F Q R = 3 < 2 9 > 4 9 > 4 3 < 2 F Q S = 3 < 2 9 < 4 9 < 4 3 < 2 T Q S T ( ). (14) P Q =(P Q) (Q P ) ( ). :,,,, A T A ( ). P Q. {(P Q) (Q R)} (P R) P Q R ( ). I O. {P (P Q)} Q P Q {P (P Q)} Q = I P P P P P = O ( ). (15) P P = I ( ) (16) P P = O ( ). P P P P 3

4 ( ). A B P,Q,R,... A P,Q,R,... B A P,Q,R,... B A B A B A B. (13) (P Q) P Q (P Q) P Q (P Q) P Q ( ). P Q Q P P Q Q P ( ). (17) P Q = Q P ( ) (18) Q P = P Q ( ) (19) P Q Q P ( ( ). P,Q,R,... A B A B P,Q,R,... A B A B A B A P,Q,R,... B A B ( ). (20) P Q P ( ) (21) P P Q ( ) (22) P (P Q) Q ( ) (23) (P Q) Q P ( ) (24) (P Q) (Q R) (P R) ( ) (25) (P Q) P Q ( ) (26) (P Q) (R S) (Q S ) P R ( ) ( ) A B A B B A A B. (22) P (P Q) Q Q P (P Q) P 1 4

5 1.2.1 ( ). x P x P P (x) x. P (x) = x P (x) P (x) x P (2) P (3) ( ). P (x) x P (x) x P (x) x, P (x) x, P (x) : x a P (a) x, P (x) x a P (a) x, P (x). x x>3 x 2 > 9 x, x > 3 x 2 > 9 x x 2 < 0 x, x x 2 < 0 x >3,x 2 > 9 x R,x 2 < 0 R ( ). P (x) x P (x) P (x) x x, P (x) x, P (x) : x a P (a) x, P (x) x a P (a) x, P (x). x 3 1=0 x, x x 3 1=0 x =5 3 x x, x x =5 3 ( ) x R,x 3 1=0 x N,x =5 3 N ( ). x, P (x) x, P (x) ( ). (27) ( x, P (x)) = x, P (x) (28) ( x, P (x)) = x, P (x) (29) ( x, P (x) Q(x)) = x, P (x) Q (x) (30) ( x, P (x) Q(x)) = x, P (x) Q (x). x sin x 1/2 x, x sin x 1/2 x R, sin x 1/2 x, x sin x>1/2 x R, sin x>1/2 x x>0 log x>0 x, x > 0 log x>0 x >0, log x>0 x, x > 0 log x 0 x >0, log x (2 ). x y P x y 5

6 P P (x, y) 2 x y (2 ) x y 2 P (x, y) x, y, P(x, y) x, y, P(x, y) x, y, P(x, y) x, y, P(x, y) 4 1 x, y, P(x, y) : x a y b P (a, b) x, y, P(x, y) x a y b P (a, b) x a y b P (a, b) x a y b P (a, b) x, y, P(x, y) 3. ε n 0 n n n 0 a n a <ε ε >0, n 0 N, n N,n n 0 a n a <ε x 3 + y 3 = z 3 x, y, z x N, y N, z N,x 3 + y 3 = z 3 ( ) ( ) ( ε n 0 n ) P P P Q P Q P Q P Q P Q P Q = P Q. ε k n n k a n a <ε ε >0, k N, n N,n k a n a <ε {a n } a ε >0, k N, n N,n k a n a ε ε k n n k a n a ε ε δ x x a <δ f(x) f(a) <ε ε >0, δ >0, x R, x a <δ f(x) f(a) <ε 6

7 f(x) x = a ε >0, δ > 0, x R, x a <δ f(x) f(a) ε ε δ x x a <δ f(x) f(a) ε ε δ x y x y <δ f(x) f(y) <ε ε >0, δ >0, x R, y R, x y <δ f(x) f(y) <ε f(x) ε >0, δ >0, x R, y R, x y <δ f(x) f(y) ε ε δ x y x y <δ f(x) f(y) ε 7

8 ( ). A B C x y z x A x A A x x A. A A 1/3 A 2 A A B B 2 B ( ) A = {x,y,z,...} 1 P (x) x A = {x : P (x)} 2 P (x) Q(x) x A = {x : P (x) Q(x)}, A = {x : P (x),q(x)} A A = {1, 2, 3, 4, 5} A = {x : x 1 5 } B B = {x : x>0} ( ). N : ( ) (natural number) Z : (integer, Zahl ( ) ) Q : (rational number, quotient ( ) ) R : (real number) C : (complex number) ( ). A B A B A B B A A B x, x A x B A B A B B A ( ). A B B A A B A = B A = B x, x A x B A B ( ). A B A B A B A B ( ). (31) A A ( ) (32) A B B A A = B ( ) 8

