Gusev Diskretnaya_matematika Konspekt Lekciy

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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