Martynyuk_A_N_Diskretnaya_matematika

171

0 ` `:

(5: 1001~ # 1~011 = , ` 1001~;

(3: 1011~ # ~01~0 = 10111

10111 # 101~1 = , ` 10~1~.

) 10010 %'& ` " ' "

33.1.3. . 5

. C 6 ( ) : 9 8 5C

8': 8 5 A

8': (A 2-z). . C 4 +

+ 5: 6 8 8,

4 8 + 5 9 5 . < + +, + 5

:+, 5 , A 4 : 5.

2 4 5 4 ++ 5 :

8 6 8 8, 4 : . M 4 5 :

1.1 8 : 5 4 C +: 5 5 4 I ( 8 5-

4 6), A 5 8 , 8 : (8': +: I).

2.= I 5 4 C 5 I

. : 5 9 8':. D A 9 5, 5

I i = I Ii 5 4 : 5 4

. ? : 5 6 Ii, A 9 5 8': +

6 C 6 6 I. D A 6 5 I i ,

I – , 5 C 8, A 9 5 ,

172

+ 5 I.

4 &. E ( " I = ~01~0. I `$' & ':

(2 ` ;

(3: ~00~0 # 000~0 =100~0 100~0 # 1~000 =10010 10010 # 10~1~ =(, ` ~0~~0;

(5 ` .

173

33.1.4. . 8':

D A 8 4 I, A 6 5 8': ,

+ 5 + C 6 8': , I C

9 8 . . 5 I C 6

A 9 5 +, I - C.

4 &. E ( " I =10~1~ %'& & ` ":

33.2. - + * .'' $+H /$

8 6 9 5 8

6 6:

1)4 8 6 ;

2)6 , 5 6 6

8 6 ;

3)4 8 5 9 8 6

, 8 4 8 4 : 4 ’ 6

, 9, 8, A + 5 +, 9 5 9 +

5 6 4.

-9 8 6 2-z, A + 5

8 5C.

33.2.1. < 8 2-z

* 5 5 – 2nn, 5 - 2n-1n. 33.2.2. > 2-z

* 5 5 – 3n/yn, , A -*

174

? 5 – 2n nlog log n, , 9 2-z

A : 9 8 2-z.

33.2.3.-C 2-z * 5 5 – 2n-1. ? 5 – 2n-1/log n log log n.

33.2.4.<

1 C 5 2-z C 2-z 9 6

+: . .C C 2-z - C .

? C Vn 5 8 C 2-z C 2-z

: 5 +: Vn 4 log n,

8 5 8 C 2-z 9 8 5 4 C, 9 C. v tn 8 C 6 2-z 9 6 8 6 +: 5 +

tn = (22n-1)log n log log n,

8 8 8 C 6 2-z 8 5C 6 n - 8.

v qn C 6 2-z 9 +

qn 4 (22n-1)Cnyn,

, 8 C 6 8 C 6 2-z

: 5 4 C 6 8 C 6 2-z. - + 5 8': ' 4

8 6 ’+ 5 .

+H . % ,

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2.3 & ` & %'&

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3.S ' & ` & %'&

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(. 0 & ( (. – _.: 3 %-

& " % b ` %, 1992. - N.190-196. 0 &

2.d % >.9. + b & " . – A.: >b ` ` , 1986. –

N.25-31.

0 ( '

3.A & & & ' ( % & «+ & » & & P P ( 6.0804, 6.0915 / +.A. A$ . – +&: +E43, 2004. - .2. – N.75-76.

176

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" %

& % & ’ % ( '

". , , % ( ' "

", & ' & ( ' &'$"

'$" P (, ’ ' ". )

' ’ ' ". 3 & & :

34.1.1

34.2.1

34.3.N ' " '

34.1 B + , ,

>. E ( " f(x1, x2, …, xk) g(x1, x2, …, xk) — & % P & k (, & , &, &$ & ( (-% &' & & ( % (

P. .&

f(x1, x2, …, xk) = g (x1, x2, …, xk)

"% '` '$ P$ % . ’

% & = (x’1, x’2, …, x’k) Bk, & > = {0, 1}, & ( & f( ) = g( ).

