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( , )

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FAQ

.

. .

,
  , ObS .,
  (.) .
  . , N . N ={1, 2, 3, }
  . , R . ( )
- . . , x Î .
X,YÎObS , X Í Y ½ x Î X Þ Î Y
CÎ ObS, UÍX , X\U X \ U = {xÎX½xÏU}Í X
X,YÎObS ., X\Y .X\Y = {xÎX½xÏY}
B, CbÎ ObS "bÎB ., ÈCb .ÈCb = {x½$bÎB , xÎCb}
B, CbÎ ObS "bÎB ., ÇCb .ÇCb = {x½"bÎB xÎCb}
CkÎObS, "k=1,¼,n ., PCk .{<x1, x2,¼, xn>½xkÎCk "k} ( , X´Y = {<x, y>| xÎX & yÎY})
CÎObS , C2 C´C
  Rn R´R´¼´R(n )
C,UÎ ObS . , S(C, U) . A:CU
TÎ ObS . () ., - . , S(T, R) . S(C, U) C=T U=R
a,bÎR, a < b . S[a, b] . S(T, R) T= [a, b]
  .l . S(N, R)
PÎ ObS . , S(P, PR) . S(C, U) C=R (. ) Y=PR
Î ObS . ,Ã() .{a½a Í }
  . , PR(PRoposition) . , ,
C,UÎObS (.) CÈU R Í C´U; (<x, y> Î R Û xRy)
CÎObS . .R Í C2 (. CÈU C=U)
CÎObS . C .R C ½ xRx "x
CÎObS . C .R C ½xRy & yRz Þ xRz
CÎObS . C .R C ½ xRy Þ yRx
CÎObS . C .R C ½ xRy & yRx Þ x=y
CÎObS (..) C, ~ . C, ,
nÎN . (..) Rn, = . . Rn: x = y Û xk = yk "k
XÎObS . . , =Ã(X) . . Ã(): A = B Û A Í B & B Í A
  . . , = . . CON: = b Û Va = Vb ()
  . . , = . . PR: p=q Û p q (.. , )
S(C,U) <Y, =Y> . . ,= . . S(C,U): A=B Û A(x) =Y B(x) "xÎC, =Y - .. Y, .. - S(C,U)
S(,R) . . ,= . . S(, R)
S(P, PR) . . , = . . S(P, PR)
CÎObS . () C, £ , .£ X
CÎObS . (), <, £ > <, £ >, £ -
<, £ >, x,yÎC , x £ y <x, y> Î £ ( )
<, £ >, x,yÎC , x < y x £ y, x ¹ y
<, £ >, x, yÎC x y <x, y> ï<x, y> Ï £ & <y, x> Ï £
CÎObS . (), < , £ > < , £ >,
  < R, £ >, R ½ x £ y Û y - x Î[0,+¥)
nÎN <Rn, £ >, Rn ½ x £ y Û xk £ R yk "k
XÎObS <Ã(X), £ >, Ã(X) ½ x £ y Û x Í y
  <CON, £ >, CON ½ x £ y Û Vx Í Vy
  < PR, £ >, PR ½ x £ y Û x Þ y (Û h(x) £ h(y))
S(C,U), <Y, £Y >- < S(C,U), £ >,S(C,U) ½ A £ B Û A(x) £Y B(x) "xÎX (.. S(C,U) )
TÎObS < S(T, R), £ >,S(T, R) S(X,Y) X=T, Y=R
a,bÎR, a<b < S[a, b], £ >, S[a, b] S(T, R) T = [a,b]
  < S(P,PR), £ >, S(P, PR) S(X,Y) X=P, Y=PR(,B ÎS(P, PR) Þ A£B Û A(p) Þ B(p) "pÎP)
X,YÎObS (o.) A X Y, A:DA Í XY .A XÈY ½xAy & xAz Þ y = z, .. Ax = y & Ax = z, y = z ( . XÈY); (<x, y>ÎA Û xAy Û Ax = y)
X,YÎObS (.), A:XY . A:DA Í XY ½ DA = X
AÎS(X,Y) . xÎC
AÎS(X,Y), xÎX . x, Ax (Û A(x)) yÎY½y =Ax (<x, y>ÎA)
A:XY . . ()
AÎS(X,Y),UÍC U, A(U) A(U) = {yÎY½ y=Ax & xÎU}ÍU yÎA(U) Ü( A(U))Þ y=Ax & xÎU
AÎS(X,Y),VÍY V, A-1(V) A-1(V) ={x ½ AxÎV}ÍX
AÎS(X,Y) . () . , im . X,imA = A(X)
B,CbÎObS"bÎB ., Cb .Cb = {xÎS(B, ÈCb)½x(b)ÎB "bÎB}
  .xÎS(T, R)
  () .h:PR{0, 1}ÍR½ Û h() = 1, Û h() = 0

-1 ( ..) . :

1 ; 2 ; 3 ; 4 ; 5 ;

6 .

