Martynyuk_A_N_Diskretnaya_matematika

(x1 x2) ( x1 x2) (( x1 x2) (x1 x2)) ((x1 x2) ( x1 x2)) ((( x1 x2) (x1

x2)) ( x1 x2)) = 0.

4 ' "&

(x1 x2) ( x1 x2) (x1 x2) ((x1 x2)(x1 x2)) ((( x1 x2) (x1 x2))(x1 x2)) = x1 x2 (x1 x2) (x1 x2) = x1 x2 = 0.

0'$ &$ 0 ' ( (, % x1 = 0 x2 = 0. +,

= (x1 = 0, x2 = 0) % & ' ’ ' ( ' & ( ".

- 9 + 8 9 ' .

>. 1 P f(x1 ... (k, y1(x1, x2, …, xk), ....., m(x1, x2, …, xk)) '

' $ 1, ..., m, 8 % P 1(x1, x2, …, xk), ....., m(x1, x2, …, xk), , 8 f(x1 ... (k, y1(x1, x2, …, xk), ....., m(x1, x2, …, xk)) = 0

. 8 4 : 6 9 6 6 ( 1, ... m), 9 5 8

4 8 6 9 + 6 6

x1, ..., 6k, A 5 + 5 5: +.

+H . % ,

1.Y $' % b ?

2.S ' " ' ’ % b( '?S

' '?

3.Y ’ % b( '?

4.S & & % b( '?

5.Y $' % $ $?

6.S ' ' ’ % b( "? S

' '?

7.S & & % b(

"?

8.S P & ( '?

9.S ' % b( ' "?

10.S ( & ' & ' % b( '

"?

11.S P $' % ' ?

% +

+

1.1 ( 0., 4 ( P „. 0 b & b. – A.: c &, 1986. - N.60-61.

0 ( '

2.A & & & ' ( % & «+ & » & & P P ( 6.0804,

6.0915 / +.A. A$ . – +&: +E43, 2004. - .2. – N.75-76.

181

? 35. B " # : ?, D #

" %

& % ( , ( & ( " & P . % ( , " ( & ",

( % ( ( & ( & P , ( , 8 & $' & & % ( P " ( & ( (.

3 & & :

35.1.> % "

35.2.A & ( & % "

35.3.0 " %

35.4.1 & P ( &

35.1. "+ G + E . C

-6 5 4 X = (61, x2,..., 6i,...,xn) 8 y = F(`) = F(61, x2,..., 6i,...,xn), 8 6 6

61, x2,..., 6i,...,xn. D A 6 6i : : 6i, 6

8 y = F(61, x2,..., 6i,...,xn). 1 9 ,

6 6 8 4 + 5 + 5, A : 8

8 .

>. 1 " P F((1,x2,...,(i,...,xn) 8 & (i (d/d(i – %

( &, ) ' &

d(x)/d(i = F(x1, (2,..., (i,...,xn) [ F(x1, x2 ,..., (i,...,(n)

3 : 2 4 8 ™ 4, 6 + 5 9 8 .

[ (d( )/d(i)(d( )/d(i)

B 9 + 5 8 6 6 8 . -8 5C 9 + + 8 : 5 1, A

6 8 4 + 5 6 5 4 (61, x2,..., 6i,...,xn) (61, x2,...,6i,...,xn) 5 0, A 6 6 8 4 + 5.

3 : C 5 8 6 4 6 6 :

9 .

>. 1 P F(„) ' & i, 8

F(„) $' i , % 8 F((1, x2,..., (i,...,xn) = F((1,

x2,..., (i,...,xn).

-6 F(`) 9 5 6i 5 , F(`)

9 5 6i 6 6 C 6 6. 1 5 :, A 6i : F(`).

1 + 8 +, 9 9

+, A : 4 4 F(F = 0.

182

. r + F(s) / / 0 i , + *

d(s)/d i = 0.

- 4 A 9 9 5

D 9 9 A :

10.? S d( )/d(i = 0, .$ (i + .$'N F( ) .

11.? S d( )/d(i = 1, .$ (i . ;* .$'N F( ) .

12.? S d( )/d(i = G( ), .$ (i G * + .$ F( ) % G( ) = 1.

4 &. E ( " ( (&. . 35.1). A , (

( &

. 35.1. ( & &

( & (1 ( &

9 P &$ % š & '

d(„)/d(i = (x1x2+x3) [ ( x1x2+x3) = x1x2 [ x3 [ x1x2x3 [ x1x2 [ x3 [ [x1x2x3 =

x2 x3

+& " ` &` & " &$ % "

d(x1,x2,x3)/d(i = d(x1x2+x3)/d(i = x3d(x1x2)/d(i = x2 x3d(x1)/d(i = x2 x3

35.2. * . ( *; , G + E . C

0 C 6 9 8 , A + 5

8 . D A F(`) , 8

. ( 9 8 5C 8 d(`)/d6i

8 .