. C

f( ) = g ( ) = 0 8 f( ) = g ( ) = 1

4 &. ’ % x1 [ x2 = x1 x2 % & ':

% c1 = (x1,= 0, x2 = 0), x1 [ x2 = x1 x2 = 0;

% c2 = (x1,= 0, x2 = 1) c3 = (x1,= 1, x2 = 0), ( x1 [ x2 = x1 x2 = 1.

2 8 6 5 5 4 f(x1, x2, …, xk) = 0 f(x1, x2, …, xk) = 1, 5

f(x1, x2, …, xk) = g(x1, x2, …, xk) ;

; f(x1, x2, …, xk) [ g(x1, x2, …, xk) = 0

B : , A + 5

9 ’5. ( f(x1, x2, …, xk) [ g (x1, x2, …, xk) = 0 : 5

5 .

D A g(x1, x2, …, xk) = 1, 8

f(x1, x2, …, xk) = 1 ; f(x1, x2, …, xk) = 0,

A g(x1, x2, …, xk) = 0, 8

f(x1, x2, …, xk) = 0 ; f(x1, x2, …, xk) = 1

D A n 5

f1(x1, x2, …, xk) = 0

… … …

fn(x1, x2, …, xk) = 0,

= (x1, x2, …, xk) Bk : ’ : , 5 A

5 : 9 +.

B 5 9 8 .

D A 8 6 5, A 6 5 , 8':

’+,

f1(x1, x2, …, xk) f2(x1, x2, …, xk) … fn(x1, x2, …, xk) = 0,

5 6 8 6 5, 5 '+ +: 0 5

, 9 ( 9 8 ) : 0.

4 &. E ( " & ' ( " ` ( ': x1 x2 = 0

177

x1 T x2 = 0

x1 [ x2 = 0

. & &'$ %'& ( ( &

(x1 x2) (x1 T x2) (x1 [ x2) = 0.

4 & & % % & '

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

4 ' "&

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

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

D A 5, A 6 5 , 8': ’+,

f1(x1, x2, …, xk) f2(x1, x2, …, xk) … fn(x1, x2, …, xk) = 1,

5 6 8 6 5, 5 ’+ +: 1 5

, 9 ( - 9 8 )

: 1.

4 &. E ( " & ' ( " ` ( '

x1 R x2 = 1 x1 N x2 = 1

x1 > x2 = 1

. & ’$ %'& ( ( "&

(x1 R x2) (x1 N x2) (x1 > x2) = 1.

4 & & % % & '

( x1 x2) (( x1 x2) (x1 x2)) ( x1 x2) = 1.

4 ' "&

x1 x2 ( x1 x2) (x1 x2) ( x1 x2) = x1 x2 = 1.

S &, ’$ &$ 1 & ( (, % x1 = 0 x2 = 0. +, ' &, & ', (x1 = 0, x2 = 0) % & ’

' ( '.

34.2. B +

>. E ( " f(x1, x2, …, xk) g(x1, x2, …, xk) — & % P & k (,

& &$ & ( % ( P. .&

f(x1, x2, …, xk) , g (x1, x2, …, xk)

$ '$ P$ % . ’

% = (x1, x2, …, xk) Bk, & ( & f( ) ( g ( ).

( 5

f( ) = g ( ) = 0 8 f( ) = 0, g( ) = 1 8 f( ) = g ( ) = 1.

4 &. ’ % x1 [ x2 , x1 x2 % & % c = (x1,= 1, x2 = 1), x1 [ x2 =0, x1 x2 = 1.

