Основные структуры классической математики
.pdf20. 0 ) A B
* !: A=B, AÌ B, BÌ A, A B
, A B 4 . $
, ) A\B, B\A AÇB . 0 )
* ) 2 * ) 4 . 4% , n ) *
). . 9- ) A, B, + * ) 8 . (
.)
)
! |
! +′ |
!′ +′ |
|
! ′+′ |
|||
|
|
||
|
! + |
|
|
! ′+ |
!′ + |
!′ ′+′ |
!′ ′+ |
|
|
|
+ |
0 ), )
16 . - , * 1
32 ( . [9, . 538]).
21. # ! ) )
n ) A1, A2, …, An,
2 )? $),
n=3 18 ). $ n=4 – 166 ) [6, 14]. 22. 0 ), n ), * *
), * 2,
22 ).
23.: ! ), n
) 2,
?
24.)
B(A) ) ) f: A®{0, 1},
* ') f-1(1).
31
1.2. 0 & 1 (
. ( ) )
) A B
( ) ρ ) 1 a A b B. , 1 a b ρ, ! aρb ), a b
! ρ. $ A=B ! ρ .
A. & %, . )
) A B ) ρ
A×B, . . )
(a; b), a A, b B36. 4 ) ) %
! )
ρ ≡ {(a; b): aρb}.
" ! ) , %,
%. " ! ρ ) ) AB / %.
) A B )
9- , aρb, a A b B, a b
a→b.
*. $ A = {1, 2, 3, 4, 5} B )
%. ( ! ρ ) )
A B : aρb , a
|
b. 9 |
|
! |
) |
|
|
|
||
0, 1– 5×9: |
|
|
|
|
|
|
|||
ρ |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
1 |
2 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
1 |
0 |
3 |
0 |
0 |
1 |
0 |
0 |
1 |
0 |
0 |
1 |
4 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
5 |
0 |
0 |
0 |
0 |
1 |
0 |
0 |
0 |
0 |
! " #
$ ! ρ ) ) A, B σ ) ) B, C. " ! ρ-1 )
) B A ! ρ,
36 $ 1 ! ρ ) A, B. 4 * ) ' !.
32
"aÎ A"bÎ B(br-1a Û arb).
1 ! r s !
rs=s°r ) ) A C, %
"aÎ A" Î +(ars Û $bÎ B(arb & brc))
( « » a®c, *
a®b b®c bÎ B).
ρσ
! |
ρ |
σ |
|
|
+ |
) ! )
) A B, . . ) ) A´B.
B(A´B), !
Í 2 È, Ç
¢. - , r, sÎ B(A´B) : rÍ s Û "aÎ A"bÎ B(arb asb),
rÈs = {(a; b): arb Ú asb}, rÇs = {(a; b): arb & asb}, r′ = {(a; b)Î A´B: arb}.
! r )
) A B. -%# ! r
)
D(r) = {aÎ A: $bÎ B arb},
, ! r
)
Im r37 = r(A) = {bÎ B: $aÎ A arb} = D(r¢).
! r :
# , D(r)=A;
37 D, Im – domain ( ) image ( ).
33
38,
a A b1, b2 B(aρb1&aρb2 b1=b2);
6,
a1, a2 A b B(a1ρb&a2ρb a1=a2);
# 6, Imρ=B;
' %, ;
, 2, 2
%.
! ρ )
) A B % , )
) A ( B).
! ρ , ) )
A . -2 ρ
, ) A
* « » B, 2 – , , )
) B ) .
! ρ , ) ) A
.
. * : ! ρ 2,
! ; ! ρ-1 %, 2, 2 ;
! σ ρσ 2; ! σ-1 (ρσ)-1=σ-1ρ-1
! .
!
$ % ! ρ )
) A B ' A B,
A B, ρ: A → B. + , ) ) A
) B – 1 A, f, B , f: A → B – % . + «)» «%»
.
())
f, g, h ( 1 ).
$ f: A → B – %. $ a A * 1 b B, afb,
b=f(a)=af 2 a f.
38 ! ρ * ' A B.
34
0 CÍ A f(C) )
1 ) C ( f). , ) CÍ B,
C ( f) )
f-1(C)= f ¢(C)={aÎ A: afÎ C}; , C={ }
f-1(c) 1 . $), f
% g: B ® C h: C ® D. #39 fg: A ® C
: a(fg)=(af)g aÎ A. ' ,
% %,
, ( ): aÎ A
((fg)h)=( (fg))h)=(( f)g)h)=( f)(gh)= (f(gh)),
. . (fg)h=f(gh) % A ® D.
$ f: A ® B – ) CÍ A. ) f C: C ® B,
* f C, ( ) f C.
) f: A ® B :
6, * ! f 2, . . f(a1)¹f(a2) a1¹a2 A;
# 6() ), ! f 2;
( ),
! f .
0 f: A ® B *
( ) f-1: B ® A, !:
"aÎ A"bÎ B(f-1(b)=a Û f(a)=b).
f
!
f-1
1A A ® A,
a1A=a aÎ A. ;, ! 1A
! ) A. , f: A ® B – , ff-1=1A f-1f=1B ( ).
