Gusev Diskretnaya_matematika Konspekt Lekciy
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Transcript of Gusev Diskretnaya_matematika Konspekt Lekciy
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_____________________________________________________________________
.. , ..
-
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