2 8 6 , 5, 5 4

B , A + 5

9 ’5. 1 + : 5 8 f = 1, g = 0. D A g(x1, x2, …, xk) = 1, 8 g(x1, x2, …, xk) = 0 ,

5 4 8

f(x1, x2, …, xk) = 0,

178

(fn(x1, x2, …, xk) gn(x1, x2, …, xk)) = 0...

B 5 6 8 6 , 5 '+

+: 0 5 , 9 ( , 9 8 ) : 0.

4 &. E ( " & & ( "

x1 x2 , x1 / x2`

x1 [ x2 , x1T x2.

. & &'$ %'& ( ( ' "&

((x1 x2) (x1 / x2)) ((x1 [ x2) (x1T x2)) = 0.

1 = (x1 = 0, x2 = 0) 2 = (x1 = 0, x2 = 1) % & ' ’ & (

".

(fn(x1, x2, …, xk) gn(x1, x2, …, xk)) = 1

5 6 8 6 , 5 ’+ +: 1

5 , 9 ( 4 5 -

4 9 8 ) : 1.

4 &. E ( " & & ( "

x1 N x2 , x1 / x2

x1 > x2 , x1 [ x2

. & ’$ %'& ( ( ' "&

((x1 N x2) (x1 / x2)) ((x1 > x2) (x1 [ x2)) = 1.

4 & & % % & '

(((x1 x2) ( x1 x2)) (x1 x2)) (( x1 x2) ((x1 x2) ( x1 x2)))=1.

4 ' "&

( x1 x2) (( x1 x2) (x1 x2)) = ( x1 x2) (x1 x2) = 1.

179

_ ’$ &$ 1 & ( (, ,

1 = (x1 = 0, x2 = 1) 2 = (x1 = 1, x2 = 0) % & ' ’ & (

".

34.3. % +H $ E , H

fm(x1, x2, …, xk) , gm(x1, x2, …, xk); fm+1(x1, x2, …, xk) = gm+1(x1, x2, …, xk);

… … …

fm+n(x1, x2, …, xk) = gm+n(x1, x2, …, xk)

> 5 m 8 6

f1(x1, x2, …, xk) g1(x1, x2, …, xk) = 0

..... … …

fm(x1, x2, …, xk) gm(x1, x2, …, xk) = 0.

2 9 8 n 8 6 5, A C

fm+1(x1, x2, …, xk) [ g m+1(x1, x2, …, xk) = 0

..... … …

fm+n(x1, x2, …, xk) [ g m+n(x1, x2, …, xk) = 0.

. 5 4 9 8': , A C, :

. ( 5 8

f1(x1, x2, …, xk) g1(x1, x2, …, xk) …

fm(x1, x2, …, xk) gm(x1, x2, …, xk)

(fm+n(x1, x2, …, xk) [ gm+n(x1, x2, …, xk)) …

fm+n(x1, x2, …, xk) [ g m+n(x1, x2, …, xk) = 0.

1 + -* 4, 9

1

(f1(x1, x2, …, xk) g1(x1, x2, …, xk)) …

(fm(x1, x2, …, xk) gm(x1, x2, …, xk))

( (fm+n(x1, x2, …, xk) [ gm+n(x1, x2, …, xk)) …

(fm+n(x1, x2, …, xk) [ g m+n(x1, x2, …, xk)) = 1.

z ’5 6 (5

8 4 6 8 6 = (x1, x2, …, xk) Bk), 4 6 :

+ +, 4 : 4. 6

6 ’ 8 6 5 : :+ 6 , A

+ 5 , , 4 6 6, 4 4.

4 &. 0 ( 8 ' ( ' & ( "

' ’

x1 x2 = 0 x1 T x2 = 0

x1 [ x2 = 0

x1 x2 , x1 / x2 x1 [ x2 , x1T x2

. & &'$ %'& ( ( " ( ' "&

(x1 x2) (x1 T x2) (x1 [ x2) ((x1 x2) (x1 / x2)) ((x1 [ x2) (x1T x2)) = 0.

4 & & % % & '

180