39 # * ,
(%). % ! , – 1
. - af f(a) ! af f.
35
$ ! 16
* !:
, , 2 2:
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ρ |
|
|
|
|
- |
|
|
- |
|
|
- |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ρ A B |
< |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1. |
+ |
|
+ |
|
+ |
|
+ |
|
|
|
||||
2. |
+ |
|
+ |
|
+ |
|
|
– |
|
2 |
||||
|
|
|
|
|
|
2 |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3. |
+ |
|
+ |
|
|
– |
+ |
|
|
2 |
||||
|
|
|
|
|
|
2 |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4. |
+ |
|
+ |
|
|
– |
|
– |
|
2 |
||||
|
|
|
|
|
|
2 |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
% |
5. |
+ |
|
|
– |
+ |
|
+ |
|
|
A={a}, B={b, c} |
||||
|
|
|
|
|
|
ab ac |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
6. |
+ |
|
|
– |
+ |
|
|
– |
|
A={a}, B={b, c, d} |
||||
|
|
|
|
|
|
ab ac |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
7. |
+ |
|
|
– |
|
– |
+ |
|
|
A={a, d }, B={b, c} |
||||
|
|
|
|
|
|
ab ac db |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
8. |
+ |
|
|
– |
|
– |
|
– |
|
A={a, d }, B={b,c,e} |
||||
|
|
|
|
|
|
ab ac db |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
9. |
|
– |
+ |
|
+ |
|
+ |
|
|
A={a, c}, B={b}, ab |
||||
10. |
|
– |
+ |
|
+ |
|
|
– |
|
A={a, d }, B={b, c} |
||||
|
|
|
|
|
|
ab |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
11. |
|
– |
+ |
|
|
– |
+ |
|
|
A={a, c, d}, B={b} |
||||
|
|
|
|
|
|
ab cb |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
12. |
|
– |
+ |
|
|
– |
|
– |
|
A={a, c, d}, B={b,e} |
||||
|
|
|
|
|
|
ab cb |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
13. |
|
– |
|
– |
+ |
|
+ |
|
|
A={a, d }, B={b, c} |
||||
|
|
|
|
|
|
ab ac |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
14. |
|
– |
|
– |
+ |
|
|
– |
|
A={a, d }, B={b,c,e} |
||||
|
|
|
|
|
|
ab ac |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
15. |
|
– |
|
– |
|
– |
+ |
|
|
A={a, c, d}, B={b,e} |
||||
|
|
|
|
|
|
ab ae cb |
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
16. |
|
|
|
|
|
|
|
|
|
|
|
|
|
A={a, c, d} |
|
|
– |
|
– |
|
– |
|
– |
|
B={b, e, m} |
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
ab ae cb |
* . A B.
* ! ρ. $ . 1. $), !
,
! 2.
36
2.0 ), ! r
2 ! r-1.
3.$ , ) f
2 g 2 h. : 1 f, 2 g, 2 h?
f
A
|
|
Im f |
B |
|
|
|
|
g |
C |
h |
|
|
|
||
|
|
|
4. $ ! r, s )
) A B d, g ) ) B +. 0 )
* !:
rÍ s Û r-1Í s-1 Û s¢Í r¢; 1Ar=r=r1B;
(r-1)-1=r; (rd)-1=d-1r-1;
rÍ rr-1r, , r=rr-1r Û r 2; (rÈs)d=rdÈsd; (rÇs)dÍ rdÇsd; r(dÈg)=rdÈrg; r(dÇg)Í rdÇrg.
5.# * !, s, d, g
* )?
6.$ r – ! ) ) X, Y
A, BÍ X. 0 ) * !:
) r(AÈB)=r(A)Èr(B);) r(AÇB)Í r(A)Çr(B);
) ("C, DÍ X r(+ÇD)=r(C)Çr(D)) Û r 2.
7. ) * )
) X? |
f −1(A ) |
8. 0 ) f-1( Ai) = |
|
i I |
i |
i I |
) f: X ® Y (Ai)i I )
) Y.
37
9. $' % . " ! r
) ) A B %,
a, cÎ A, b, dÎ B arb, ard, crb crd ( ,
rr-1rÍ r). ! % ! )
) ( . [1, . 17]).
10. 7 . 0 ), !
)
.
$ #
$ ! r ) A ) * *
("a, b, cÎ A):
1) %: ara ( . . 1AÍ r); 2) : arb bra (r-1Í r);
3) : arb&brc arc (rrÍ r);
4) %: ara ( , ara);
5) : arb&bra a=b (rÇr-1Í 1A). + ) 25=32 1 . .
23=8 )
% %. $1
32–8=24 . , ! r
, )
1, . . ) ( ) )
! 1A; ! . $1
* « » , ) *
, –
. $ 21 ) . (,
( ) ) !, *
. $ !
– , %. (.
) ! !, *
.) !
, %. ,
r % (%), r=1A (r=Æ). 0, % !
, ) * %
!, * , –
. 19 ,
38
) , . . . 4 *
).
. ) !
