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### Transcript of Gusev Diskretnaya_matematika Konspekt Lekciy

• _____________________________________________________________________

.. , ..

-

2003

• 519.1(075.8) 962

: . .-. , . ..

I ,

.., .. 962 : . : - , 2003. 72 .

, I

. , , , , . , .

519.1(075.8)

, 2003

• ,

, . : , , .

, .

, , ( , ) . . - , , . . , . , , , .

.

I.

1. ,

1.1. , , , ,

. - .

. , .

( ). a A a A a A b , , , b A . A 1 1) . . A2) C . 3) D ; 2D . 4) N ; 1, ; 2N 2N . ,

. . . , .

A B AB A B

, , .. . .

BA B=

. A B

, ( ). B A B

3

• , . .

A B A B A B A B ( ) . A A A .

. 0, .. , .

, . , , , .

1.2. : ;

, .

. , , , . { , , , }A a b c d= A , , ,a b c d

. , .

2 1) si . { | / 2 2 , 0, 1, 2,A x x k k= = + = }

}

}

n 1x =2)

{ }1 2 0 1, 2,3, ; 1, 1n n nF f f f n f f = = + = = = . , , .

3

{ 2 2( , ) | 1K x y x y= + ; { | sin 0.5}Z x x= , 0.5.

1.3. , ,

. . A B{ |A B x x A= }.x B

, ( ) , . :

A B C D A

, , , D;

A S , S. ,

, , , .

1

ni

iA

= n

1i

iA

=

4 1) , , . { }1 , ,B a b c= {2 , ,B c d e= 1 2B B =

| 1x ={ }, , , ,a b c d e=

2) | : sin

2 , 0, 1, 2, 2 , 0, 1, 2,2 2

A x x k k x x k k = = + = = + = .

4

• 3) D 2

{ }2 ( , ) ,x y x y= D D =D {( , ) , , 0} {( , ) , , 0}.x y x y x x y x y x= C x= .

A B

( / ) ( / )A B A B B A = . . 1.1 , .

, .

5

• . 1.1.

1.4. 1. : A B B A= , ; A B B A= 2. :

( ) ( )A B C A B C= , A (B C) = (A B)C; 3. : A(BUC)=(AB)U(AC), AU(BC)=(AUB)(AUC) ;

4. : AUA=A, AA=A; 5. : ( )A B A A= , ; ( )A B A A= 6. : A = A , ; A = 7. : A U U= , ; A AU =

8. : A A= ; 9. : A B A B= , A B A B= ; 10. : A A = , A A U= .

1.5.

, .

, .

{ }i i IB =iB

A iB A A x A

6

• , , i .

A ,i jB B i jB B = j

}

))

)

}

}

))))

))))

8 , . .

. .

C DC D

A C D= { },C D{ , A \ , \C D D C

A

2. , ,

. . . , .

n( 1 2, , , nx x x1 1,x y= x y=

ix

n nx y

i( )1 2, , , nx x x

B

( 1 2, , , ny y y2 2, , =

, .

A A B( ,a b ,a A b BA B= , ,

. , , .

. . ,

n - .

A

nA

2A 1, , nA An

1 2 nA A A 2 , n na A

nA

( )1 2, , , na a aA

A

1 1 2, ,a A a A .A..n

1 2=D D D , .. D D . ( ){ }2 , | ,a b a b=

2 , .

. { }, , , , , , , A a b c d e f q h= { 3,4,5,6,7B = 1,2, , 8 A x B

3 .

.

{0, 3,4,5,6,7 ,9M = 1,2, , 8999

3M

0 : 1. , ( ) ( ) (1 2 1 2X X Y X Y X Y = ( ) ( ) (1 2 1 2X X Y X Y X ; ( ) ( ) (1 2 1 2X X Y X Y X = Y( ) ( ) (1 2 1 2Y X X Y X Y X =

, ;

( ) ( ) (1 2 1 2\ \X X Y X Y X Y = , ( ) ( ) (1 2 1 2\ \Y X X Y X Y X = ; ( ) ( ) (1 2 1 2X X X Y X Y , ( ) ( ) (1 2 1 2X X Y X Y X . , , ,A A A

1. 1 2 n 1 1A m= , 2 2A m= , , n

1 2, , ,A A nA m=

nA nA.

, .. 1 2A A 1 2A A A m = 1 2, ,n nm m .

