Аналитическая геометрия. Шурыгин В.В. - 6 стр.

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M
1
, M
2
, . . . , M
n
n ∈ N
M
1
× M
2
× . . . × M
n
= {(x
1
, x
2
, . . . , x
n
) : x
i
∈ M
i
, i = 1, 2, . . . , n}.
M
i
i = 1, 2, . . . , n
M
i
= M i = 1, 2, . . . , n
M
n
R
2
= {(x, y) : x ∈ R, y ∈ R}
M N M N
f x ∈ M
y ∈ N x f y
x f
y = f(x) x 7→ y f
f : M → N f : M 3 x 7→ y ∈ N
f : M → N
M N y ∈ N
y ∈ N y = f(x) f : R 3 x 7→
x
2
∈ R f : R 3 x 7→ 2
x
∈ R
f : R 3 x 7→ x
3
∈ R
f : R 3 x 7→ x
3
−x ∈ R
x
3
= a x
3
− x = a a
f : M → N
M N x
1
6= x
2
f(x
1
) 6=
f(x
2
) ⇒
x
1
6= x
2
⇒
f(x
1
) 6= f(x
2
) f : R 3 x 7→ x
2
∈ R
f : R 3 x 7→ x
3
− x ∈ R
f(−a) = f(a) a ∈ R
f(−1) = f(0) = f(1) = 0 f : R 3 x 7→ x
3
∈ R
f : R 3 x 7→ 2
x
∈ R
f : M → N
M N
5,67-:0),- .3-0[b*L*,0J ‚•ƒ Lb2N ;,-|*/8b -.3*L*79*8/9 .3-0[b*L*,0*
7J+-:- 1-,*),-:- )0/76 ;,-|*/8b M , M , . . . , M \ n ∈ N †
                                                         1       2       n


    M × M × . . . × M = {(x , x , . . . , x ) : x ∈ M , i = 1, 2, . . . , n}.
         1       2                   n       1   2           n       i       i
                                                                                ‚]ƒ
{ )6/8,-/80\ */70 b/* ;,-|*/8b6 M \ i = 1, 2, . . . , n \ b .3-0[b*L*,00 ‚]ƒ
/-b.6L6J8\ 8- */8?\ M = M \ i = 1, 2, . . . , n \ 8- 0N .3-0[b*L*,0* -+-[,6)6K
                                                         i


*8/9 161 /8*.*,? M = €6.30;*3\ R = {(x, y) : x ∈ R, y ∈ R} · )0/7-b69
                             n
                                     i
                                                     2
.7-/1-/8?=
    H2/8? M 0 N · ,*1-8-3* ;,-|*/8b6= ¿žŠ¬»ŽŒŽ’ Œ¢ M ” N ,6[K
b6*8/9 /--8b*8/8b0* f \ -8,-/9}** 16|L-;2 ~7*;*,82 x ∈ M ,*1-8-3M ~7*K
;*,8 y ∈ N \ ,6[b6*;M Š¬¢Š’ ~7*;*,86 x .30 -8-+36|*,00 f = „-\ )8- y
9b79*8/9 -+36[-; ~7*;*,86 x .30 -8-+36|*,00 f \ -+-[,6)6*8/9 /7*L2J}0;
-+36[-;† y = f (x) 070 x 7→ y = ‡79 /6;-:- -8-+36|*,09 f .30 ~8-; 0/.-7?K
[2J8/9 /7*L2J}0* -+-[,6)*,09† f : M → N 070 f : M 3 x 7→ y ∈ N =
    ˆ8-+36|*,0* f : M → N ,6[b6*8/9 À½Ž‰žŒ”‘’ 070 ŠžŠ¬»Žº
ŒŽ’ M  N \ */70 L79 16|L-:- y ∈ N ,6ML*8/9 N-89 + -L0, ¡ŠŠ¬¢\
8- */8? 861-M y ∈ N \ )8- y = f (x) = €6.30;*3\ -8-+36|*,09 f : R 3 x 7→
x ∈ R 0 f : R 3 x 7→ 2 ∈ R ,* 9b79J8/9 /J3*180b,;0\ .-/1-7?12 2
-830 68*7?,N )0/*7 ,*8 .3--+36[-b\ 6 -8-+36|*,09 f : R 3 x 7→ x ∈ R
 2                                       x

                                                                                         3
0 f : R 3 x 7→ x − x ∈ R 9b79J8/9 /J3*180b,;0\ .-/1-7?12 236b,*,09
                         3

x = a 0 x − x = a 0;*J8 3*I*,09 .30 7J+-; a =
                                                                               ªŠº
 3           3

    ˆ8-+36|*,0* f : M → N ,6[b6*8/9 Œ½Ž‰žŒ”‘’ 070 ”¢Œ’ŠŠ
¢¤‘’ ŠžŠ¬»ŽŒŽ’ M ” N \ */70 0[ x 6= x /7*L2*8\ )8- f (x ) 6=
        {
f (x ) = ;*/8- /7-b6 4/7*L2*8< +2L*; 0/.-7?[-b68? 861|* /0;b-7 ⇒ = /7-K
                                                                     1       2
                                                                              (               1


b0* 0,*180b,-/80\ 8610; -+36[-;\ ;-|,- [6.0/68? b b0L* x 6= x ⇒
     2

                  —
f (x ) 6= f (x ) = [ 36//;-83*,,N bI* -8-+36|*,0M\ f : R 3 x 7→ x ∈ R
                                                                                 1       2
                                                                                          2
0 f : R 3 x 7→ x − x ∈ R ,* 9b79J8/9 0,*180b,;0\ .-/1-7?12
     1           2


b .*3b-; /72)6* f (−a) = f (a) L79 7J+-:- a ∈ R \ 6 b- b8-3-; /72K
                                 3


)6* f (−1) = f (0) = f (1) = 0 = ˆ8-+36|*,09 f : R 3 x 7→ x ∈ R 0                    3

f : R 3 x 7→ 2 ∈ R 9b79J8/9 0,*180b,;0 =
    ˆ8-+36|*,0* f : M → N ,6[b6*8/9 ¬ŒŽ‰žŒ”‘’ 070 ”¢Œ’ŠŠªŠº
                     x



¢¤‘’ ŠžŠ¬»ŽŒŽ’ M  N \ */70 -,- 9b79*8/9 0 /J3*180b,; 0
0,*180b,;\ 8- */8? 2/86,6b70b6*8 b[60;,- -L,-[,6),-* /--8b*8/8b0*
                                      Á

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