-2 ( ) : R, PR.

-3 ( ) ( ) : Rn, Ã (), CON, S (X, Y), S (T, R), S [a, b], S (P, PR).

-4 , BÎ ObS, XbÎ Ã () "bÎB. :

X\ÈXb = Ç(X\Xb), .. ;

X\ÇXb = È(X\Xb), .. .

 

-1 X,YÎ ObS, R X ( XUY). , R: 1 ? 2 ? 3 ? 4? 5 . ? 6 ? 7 X Y?

 

͖2 A, BÎ ObS; O Î{ R, PR, Rn, Ã (), CON, S (X, Y), S (T, R), S [a, b], S (P, PR)}.

, .. :

AÎ O & BÎ O & A > B Þ (A)

AÎ O & BÎ O & A < B Þ (B)

AÎ O & BÎ O & A = B Þ (C)

AÎ O & BÎ O & A B Þ (D)

AÏ O BÏ O Þ (E)

͖3 X,YÎ ObS, AÎ S (X,Y). :

1 , x 2 , Ax 3 U Í X A(U)

4 V Í Y A-1(V) 5 . . , imA.

 

-1 X=Y= N 2; <n,m>R<p,q> Û n + q = p + m. , R: 1 ? 2 ? 3 ? 4? 5 . ? 6 ? 7 X Y?


¨ 1 n + m = n + m Þ <n,m> R <n,m> Þ R ! 2 <n,m> R <p,q> & <p,q> R <v,w> Û n + q = p + m & p + w = v + q Þ n + w = (p + m - q) + (v + q - p) = m + v = v + m Û <n,m> R <v,w> Þ R ! 3 <n,m> R <p,q> Û n + q = p + m Û p + m = n + q Û <p,q> R <n,m> Þ R ! 4 <1,2> R <3,4> & <3,4> R <1,2>, <1,2> ¹ <3,4> Þ R ! 5 R - . N 2 6 R N 2 7 <4,3> R <2,1> & <4,3> R <3,2>, <2,1> ¹ <3,2>, .. R N 2 N 2 ¨

-2 15 -2 .

.
  3.75 -4 , R A
  [a, b] (a, b) ; Ã(R); a,bÎR, a<b A
  A(t)=sint "t B(t)=cost "t ; S[0, 1] D
  5Î{2.75, 5.01} -3ÎZ , PR B
  A =S B & B =S C A(x) =Y (x) "x , A,B,CÎS(X,Y)
  (-¥, 2] \ [-2, +¥) (-1, 1) Ç [0, 10) , Ã(R) D
  [0, 1] ´ [0, 2] [0, 2] ´ [0, 1] , Ã(R2) D
  {x=<x1, x2, x3>ÎR3½ x12 + x22 + x32 < 1} {x=<x1, x2, x3>ÎR3½ x1 < 1} , Ã(R3) B
  g-1([0, 1]) g([0, 1]) , Ã(R); g(x) = (x-1)2 - 1 D
  gf(1) fg(1) ; R; f(t) = sin(t), g(t) = t2 B
  . R3 A
  B
  N B
  D
    1- , PR E

. 5. = A =S B & B =S C, : A,B,CÎS(X,Y); = A(x) =Y (x) "x .