>. 1 " $ P F((1,x2,...,(i,...,xn) 8 & (i '

&

d(x)/d(i = F(x1, (2,..., 1,...,xn) [ F(x1, x2 ,..., 0,...,(n),

& & % &'-( (i, 1 , i , n.

? 8 d(`)/d6i : 5 2 6 , C 6 : F(`)

6i = 1, 4 F(`) 6i = 0.

= (F(x1, 62,..., 1,...,xn)) [ (F(x1, 62,..., 0,...,xn)).

4 &. E ( " & P F((1, x2, x3) = x1 x2+x3, % d((1, x2,

x3)/d(2.

d((1, x2, x3)/d(2 = ( x10 + x3) [ ( x11 + x3) = x3 [ ( x1 + x3) =

=x1 [ (1 x3 = (1 x3.

35.2.1.*

. C + 8

d(x)/d6i = F(x1, 62,..., 6i,...,xn) [ F(x1, x2 ,..., 6i,...,6n)

183

* 9 8 : F(x1, 62,..., 6i,...,xn), C F(x1, x2 ,...,6i,...,6n). 1 + 5 + 2. 0 5 :

d(x)/d6i.

: 5 + 8 + , A + :

6 , 8 5- 4 9 F G 9 8

4 + 9 F G.

4 &. E ( " & P F((1, x2, x3) = x1 x2+x3, % d((1, x2, x3)/d(2 & _ .

0 (% .. 35.1 35.2) &$' & 2 % % . 35.3.

x1 x3

+, d((1, x2, x3)/d(2 = x1 x3.

35.2.2. * ` >

* : 8 5- 8 F(x) 6 , A + 5 + 2. 0 5 + 5 : 5 5 + +, A

+ : (¥-z).

2 9 ¥-z 4 + 5 4, 5 9 4 : 5 5 . .5 d(`)/d6i

5 6 :

1.F(`) : 5 ¥-z 4 + 4 9 4 4

F(`) .

2.2 ¥-z + 5 9 8 ,

, 4 4 6 5 5 d(`)/d6i.

4 &. 0 P$ Cn = an bn + (an+ bn)Cn-1, % dNn/dCn-1. E ` Cn $' ŒE- (% . 35.4).

+, Cn = an bn [ anCn-1 [ bnCn-1,

dCn/dCn-1 = d(an bn [ d(anCn-1)/dCn-1 [ d(bnCn-1)/dCn-1)/dCn-1 = = 0 [ an [ bn = an [ bn.

184

D, , 9 A 5 8

13. ? S F(x) = A(X) +xi(X) + xi(X), * A(X), B(X) C(X) . + ; H * xi, % *+

d( )/d(i = A(X)(B(X) [ C(X))

35.3. * E G + . C,

> 5 6 + 5 6

6 . < 6 6 9 8 4 +

8 .

>. 4 & "$ % $ $ P F((1,x2,...,(i,...,(j,...,xn) 8 & (

(i (j ' &

+, d2F(x)/d(id(j = x3( x1 x2+ x1x2).

3 4 4 , ,

d(d(x)/d6j)/d6i, 8

d2F(`)/d6id6j d(d(x)/d6j)/d6i

>. 1 P F((1,x2,...,(i,...,(j,...,xn) ' & ( (i (j, 8 F((1,x2,...,(i,...,(j,...,xn) $' (i (j , %, 8

F((1,x2,...,(i,...,(j,...,xn) = F((1,x2,..., (i,..., (j,...,xn)

. r + F(s) / / i j, + *

d2F(s)/d id j = 0.

1 5 6i 6j, A , A 5 8 9

6i 8 6j.

? , 9 :

1.0 6i 6j

185

=F(61,x2,...,6i,...,6j,...,xn) [ F(61,x2,..., 6i,..., 6j,...,xn).

2.0 8 6i, 8 6j

d(`)/d(6i[6j) = = d(`)/d6i + d(`)/d6j.

3. 0 8 6i, 8 6j, 8 6i 6j

d(`)/d(6i+6j) = = d(`)/d(6i[6j) + d(`)/d(6i6j).