7

• . . , . ,

n k . 1n = n k=

1= + 1 2 1 2 , ,kA A A m m m =

1ka + 1kA +1 2A A

1 2 1kA A A + n

k

11

. . . . . , . , , , .

( )1 2, , , ka a a1 2 km m m

1 2 km m m + 1n k= +

1km +1km + kA +

1 1k ka A+ +

n. nA A= .

3. . A B

. , .

A B R

R

A B) R

A B= RA , , .

. ( , a b a b R ( , ) a b

aRb 1 , { }( , ) | , ; , R a b a A b B a b= . 2 . , .. .

(5, 7), (2, 2), (5, 4). A B N= = ( ){ }, | , ;R m n m n m nN=

A B= =N( , 1)n n +

{ , , }A a a= { , , }B b b= k l r

{1,2,3,4,5,6}A B= =

3 . ,

(2,4), (3,15), , , .

.

, , i- j- , :

1 k 1 lij

0

1, a Rai jr =ij

, ,

1 1 1 1 1 10 1 1 1 1 10 0 1 1 1 10 0 0 1 1 10 0 0 0 1 10 0 0 0 0 1

R =

R

8

• ( ){ }| , ,a b a bR = R

+

R

+

}

.

4 , , . {1,2,3,4,5,6}A B= = " "R = { }1,2,3,4,5,6R = 5 , , . {1,2,3,4,5,6}A B= = ( ){ , | , , 2R a b a b A a b= = } { }1,2,3,4R =

R( ){ }| , ,b a a bR = .

6 , , . {1,2,3,4,5,6}A B= = ( ){ }, | , , 2R a b a b A a b= = { }1,2,3,4R = { }3,4,5,6R = . . a A {( ) | , R a b b B aRb= a B 7 , , a , . {1,2,3,4,5,6}A B= = " "R = 2= { }(2) 2,3,4,5,6R =

1 b B. ( ) { , }R b a a A aRb= b . A 8 , , b , . {1,2,3,4,5,6}A B= = " "R = 2= { }1(2) 1,2R =

.

C A

R

C RC

( ) { | , }a C

R C b b B aRb

= 9

{1, 2, 3, 4, 5, 6}A B= = , , C , . " "R = {2, 3}= { }( ) 2,3,4,5,6R C = ,

D D B R

1( ) { | , }b D

R D a a A aRb

= . 10

{1, 2, 3, 4, 5, 6}A B= = , , , . " "R = {2, 3}D = -1( ) {1,2,3}R D =

R . , R. - , , , .

A B 1R 1R

R A B \R A B R= . 11

A B= =D , . . " "R = _

" "R = >

9

• R 1 {( , ) ( , ) }R b a a b R = . 12 A B= =D . . " "R = 1 " "R = 13

{1,2,3,4,5,6}A B= = . 1 .

1R A

( , )R z yB C

x

2

2

)=

x

2R B }1 2 1 2{( , ) | , ( , )R R x y z B x z R= D .

14

{1,2,3,4,5,6}A B = = ={(2,2), (2,4), (2,6),R =

. 1, . 1R (3,6

( ){ }2 , | 2R x y y= =1 (3,3), ), (4,2), (4,4), (4,6), (5,5), },2), (6,4), (6,6) {(1,2), (2,4), (3,6)}R = {(2,R R =D

, , . (6 2 1 2 4), (3,6), (4,4), (6,4), (6,6)}

f ,

( , . A

A

B

B

f A =

f A =

f B

f B =

1, , x y y

1, , x y y1( , )x y f 2 ) x y f 1 2y y= f ,

. , . - , ( ) y x. , , .

2

1( )y f x=

1( , )x y ff

2( , ) x y ff

f

1y y=y f x=

:f A( , ) x y f

B,x x1 2, y 1-1 , ,

. 2(y f x 1 2x x= 15

1) 1-1 , siny = ,2 2

A =

:f A B A =B = f

exp( )y x=

1

:f A B b:f A B

f

, 1-1 ,

. 2) , , 1-1

. - , ,

1-1 . f

f 16 1) ,

, , .

2) , , .

f , ,

, . f f ,

, , . , , .

f

10

• 4.

2. -

, A B

A B= . , ,

, , , f 1-1 . , :f A B f A = f B = A B . A B> . , ,

. 1-1 . A

A B