=( =S)Þ A =S Ü( =S)Þ A(x) =Y (x) "x = , Þ Ü( S (P, PR))Þ A £ B. , .. A,B,CÎS(X,Y) , A =S , A ¹S B, , , Ü( S (P, PR))Þ , . , A £ B, ¹ Þ < B, (B) ¨

-3 X = S [0, 1], Y= R, AÎ S (X, Y), Ax = x(0). :

1 , x 2 , Ax 3 U Í X A(U)

4 V Í Y A-1(V) 5 . . , imA.

1 x (t) = cos t 2 Ax = x(0) = cos 0 = 1 3 U = {xÎX½q £ x £ y}Ì X, q(t) = 0 "tÎ[0, 1], y(t) = t + 0.3, tÎ[0, 1] Þ A(U) = {Ax = x(0)½xÎ U } = [0, 0.3] Ì Y 4 V = [-1, 1]ÌY Þ

A-1(V) = {xÎX½Ax = x(0) Î [-1, 1]}Ì X 5 imA = A(X) = R, .. "yÎY (t) º y X A(c) = y ¨

, .

_______________

( ) () <, £>. <, £> . . :

A,BÎO & A>B Þ (A); A,BÎO & A<B Þ (B); A,BÎO & A=B Þ (C);
A,BÎO & A B Þ (D); AÏO BÏO Þ (E)

:

O = < R,£>, A £ B Û B A Î [0, +¥), ;

O = < Ã (X),£> XÎ ObS, A £ B Û AÍB, ;

O = < S(T,R),£> TÎ ObS, A £ B Û A(t) £ B(t) "tÎ T (A=B Û A(t) = B(t) "tÎ T), ;

O = < PR,£> - , h: PR {0,1} , A £ B Û [h(A) £ h(B)] Û [A Þ B] (A = B Û h(A) = h(B) Û [AÛB]), ;

O = < S(P,PR),£> - . P Î ObS,
A £ B Û [hA(p) £ hB(p) "pÎP] Û [A(p) Þ B(p) "pÎP] (A = B Û [hA(p) = hB(p) "pÎP] Û
[A(p) ÛB(p) "pÎP]), ;

O = < CON, £> - , A £ B Û VA Í VB, VA,VB (A = B Û VA = VB), .

  . .R C ½xRy & yRz Þ xRz . . Rn: x = y Û xk = yk "k
  . A:CU . () ., . , S(T, R)
  . .R C ½ xRx "x . C
  . .R C ½ xRy & yRx Þ x=y < S(C,U), £ >,S(C,U)
  A(U)={Ax½xÎU}ÍU V ( A)
  [0,1]´[3,4] A([0,1]´[3,4]) , Ã(R) A(x1,x2) = x1
  A =S B & B =S C A(x) =Y B(x) "x - , A,B,CÎS(X,Y)
  A £S B A(x) <Y B(x) "xÎX , A,BÎS(X,Y)
  A-1([0,1]) [0,1]´[3,4] , Ã(R2) A(x1,x2) = x2
  A({1,5}) Ax = =x(mod5):{1,2,..,20}{1,2,..,5} A-1({1,5}) Ax = x(mod5): {1,2,..,20}{1,2,..,5}

:

1. A

2. A (. , () = C(S(X,Y)). C(B) = C(S(X,Y)) È {X=T, Y= R } Þ C(A) Ì C(B) =( £CON)Þ B < A Þ ())

3. D (. , () = (.. ) È {}. C(B) = (.. ) È È{}, .. C(A) Ë C(B) & C(B) Ë C(A) =( £CON)Þ Þ (D))

4. D

5. D

6. E (. A = [0,1]´[3,4] Î Ã(R2), R, , ())

7. B (. -2)

8. A (. A,BÎS(X,Y) B = A(x) <Y B(x) "xÎX ( ) 1 = A(x) £Y B(x) "xÎX () Ü( £S(X,Y))Þ A £S B = A Ü( S(P,PR))Þ B£A. A , .. A,BÎS(X,Y), A £S B, xÎX, B = A(x) <Y B(x) "xÎX ( ) , B£A B¹A =( )Þ B<A Þ ().

9. D (. = A-1([0,1]) =( )= {xÎ R 2½ AxÎ[0,1]} =(A(x1,x2) = x2)=
={< x1,x2>Î R 2 ½ x2 Î[0,1]} = R ´[0,1]. B = [0,1]´[3,4] Þ AËB & BËA =( £ R2 . Ã(R2)) Þ (D))

10. B (. = A({1,5}) =( )= {Ax ½ xÎ{1,5}} =( . )= {1,5}. = A-1({1,5}) =( )= {xÎ{1,2,..,20}½ AxÎ{1,5}} =(Ax = x(mod5))= {xÎ{1,2,..,20}½ x(mod5)Î{1,5}} = {1, 6, 11, 16, 5, 10, 15, 20} Þ AÍ B & A¹B, .. A<B Þ ())

1. X, Y, Z Î ObS. AÎ S (X,Y), BÎ S (Y,Z). .

1. . .