35.4. B + % ( * E * / C +

>. o$ % $ ( &$ df(x)/dxi P f(x) & ( (0, (1, .. ., (i,

..., ( " (i & & & $ 2 & f(x):

ðf(x)/ ð(i = f(x0, (1,..., (i,...,xn) [ f(x0, x1 ,..., (i,...,(n)

? 9 + : 3.:

ðf(x)/ ð6i = f(x0, 61,... , 6i-1, l, 6i+1,…, xn) [f(x0, 61,... , 6i-1, 0, 6i+1,…, xn)

>. o % & P (10) P dXi f(x) '

P f(x) xi, dxi:

dXi f(x) = f(x) [ f(xi [ dxi )

= : 9 5 9 32 + 8 + 6 +:

dXi f(x) = (ðf(x)/ ð6i)d6i.

1 8 6 (3.) + 2

. v 32 : + + 8 6 dxi

: 9 .

-8 5C 9 5 3. 4 : , A 3.

+: , f(x) + 5 4 8 4

xi. 3. +: +, f(x) : 4

8 4 xi.

? 9 C 8 .

>. 4 % & P P f(x) '

P ( & X d: ðf(x)/ðX = df(x) = f(x) [ f(X [ d)

= : 4 32: ðf(x) = [ al (ðf(x)/ðXl)dXl

D A (X0, X1) : 8 + 6 4 X, ðf(d0 = 1, d1 = 0) = f(X0, X1) [ f(X0, X1) 9 Rf(x)/RX0 f(x) 6 . <

4 + Rf(x)/RX0 8 : 5 '+ 6

:

df(x) = l (Rf(x)/RXl) d1 (0<l<=2n-1)

= + 5 9 9 6:

p — 6 „0 .

3 , C, : : + 5

1 0, 8 0 1.

>. 1 & P , " % '` df(x), % $ P$, 8 &$ 1, & " ' &, f(x) $' „ 0 1. 1 & P , " ` df(x), % $ P$, 8 &$ 1, & "

' &, f(x) $' X 1 0.

<5 : ' 9 : : 32:

>df(x) = f(x)df(x); Tdf(x) = f(x)df(x).

186

* 9 4 : 32 (32) 6 : 6

8 œ6 6 6 (3.) 9 6 . < 6 ’ 9

: : 32 9 C 9 32 32:

>dxi f(x) = f(x)dxi f(x);

Tdxi f(x) = f(x)dxi f(x) .

1 + 9 9 3. 32, C 5 9

32 32 9 4

>dxi f(x) = f(x) (ðf(x)/ð6i) dxi; Tdxi f(x) = f(x) (ðf(x)/ð6i) dxi.

>. o$ % $ ( &$, $ % '` >ðf(x)/ð(i,

' & f(x) (ðf(x)/ð(i). o$ % $ ( &$, $

` Tðf(x)/ð(i, ' & f(x) (ðf(x)/ð(i).

<6 5 C 5 9 32 32 32 9 4

>dxi f(x) = (>ðf(x)/ð6i) dxi;

Tdxi f(x) = (Tðf(x)/ð6i) dxi.

. C 32 9 3., 9 9 :

>ðf(x)/ð6i = f(xi)f( xj); Tðf(x)/ð6i = f(xi) f( xj) .

0’ 6 5, A +: 1, : 8 6 6 6, 6

xj 9 5 0 1 8 1 0. . 5 xi 9 0 1 , 1 0, 8 : : +.

: (i 5 6 C 5 12:

>dxi f(x) = f(xi=1) f(xi=0) >dxi f(xi=1) f(xi=0) Tdxi; Tdxi f(x) = f(xi=1) f(xi=0) Tdxi f(xi=1) f(xi=0) >dxi.

>. 4$ $ % $ ( &$ [ðf(x)/ð(i] '

& f(xi=1) f(xi=0), 8 , ( & P dxi " dxi f(x)$' & $. +%$ $ % $ ( &$ [ðf(x)/ð(i]0

& f(xi=1)f(xi=0), 8 , ( & P dxi " dxi f(x)$' $.

< 5 32 :, A ’ C 4 9, , A 3. +: , : 4 6, A : 8

+, C 6 , ’ , 4 4

, A 3. +: 1, : C 6 : [ðf(x)/ð6i] = f(xi = 1) f(xi=0) = 1;

[ðf(x)/ð6i]0 = f(xi = 1) f(xi=0) = 1.

D A + f(x) 4 f(x) = xi f(xi = 1) xi f(xi = 0), 9

9 9 3. 8 3.:

Tðf(x)/ð6i = 6i [ðf(x)/ð6i] 6i [ðf(x)/ð6i] ;

>ðf(x)/ð6i = 6i [ðf(x)/ð6i] 6i [ðf(x)/ð6i] .

: 8 6 + 5 5 3. + 5

6 . ( ' 9 5 3..

187

7. Tð(f(x) [ g(x))/ð(i = g(x) g( x)Tðf(x)/ð(ig(x) g( x) >ðf(x)/ð(i f(x) f( x)Tðg(x)/ð(i

f(x) f( x) >ðg(x)/ð(i.