9 (33) A B B C A C ( ) ( ). 1 X X 1 A B C X ( ). A B A B A B A B = {x : x A x B} = {x : x A x B} ( ). A B A B A B A B = {x : x A x B} = {x : x A x B} = {x : x A, x B} A B A B = A B ( ). (34) A B = B A A B = B A ( ) (35) (A B) C = A (B C) (A B) C = A (B C) ( ) (36) A (B C) =(A B) (A C) A (B C) =(A B) (A C) ( ) ( ). (37) A B = B A B A B = A ( ). X A A A A = {x : x A} A B A B A B A \ B A B = {x : x A x B} = {x : x A x B} = {x : x A, x B} A B = A B ( ). (38) X = = X (39) (A B) = A B (A B) = A B ( ) ( ). X Y 2 X x Y 1 y f X Y f : X Y X 9

10 x Y f(x) x f X f Y f X Y. X = {1, 2, 3, 4, 5},Y = {1, 4, 9, 16, 25} f(1) = 1,f(2) = 4,f(3) = 4,f(4) = 9,f(5) = 25 X = Y = R x X y 2 = x 2 +1 y ( ). f X Y X A A x f f(x) A f f(a) f(a) ={f(x) :x A} Y B f(x) B X x f B f 1 (B) f 1 (B) ={x : f(x) B} f( ) = f 1 ( ) =. f 1 (B) f f 1 B ( ). f : X Y X A 1 A 2 Y B 1 B 2 (40) A 1 A 2 f(a 1 ) f(a 2 ) (41) B 1 B 2 f 1 (B 1 ) f 1 (B 2 ) (42) f(a 1 A 2 )=f(a 1 ) f(a 2 ) (43) f 1 (B 1 B 2 )=f 1 (B 1 ) f 1 (B 2 ) (44) f(a 1 A 2 ) f(a 1 ) f(a 2 ) (45) f 1 (B 1 B 2 )=f 1 (B 1 ) f 1 (B 2 ) (46) A f 1 (f(a)) (47) B = f(f 1 (B)) ( ). f : X Y f(x) =Y f X Y f y Y, x X, y = f(x) f x 1 X, x 2 X, x 1 x 2 f(x 1 ) f(x 2 ) 1 1 f x 1 X, x 2 X, f(x 1 )=f(x 2 ) x 1 = x 2 10

11 f ( ). f : X Y g : Y Z X x f f(x) g g(f(x)) X Z f g g f ( ). f : X Y g : Y Z h : Z W (48) (h g) f = h (g f) ( ) ( ). f X Y Y y 1 X x f 1 f ( ). X X 2 X 2 X = {A : A X} ( ). X A 1,A 2,...,A n {A 1,A 2,...,A n } A 1,A 2,... {A 1,A 2,...} {A i : i =1, 2,...} {A i } i=1 {A i } i N A = {A µ : μ M} A = {A µ } μ A M ( ). X A = {A µ } X A µ X {A µ } {A µ : μ M} A µ A µ = {x : μ M,x A µ } M = A µ = A µ X {A µ } {A µ : μ M} A µ A µ = {x : μ M,x A µ } M = A µ = X {A i : i =1, 2,...} A i, i=1 i=1 A i 11

12 2.3.4 ( ). {A µ } ( ) (49) A µ B = (A µ B) ( ) (50) (51) (52) (53) (54) ( ( ( ( ( A µ ) B = A µ ) B = A µ ) B = A µ ) = A µ ) = A µ A µ (A µ B) ( ) (A µ B) ( ) (A µ B) ( ) ( ) ( ) ( ). f : X Y {A µ } X {B µ } Y ( ) (55) f A µ = f(a µ ), f 1( ) B µ = f 1 (B µ ) ( ) (56) f A µ f(a µ ), f 1( ) B µ = f 1 (B µ ) ( ). 2 : X Y X x Y y (x, y) X Y X Y X Y = {(x, y) :x X, y Y } X Y 2 (x 1,y 1 ) (x 2,y 2 ) x 1 = x 2 y 1 = y 2 (x 1,y 1 )=(x 2,y 2 ) X A Y B A B = {(x, y) :x A, y B} A B A B A = B = = n : n X 1,X 2,...,X n A 1,A 2,...,A n n X i = {(x 1,x 2,...,x n ):x 1 X 1,x 2 X 2,...,x n X n } i=1 n A i = {(x 1,x 2,...,x n ):x 1 A 1,x 2 A 2,...,x n A n } i=1 A i (i =1, 2,...,n) n i=1 A i = X 1 = X 2 = = X n = X A 1 = A 2 = = A n = A n i=1 X i n i=1 A i X n A n 12

13 : {X µ } X µ A µ {A µ } X µ = {x : x M X µ μ M x(μ) X µ } A µ = {x : x M X µ μ M x(μ) A µ } A µ (μ M) A µ = μ M X µ = X A µ = A X µ A µ X M A M ( ). {X µ } X µ A µ {A µ } ( μ M,A µ = ) A µ = A µ = x : M X µ x(μ) A µ A µ ( ) ( μ M,A µ ) A µ (AC) ( μ M,A µ ) A µ 1 (AC) (axiom of choice) {A µ } x : M X µ μ M x µ = x(μ) A µ A µ x µ ( ). X Y 2 f : X Y g : Y X 13

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x

, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x 1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d