2. . .

3. . .

4. .

5. : UÍV Þ A-1(U) Í A-1(V).

6. : UÍV Þ A(U) Í A(V).

7. . .

8. "UÍX UÍA-1(AU).

9. "VÍY A(A-1V)ÍV.

10. . .

11. "WÍZ (BA)-1(W) = A-1[B-1(W)].

12. "UÍX BA(U) = B[A(U)].

13. UÍX; VÍY; PÍY; A(U) = V Þ A(U Ç A-1(P)) = VÇP.

( )

, O = < Ã (X),£> ( XÎ ObS, A £ B Û AÍ B) .

-. 1) A ÎÃ(X) =( Í)Þ AÍA Ü( £ Ã (X))Þ A £ A Ü( )Þ £ Ã (X) !

2) A £ B & B £ C Ü( £ Ã (X))Þ AÍ B & BÍ C =( Í)Þ AÍ C Ü( £ Ã (X))Þ A £ C Ü( )Þ £ Ã (X) !

3) A £ B & B £ A Ü( £ Ã (X))Þ AÍ B & BÍ A Ü( = Ã (X))Þ A = B Ü( )Þ £ Ã (X) !

1), 2), 3) Ü( )Þ£ Ã (X) Ü( )Þ O = < Ã (X),£> ¨

 

2. "" ( , , - 1). () (mod12) , .

1. - 1.

-
  ½ xÎ X Þ Î Y , X Í Y
  . S(T, R) T= [a, b] .l
  (o.) A X Y, A:DA Í XY (.), A:XY
  Ax0 x0(t) = t, tÎ[0,1] , tÎ[0,1] (Ax)(t)=ò[0,1]tsx(s)ds
  imA Z - Ax=[x]:RR
  F({xÎC[0,1]| sup|x(t)|£1}) (- ¥, F(1)] Fx=ò[0,1]x(t)dt
  {<1,1>, <1,2>} - {1,2} {<1,1>, <1,2>, <2,2>} - {1,2}
  A =S C A(x) =Y B(x) & B(x) =Y C(x) "x , A,B,CÎS(X,Y)
  x =Ã(X) y & y =Ã(X) z xÍy & yÍz , x, y, z Î Ã(X)
  [0,1]´[3,4] A([0,1]´[3,4]) , Ã(X) A(x1,x2) = x1
  A-1([0,1]) [0,1]´[3,4] , Ã(X) A(x1,x2) = x1
  [3,4] ´ [0,1] A-1([0,1]) , Ã(X) A(x1,x2) = x2

2. - 2.

-
  . , S(P, PR) . , S(C, U)
  (. .) CÈU .
  (..) C,~ . (..) Rn, =
  . () C, £ , ..£ X
  Ax0 x0(t) = t, tÎ[-1,1] , tÎ[-1,1] (Ax)(t)=ò[-1,1]tsx(s)ds
  A([0, 1]) imA Ax=x2 + 1:RR
  S[a, b] C[a, b]
  A =S C A =S B & B =S C , A,B,CÎS(X,Y)
  A £S B & B £S C A(x) £Y C(x) "xÎX - , A,B,CÎS(X,Y)
  im A A([3,4]´R) , Ã(X) A(x1,x2) = x2
  ([0,2] \ (0,2))È([0,2] \ (1,2])È([0,2] \ [0.5, 2)) [0,2] \ ((0,2)Ç(1,2]Ç[0.5, 2)) , Ã(R)
  Ã({1,2,3})³{1, 2, 3, {1,2},{1,3}} Ã({1,2}) = {1, 2, {1,2}, Æ}
         

 

 