8. >ð(f(x) [ g(x))/ð(i = g(x) g( x)>ðf(x)/ð(ig(x) g( x)Tðf(x)/ð(i f(x) f( x)>ðg(x)/ð(i

f(x) f( x)Tðg(x)/ð(i.

1 8 3. 9 6 C 5.

11.[ð(f(x)g(x))/ð(i]% = g(x=1)(ðf(x)/ð(i]n f(x=1)(ðg(x)/ð(i]n

12.[ð(f(x) g(x))/ð(i]% =g(x=0)(ðf(x)/ð(i]%

f(x=0)(ðg(x)/ð(i]%

13.[ð(f(x) g(x))/ð(i]% = g(x=0)(ðf(x)/ð(i]n

f(x=0)(ðg(x)/ð(i]n.

14.[ð(f(x) g(x))/ð(i]% = g(x=1)(ðf(x)/ð(i]%

f(x=1)(ðg(x)/ð(i]%

15.[ð(f(x) [ g(x))/ð(i]% = g(x=1) g(x=0)(ðf(x)/ð(i]n

g(x=1)g(x=0)(ðf(x)/ð(i]% f(x=1) f(x=0)(ðg(x)/ð(i]n

f(x=1)f(x=0)(ðg(x)/ð(i]%

16.[ð(f(x) [ g(x))/ð(i]o = g(x=1) g(x=0)(ðf(x)/ð(i]o

g(x=1)g(x=0)(ðf(x)/ð(i]n f(x=1) f(x=0)(ðg(x)/ð(i]o

f(x=1)f(x=0)(ðg(x)/ð(i]n

4 &. A % +10 " +14 $ (2 P

f(x) = (1(2 (2(3.

1 ( &, `, $ (2 &$ Tðf(() /ð(2 = (xtx2

(2x3) (x1 x2 (2(3) = (1(2 (3, (1 (2(3. ) & % (1 = 1, (2 = 1, (˜ = 0 (1 = 0, (2 = 0, (3 = 1 $' P & (2. 4 ' (2

( 1 0 % 110 % 0 1 % 001) & ' & P 1 0. 1 ( &, % '`, " (2 &$ >ðf(() /ð(2 = (xtx2(2x3) (x1 x2 (2(3) = (1(2(3, (1 (2 (3. ) & (2 1 0 % 011 %

0 1 (1 = 0, (3 = 1. +10, " % '`, &$ 1 (2 1 0 x1 = 0, (3 = 1 % (2 0 1 (1 = 1, (3 = 0.

+H % ,

1.Y % $ $?

2.S % ?

3.o &$' _ „ N & % ?

4.Y & "$ % $ $?

5.Y ' & % ( &$ & P ?

6.o &$' % & P & (?

7.S ' P " ( &?

188

8.Y ' % & P " ( &?

9.3 ( ( ( % ( ( & (

& P ?

% +

+

1.N -. A &b % ` % % c>A. – A.: A, 1974. -

N.32-41.

0 &

2.1 ( 0., 4 ( P „. 0 b & b. A.: c &, 1986. - N.66-167.

0 ( '

3.A & & & ' ( % & «+ & » & & P P ( 6.0804, 6.0915 / +.A. A$ . – +&: +E43, 2001. – N.45-50.

189

? 36. A # "

" %

% &.

&, - , , &,

. ) & & ( P, P ,

&.

3 & & :

36.1.> &

36.2.&

36.3.4

36.1. " + + , % *

-6 M 1 - , A 9 5 8 («1»)

(«0»). D A , 9

4 + 6 4 6 .

4 &. 9 - «S & »,

N - «A ' » 9£>£N.

< 4 : 5 5

8':, 6 , : 5 &. . : 8 + 4 + 0(6), A 8 9 «0» «1»,

6 4 9 5 8 9 * 8':,

+ 5 8 + , 8 6 *.

( 4 5 9 8 9 + 8 4 5 6 6 61, 62,..., 6n, A + 5 9 X 6 9 X1, X2,…, 9

0(61, 62,..., 6n).

36.2. F % *

M 4 6 , 6 9 5

8 5-6 6 6 9, 8 5 5 6

8 9 9.

>. N- & ' & & &

P & n & ((1, (2,.., (n) % $ ' {0, 1}.

M 4 N-4, 8 N-4 + 5 8 + 8':

8 ` (6 `) + 5 & .

6 + 5 & ". .

. A 4 «6 » 4 « » N- 0(61, 62,..., 6n) +: 5 (N-1)- 0(61, 62,..., ,..., 6n) 6 9

190