. g○f f: S → T g: T → S 1S S, , g f, f g, f - , g - . , , f ○ g = 1T, , g f , , f g. , , .
2. , . , .
. , - . g: T → S , f:S → T. , f (s) = f (s'). ,
s=1S(s) =g(f ( s )) = g (f (s')) = 1S(s') = s'
, , f (s) = f (s' ), , s = s'. 풺 f. , : , f 풺, g 1 , g 1 f = 1S. s1 ÎS g 1:T → S :
(g 1 f)(s)= g 1(f (s)) = s sÎS, , , g 1 f = 1S g 1 f.
. tÎT fh = 1T, t = 1T(t) = f (h (t)), - tÎT f (s) s = h (t) Î S, f . , f:S → T , tÎT t = f (s) s Î S. t Î T s, t, , , h (t). h:T → S,, f (h (t)) = t tÎT, fh = 1T, . .
. f: S → T ᳺ , .
3. f: S → T :
(i) f ᳺ;
(ii) f g h.
,
(iii) f (, ) . f f 1, ᳺ,
(f 1) 1 = f. (2)
. (i) (ii) . (ii)
g = g 1T = g (fh) = (gf) h = 1s h = h.
, f , (iii). , ᳺ f g = h, gf =1s fh = 1T. , ᳺ f , (f 1)1 = f.
. - f: S → S f 2 = f f = f ◊ f, f. f 2 = f, f .
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, (x, y)- x- y- . ³ , f○g=g○f (, , g ◊ f = f ◊ g), f○g g○f - (0,0).
. f, g fg=gf .
- g ÎS2 f Î S2, fg = 1S. , f h , fh = 1S. 2, f , , f 풺. , f . , f , f - .
ᳺ , ( ) . , f, f 1. , f ᳺ,
f 1 f = f f 1 = 1s ( ).
. : ; ; , . ֳ , , , .
. ϳ - ᳺ .
ϳ . , .
|| = n Î N. nPn . nPn . ᳺ , 1 n, a 1,..., a n (.8). , , (- ). . n , . , n 1
1 2 n 1 n
... |
. 8.
. , n 1 , n 2 .. , , (n - 1)- , n , . ,
n × (n1) × (n - 2) ×... ×3 × 2 ×1.
n ( n!). , nPn = n!.
Nn n N. Nn , Nn. - Nn Nn (, , ). Nn. n :
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{(1, x1), (2, x2),...,(n, xn)}, {x1,...,xn} = Nn.
, , x1 = 1 .. :
1. N6:
s(1) = 5, s(2) = 6, s(3) = 3 ..
, . , - Nn, , . , x1, x2,...xn. ( ), .
2. 1
r
r
r
r s :
r
r s
, , r s(2) = r(s(2)) = r(6) = 5 . . , () , . , , , . , .
. = {1,,n}. ϳ r ( ),
r
, Í . r , s ,
r(x), Î ,
s: x, Î \ .
s r , . s , , , (a1, a2,,n). n.
3.
r
ϳ 5 (1, 3, 6, 5, 4).
. , s 1 . , s
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s(1) = 5, s(5) = 4, s(4) = 1, , s (1, 5, 4). 2, (2, 6). ,
s = (1, 5, 4) (2, 6) s = (2, 6) (1, 5, 4).
- , - . , , .
4. r , .
. | A | = n Î N, , Nn. r Nn. , r = s1 s2 sr, si, i = 1,2,,r, . .
1Î Nn , r(1) ¹ 1 r() = , 1 £ < 1. 1 , r = I ( r ). 1, r(1), r2(1), r3(1) . . Nn. . , rk(1) . , rk(1) = 1. , . rl(1) = rk(1) l, 0<l<k. rl-1(1) = r-1 rl(1) = r-1 rk(1) = rk-1(1) . .
rl-1(1) = rk-1(1), rk-l(1) = r0(1) = 1, k ( k-l < k). , rk(1) = 1 s1 = (1, r(1), r2(1),,rk-1(1)) r.
Î N n , r() ¹ ( ), s1, r = s1 ( ). 2 Î N n , r(2) ¹ 2 2 s1. 2 r: s2 = (2, r(2), r2(2),, rm(2)). 2 s1.
, r = s1 s2 = s2 s1. , , , , Æ. , r = s1 s2 s3 sr r Î N.
: |A| = n B Í A, |B| = r £ n, ᳺ , , 풺 .
( ) n r nPr , nPn, , r . ,
nPr = n×(n1)××(nr+1).
, , , n r, , n r n-rPn-r .
nPr = (nPn)/ (n-rPn-r) = n! / (nr)!
nPr ᳺ . ϳ .
. B Í A, || = n ³ r = |B |. 璺 ( ) n r. 璺 rn.
rn . |A| = n. ³ B Í A, || = r. n r. 풺 , B , nPr. f g , , g f g = jf, j .
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g f , , , j B. , nPr = rn× rPr, rn = (nPr)/(rPr) = n!/r!×(nr)!
| A \ B | = n r, , rn= nrn.
ᒺ.
. S N Z.
s: N Z s(n) = sn, s, (sn) (s1,s2,,sn,). sn n- .
, , . , .
. , C. [ C ] C. F: [ B C ] .
4. , , . I , . ( ) P [ I O ]; Î , i Î I Î .
5. R. f: a[xa + x], f(2): x2 + x f(2)(3) = 5, f(3): x3 + x f(3)(3) = 6 . .
, .
, . .
. , r. r r - ; r- /r.
. Y , r ry f: ծY . `f `f: X/r Y/ry , `f ={([x], [f(x)]): x Í X}, [x] . `f 1r2 Þ`f([1]) =`f([2]) f , . , f: X Y `f: X/r Y/ry.
. 9.
f, , 1 1 2, y2 = f(2) 1r2, y2, y1 = f(1) y2ry1.
6. = {1, 2, 3}, Y = {1, 4, 9} r ry , X/r = {{1},{2,3}}, Y/ry = {{1},{4,9}}, f: X Y , 2.
`f([1]) = [f(1)] = [1] = {1},
`f([2]) = [4] = {4, 9},
`f([3]) = [9] = {4, 9}.
{2,3} Î X/r Þ 2r3 Þ [2] = [3] `f ([2]) =`f([3]). `f f .
7. , Y f , 6, s x sy {{1},{2,3}} {{1,4},{9}} .
`f([2]) = [f(2)] = [4] = {1,4},
`f([3]) = [f(3)] = [9] = {9}.
, 2s3, [2] = [3] X/s , (4, 9) Ï sr, [4] ¹ [9] Y/s r. .9 , .10. , 璺 , .
rx s
x1 x2 2 3
f f s
f sr f
y1 y2 4
ry 4 sr 9
. 9 10