3. -3

-
  Rn ., ÇCb
  .{a½a Í } . , S(C, U)
  . C . .R C ½ xRy & yRx Þ x=y
  . . S(C,U):A=BÛA(x)=YB(x) "xÎC, =Y - .. Y . .
  < , £ >, <Ã(X), £ >, Ã(X)
  imA D(A) (Ax)(t)=ò[0,1]tsx(s)ds, tÎ[0,1]
  x(1); x(t) = t Fx; x(t) = t2 Fx=ò[0,1]x(t)dt
  A =S B & B =S C A(x) =Y C(x) "x , A,B,CÎS(X,Y)
  x =Ã(X) y & y =Ã(X) z xÍy & yÍx & yÍz & zÍy , x, y, z Î Ã(X)
  A([0,1]´[3,4]) im A , Ã(X) A(x1,x2) = x1
  [3,4] ´ [0,1] A-1([0,1]) , Ã(X) A(x1,x2) = x2
  [0,2] \ ((0,2)È(1,2]È[0.5, 2)) ([0,2] \ (0,2))Ç([0,2] \ (1,2])Ç([0,2] \ [0.5, 2)) , Ã(R)

4. -4

-
  ., ÈCb .{<x1, x2,¼, xn>½xkÎCk "k}
  . , Ã() , ObS
  . . X = Ã(M): A = B Û A Í B & B Í A . C
  . .A XÈY½xAy &xAzÞy=z, .. Ax = y & Ax = z, y=z ( .. XÈY); (<x, y>ÎA Û xAy Û Ax = y) R Í C´U; <x, y> Î R Û xRy
  . x, Ax (Û A(x)) yÎY½y =Ax (<x, y>ÎA)
  x(1); x(t) = t Fx; x(t) = t2 Fx=ò[0,1]x(t)dt
  {<0,1>, <1,1>} - {0,1} {<0,1>, <1,1>} - {0,1}
  S(T,R) - T 2- , TÎObS
  h(x) =R h(y) x =PR y & y =PR z , x, y, z Î PR
  A £S B & B £S C A(x) £Y B(x) & B(x) £Y C(x) "x , A,B,CÎS(X,Y)
  A-1([0,1]) [0,1]´[3,4] , Ã(X) A(x1,x2) = x1
  ([0,2] \ (0,2))Ç([0,2] \ (1,2])Ç([0,2] \ [0.5, 2)) [0,2] \ ((0,2)È(1,2]È[0.5, 2)) , Ã(R)

 

 

5. -5

-
  . . R C ½ xRy Þ yRx . C
  . .R C ½xRy & yRz Þ xRz . . Rn: x = y Û xk = yk "k
  . . , x Î
  . S(C, U) C=T U=R
  .xÎS(T, R) .h:PR{0,1}ÍR½- Ûh()=1, - Û h() = 0
  A-1(-1, 1) A(-1, 1) Ax=[x]:RR
  imA D(A) (Ax)(t)=ò[0,1]tsx(s)ds,tÎ[0,1]
  A =S B & B =S C A(x) =Y B(x) "x , A,B,CÎS(X,Y)
  x =PR y & y =PR z h(x) =R h(y) & h(y) =R h(z) ,x, y, z Î PR
  A £S C A(x) £Y B(x) & B(x) £Y C(x) "x - , A,B,CÎS(X,Y)
  [3,4] ´ [0,1] A-1([0,1]) , Ã(X) A(x1,x2) = x2
  ZÌ QÌR Z£ Q £ R

6. -6

-
  . S(C, U) C=R (. ) Y=PR .
  .l . S(N, R)
  . . S(C,U):A=BÛA(x)=YB(x) "xÎC, =Y - .. Y . .
  <x, y> Î £ <, £ >, £ -
  A({1,5}) Ax = =x(mod5):{1,2,..,20}{1,2,..,5} A-1({1,5}) Ax = x(mod5): {1,2,..,20}{1,2,..,5}
  [1, 2]´[1, 4] [1, 4]´[1, 2]
  A £S B A(x) <Y B(x) "xÎX , A,BÎS(X,Y)
  A =S B & B =S C A(x) =Y C(x) "x , A,B,CÎS(X,Y)
  x =Ã(X) y & y =Ã(X) z xÍy & yÍx & yÍz & zÍy , x, y, z Î Ã(X)
  A([0,1]´[3,4]) im A , Ã(X) A(x1,x2) = x1
  A-1([0,1]) [0,1]´[3,4] , Ã(X) A(x1,x2) = x2
  [0,2] \ ((0,2)È(1,2]È[0.5, 2)) ([0,2] \ (0,2))Ç([0,2] \ (1,2])Ç([0,2] \ [0.5, 2)) , Ã(R)

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