Аналитическая геометрия. Часть I. Шурыгин В.В. - 7 стр.

UptoLike

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M N
y ∈ N x ∈ M
y = f(x) f
−1
: M → N
y ∈ N x ∈ M
f : M → N
f : R 3 x 7→ x
3
∈ R
f : R 3 x 7→ 2
x
∈ R
+
R
+
= {x ∈ R : x > 0}
a ∈ M f : M → N
f
−1
(a) = {x ∈ M : f(x) = a} ⊂ M
A ⊂ M f : M → N
f
−1
(A) = {x ∈ M : f(x) ∈ A} ⊂ M
f : R 3 x 7→ x
3
− x ∈ R f
−1
(0)
f
−1
(0) = {−1; 0; 1} f
−1
(R
+
) = (−1; 0)∪(1; ∞)
M R ⊂ M ×M
(x, y) ∈ R x R y
(x, y) ∈ R xRy R M
R xRx x ∈ M
R xRy ⇒ yRx
R xRy yRz ⇒ xRz
∼
x y ∼
x ∼ y [x] = {y ∈ M : y ∼ x}
M x ∈ M
x ∈ M [x] [z]
w ∈ [x] ∩[z]
y ∈ [z] x ∼ w w ∼ z z ∼ y x ∼ y y ∈ [x]
y [z] [z] ⊂ [x]
[x] ⊂ [z]
∼ M
;*|L2 ~7*;*,86;0 ;,-|*/8b M 0 N = { ~8-; /72)6* 16|L-;2 ~7*;*,K
82 y ∈ N /--8b*8/8b2*8 -L0, 0 8-7?1- -L0, ~7*;*,8 x ∈ M 861-M\ )8-
             H
y = f (x) = 30 ~8-0 -8-+36|*,0* f : M → N \ -8,-/9}** 16|L-;2 ~7*K
                                     −1
;*,82 y ∈ N *:- *L0,/8b*,,M .3--+36[ x ∈ M \ ,6[b6*8/9 Š¬ž‘’
ŠžŠ¬»ŽŒŽ’ 1 -8-+36|*,0J f : M → N = —[ 36//;-83*,,N bI* .30K
;*3-b\ +0*180b,; 9b79*8/9 8-7?1- -8-+36|*,0* f : R 3 x 7→ x ∈ R =
Â0*180b,; 9b79*8/9 861|* -8-+36|*,0* f : R 3 x 7→ 2 ∈ R \ :L*
                                                                      3


                                                                     )
                                                                x         +

R = {x ∈ R : x > 0} · ;,-|*/8b- .-7-|08*7?,N b*}*/8b*,,N 0/*7 =
  +
   ›Š‹‘’ ¡ŠŠ¬¢Š’ ˎ’Žž a ∈ M .30 -8-+36|*,00 f : M → N
,6[b6*8/9 ;,-|*/8b- f (a) = {x ∈ M : f (x) = a} ⊂ M = ›Š‹‘’ ¡Šº
Š¬¢Š’ ¡Šª’Š»Žž” A ⊂ M .30 -8-+36|*,00 f : M → N ,6[b6*8/9
                           −1



;,-|*/8b- f (A) = {x ∈ M : f (x) ∈ A} ⊂ M = €6.30;*3\ L79 -8-+36K
             −1
|*,09 f : R 3 x 7→ x − x ∈ R .-7,M .3--+36[ f (0) /-/8-08 0[ 83*N
                       3                               −1
~7*;*,8-b† f (0) = {−1; 0; 1} \ 6 f (R ) = (−1; 0)∪(1; ∞) · -+*L0,*,0*
             −1                    −1     +
Lb2N 0,8*3b67-b=
   ¿ žŠÄŽŒŽ’  ’Š»Žž”Ž M ,6[b6*8/9 .-L;,-|*/8b- R ⊂ M × M =
¼/70 (x, y) ∈ R \ 8- :-b-398\ )8- ~7*;*,8 x ,6N-L08/9 b -8,-I*,00 R 1 y =
{;*/8- (x, y) ∈ R .0I28 861|* xRy = ˆ8,-I*,0* R ,6 ;,-|*/8b* M ,6[K
b6*8/9 ŠžŠÄŽŒŽ’ ɔŒ”‹ŽžŠžŒ\ */70 -,- 2L-b7*8b-39*8 /7*L2J}0;
83*; 2/7-b09;†
   •ƒ R · 3*O7*1/0b,-\ 8- */8? xRx L79 7J+-:- x ∈ M Å
   ]ƒ R · /0;;*830),-\ 8- */8? xRy ⇒ yRx Å
   ƃ R · 836,[080b,-\ 8- */8? xRy \ yRz ⇒ xRz =
   ˆ)8,-I*,0* ~1b0b67*,8,-/80 )6/8- -+-[,6)6*8/9 /0;b-7-; ∼ = H30 ~8-;
8-\ 8- ~7*;*,8 x 0 y ,6N-L98/9 b -8,-I*,00 ~1b0b67*,8,-/80 ∼ \ -+-K
[,6)6*8/9 /7*L2J}0; -+36[-;† x ∼ y = °,-|*/8b- [x] = {y ∈ M : y ∼ x}
b/*N ~7*;*,8-b 0[ M \ ~1b0b67*,8,N ~7*;*,82 x ∈ M \ ,6[b6*8/9 ‰‹º
Š’ ɔŒ”‹ŽžŠžŒ ˎ’Žž x ∈ M = ˆ)*b0L,-\ Lb6 176//6 [x] 0 [z]
70+- ,* .*3*/*16J8/9\ 70+- /-b.6L6J8= ‡*M/8b08*7?,-\ */70 w ∈ [x] ∩ [z] 0
                                    ­
y ∈ [z] \ 8- x ∼ w \ w ∼ z 0 z ∼ y = 7*L-b68*7?,-\ x ∼ y \ 0 .-~8-;2 y ∈ [x] =
H-/1-7?12 y · .3-0[b-7?,M ~7*;*,8 0[ [z] \ 8- [z] ⊂ [x] = Ç6//2|L69 6,6K
7-:0),; -+36[-;\ .30N-L0; 1 8-;2\ )8- [x] ⊂ [z] = ˆ8/JL6 /7*L2*8\ )8- -8K
,-I*,0* ~1b0b67*,8,-/80 ∼ ,6 ;,-|*/8b* M 36[+0b6*8 ~8- ;,-|*/8b- ,6
                                       È

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