! 1, , ,
( ). " ! )
:
! 2 ( 2 %#,
), %, ;
! ( ), %,
;
! ( ),
% ;
! ( , ),
% ;
! * (*, ),
% .
|
4 |
|
|
- |
|
|
- |
|
|
- |
|
|
- |
|
|
- |
|
|
|
|
|
|
r |
|
|
|
|
|
|
|
|
|
|
|
|
! ρ |
|
||||||
|
|
|
|
|
|
|
|
|
|
- |
|
|
- |
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
< |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1. |
|
+ |
|
+ |
|
+ |
|
|
|
+ |
|
+ |
|
|
|
||||||
2. |
|
+ |
|
+ |
|
+ |
|
|
|
+ |
|
|
– |
|
|
||||||
3. |
|
+ |
|
+ |
|
+ |
|
|
|
– |
+ |
|
|
1A |
|||||||
4. |
|
+ |
|
+ |
|
+ |
|
|
|
– |
|
– |
|
1¹1A |
|||||||
5. |
|
+ |
|
+ |
|
|
– |
|
|
+ |
|
+ |
|
|
|
||||||
6. |
|
+ |
|
+ |
|
|
– |
|
|
+ |
|
|
– |
|
|
||||||
7. |
|
+ |
|
+ |
|
|
– |
|
|
– |
+ |
|
|
|
|||||||
8. |
|
+ |
|
+ |
|
|
– |
|
|
– |
|
– |
|
(aabbacca), |
|||||||
|
|
|
|
|
|
|
|
|
|
1 |
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
9. |
|
+ |
|
|
– |
+ |
|
|
|
+ |
|
+ |
|
|
|
||||||
10. |
|
+ |
|
|
– |
+ |
|
|
|
+ |
|
|
– |
|
|
||||||
11. |
|
+ |
|
|
– |
+ |
|
|
|
– |
+ |
|
|
, |
|||||||
|
|
|
|
|
|
|
|
|
|
) * ab |
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
12. |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
||
|
|
|
+ |
|
|
– |
+ |
|
|
|
– |
|
– |
|
|
||||||
|
|
|
|
|
|
|
|
|
|
1 |
|||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
(aabba cc) |
|
13. |
|
+ |
|
|
– |
|
– |
|
|
+ |
|
+ |
|
|
|
||||||
14. |
|
+ |
|
|
– |
|
– |
|
|
+ |
|
|
– |
|
|
||||||
15. |
|
+ |
|
|
– |
|
– |
|
|
– |
+ |
|
|
aabbcc |
|||||||
16. |
|
+ |
|
|
– |
|
– |
|
|
– |
|
– |
|
baabbcc |
|||||||
17. |
|
|
– |
+ |
|
+ |
|
|
|
+ |
|
+ |
|
Æ |
|
||||||
18. |
|
|
– |
+ |
|
+ |
|
|
|
+ |
|
|
– |
|
|
||||||
19. |
|
|
– |
+ |
|
+ |
|
|
|
– |
+ |
|
|
ƹrÌ1A |
|||||||
20. |
|
|
– |
+ |
|
+ |
|
|
|
– |
|
– |
|
baabb c |
|||||||
21. |
|
|
– |
+ |
|
|
– |
|
|
+ |
|
+ |
|
|
|
39
22. |
– |
+ |
– |
+ |
– |
aba |
23. |
– |
+ |
– |
– |
+ |
|
24. |
– |
+ |
– |
– |
– |
aaba |
25. |
– |
– |
+ |
+ |
+ |
|
26. |
– |
– |
+ |
+ |
– |
|
27. |
– |
– |
+ |
– |
+ |
aab |
28. |
– |
– |
+ |
– |
– |
aabba cd |
29. |
– |
– |
– |
+ |
+ |
abc |
30. |
– |
– |
– |
+ |
– |
abac |
31. |
– |
– |
– |
– |
+ |
abbc |
32. |
– |
– |
– |
– |
– |
abbac |
*. # * ,
! r
) A. +, abbac ( 32) ,
A = {a, b, c} r = {(a; b), (b; b), (b; a), (a; c)}.
! # %
! 1 )
% ) *
. ! « »
) 1, ! «
» ! )
, * 1. !
1 ) !
) ) ( ,
, ). . $ 1 !
), .
) A )
* ) ) A, 2
A. - , )
A – 1 ) S=(Ai)i I, , |
|
|
|
|
|
|
||
1) |
Ai ¹ Æ iÎ I; |
a∙ b∙ |
|
|
|
|
|
! |
|
|
|
|
|
||||
2) |
AiÇAj = Æ i¹j I; |
|
|
!5 |
||||
|
|
|||||||
3) |
Ai = A. |
!4 |
!1 |
!6 |
||||
|
i I |
!3 |
∙c |
|||||
) Ai |
|
|
|
|
|
|
||
|
|
|
|
|
|
|||
. |
!2 |
|
|
!7 |
|
|||
) A |
|
), |
||||||
|
||||||||
) ) A 1 A |
. $ (Ai)i I
) A. 0 1 x, yÎ A ):
40