. .
N- G⊂n
1 ; G ¥V a M (,) G
; ¥V a M (,) G
2 ; G (a,b) G=> (b,a) G
; (a,b) G, (b,a) G a=b
3 ; G (a,b) (b,c) G => (a,c) G
G M , , , a ~ b
: G , m=UMi, Mi a,b Mi <=> a~b
: Ga={b:a~b}
1 U Ga =M .. a~a, a Ga
2 Ga∩Gb≠0 => Ga=Gb
Ga∩Gb, d Ga, a~d, a~c, b~d?
b~c, c~a => b~a, a~d => b~d => Ga⊂Gb
3 c~d, d Gc, c Gc
4 c,d Ga c~a, d~a, c~a, a~d => c~d
G , , , .
C\{(a,a):a M}
G , , , .
CU{(a,a):a M}
.
f(A) B .
f: A -> B f(A)=B
f(a1)=f(a2)=>a1=a2
(fog)(a1)=(fog)(a2)
g(f(a1))=g(f(a2)) f(a1)=f(a2) a1=a2
f:A->B g:B->A fog=iA, gof=iB => g=f -1
f -1=g -1 (fog)-1=g -1of -1
(fog)-1*(g -1of -1)=fo(gog -1)of -1=(foiB)of -1=fof -1=iA
: f f -1
1 fof -1=iA f -1of=iB
(f -1of)(b)=b
f(f -1(b))=b
f(a)=b
2 f(a1)=f(a2) f -1(f(a1))=f -1(f(a2)) a1=a2
.
f: A->B
:
1 iA: A->A
2 f:A->B
f -1: B->A
3 f:A->B
g:B->C fog: A->C
5. , 1<, ||=|1|
: ,
1 A1⊂A, A1=A
2 => |A|=|B|
3 , 2=g(1) fog: A->C2
|A|<|B|, ||<=||, ||≠||
{x} , , , .
|
|
: |N|<=|[0,1)|
1 |N|<=|[0,1]|
1 ->0.10
2->0.010
3->0.0010
2 |N|≠|[0,1)|
f:N->[0,1]
f(1)=0,a11a12a1n
f(2)=0,a21a22a2n
f(n)=0,an1an2ann
b1={1, a11≠1 b2={1, a22≠1
{2,a11=1 {2,a22=1
0,b1b2bn
.
, f:N->A
f(1)=a1
f(n)=an
A={a1,a2,,an}
1
2
3
4 , , |U|=||