.
; , , , (, ) ..;
; - - ; - .
, . , , , , .
, ( ) , . .
, , , . , , , .
.
: (+),
(), (×) (/). .
:
N={1,2,¼},
Z={¼,2,1,0,1,2,¼},
, .. Q={ , m ÎZ, n ÎN},
() R,
C.
:
N Ì Z Ì Q Ì R Ì C.
N () : 53 , 35 ( 2 ). N (/): 6/3 , 6/5 ( , ).
, N , .
Z , - .
Q, () R, C ( 0).
|
|
, . , , div, mod. ,
13 div 5=2, 13 mod 5=3, 13=2×5+3 3 5.
div mod , 0, , , , " " . , div mod N.
, , , : () (). , (20,24)=4, (20,24)=120.
MATLAB gcd (greatest common divisor) lcm (least common multiple) .
n.
: (+) (×) " ". , : (/) (\). B , A / B = A × B 1, B \ A = B 1× A; , .
- , x. , , Z[ x ], Q[ x ], R[ x ], C[ x ].
: (+), (×), (). , , . ( 0), . , " "
(x 2+ 1 ) div (x ) = x, (x 2+ 1 ) mod (x ) =1,
(x 2+ 1 ) = x × x +1 - 1 ( 0) - x ( 1).
(2 x 2+ x ) div (3 x + 1 ) = , (2 x 2+ x ) mod (3 x + 1 ) = ,
(2 x 2+ x ) =(3 x + 1 )× () - ( 0) - 3 x + 1 ( 1).
. f (x) g (x), h (x), , f (x) x g (x): h (x)= f (g (x)). ( " " " ".)
, f (x)= x 2, g (x)= x 3+1, h (x)= f (g (x))=(x 3+1)2.
"" () (×), , .. h = f g h = f × g. " ", .. . f (x) g (x) :
f (x)× g (x)= x 2×(x 3+1)= x 5+ x 2.
(, , ln(sin(x)) ..), , g (x) f (x)? "" x ÎZ ( x ÎQ, x ÎR, x ÎC).
|
|
: V R.
:
,
( ),
( ),
( ),
( ),
( ).
, , .
: "" "", 1 0 , M ={0,1}.
"" (), "" (), Ù "", Ú "" ( Ù & , ).
: " " ( " "), XOR, (), .
.
1.1
x y | x Ù y | x Ú y | x XOR y | x y |
0 0 0 1 1 0 1 1 |
. (), Ø x. x =0 =1, x =1 =0.
, .
1.7. n
n: Z n ={0,1,¼, n 1}, .
: n, + n, n ; n, × n, n .
mod ( 2)
x + n y =(x + y) mod n, x × n y =(x × y) mod n.
, , n =5. Z5 ={0,1,2,3,4}. " ": 1 +5 2 = 3, 1 +5 4 = 0, 3 +5 4 = 2, 1 ×5 2 = 2, 2 ×5 2 = 4, 3 ×5 2 = 1.
XOR , 2, Ù 2.
( ), , . , :
1.2
Ù | Ú | XOR | ||||||||||||
5 1.3
+5 | ×5 | |||||||||||
|
|
, ( , ) . , , M ={ a, b, c }, a, b, c . . , , .. - .
U, . M ( U). : M =2 U. , , U ={ p, q, r }, M =2 U 23=8 :
M ={Æ,{ p },{ q },{ r },{ p, q },{ p, r },{ q, r },{ p, q, r }}.
: (È), (Ç), (\). (D):
A D B =(A \ B)È(B \ A)=(A È B)\(A Ç B).
, :
{ p, r }È{ q, r }={ p, q, r }, { p, r }Ç{ q, r }={ r },
{ p, r }\{ q, r }={ p }, { q, r }\{ p, r }={ q }, { p, r }D{ q, r }={ p, q }.
, : A . ..
1.9.
R.
n -
n : x min=min(x 1, x 2,¼, xn),
n : x max=max(x 1, x 2,¼, xn),
n : x . = .
n : x , xi, xi £ x xi, xi ³ x . xi , , ( n) , , ( n), . , 4,2,5 x =4 ( 2<4<5), 4,2,5,2 x =3 ( 2=2<4<5, ).
, , . . - ( - ), (), "" . . .
M , . x, y Î M z = x y, z Î M, .. M . 19 ( ), .
|
|
, x, y Î M x y = y x.
, , , , , , XOR, n. , ( , A×B = B×A). () x ´ y = y ´ x. ( ). , f g f (x)= x 2, g (x)= x 3+1, f (g (x))=(x 3+1)2= x 6+2 x 3+1. g f, g (f (x))=(x 2)3+1= x 6+1.
( ).
M ={ a, b, c }, 1.4, , .
, , , , .
e Î M , x Î M x e = x
e x = x.
0, - 0, (.. , ), 1, E .
N? , (x, y)£min(x, y). 1, (x,1)= x.
´ .
, , 1. e (x)= x ( 1).
Ù " ", .. 1, Ú XOR "", .. 0. .
n 0, n 1.
M ={ a, b, c }, 1.4, a, , , , , , .
Æ, U.
. Z ( Q R C) : x Å y = x + y 1. : e? x Å e = x x. x Å e = x + e 1, x. , x + e 1= x, e =1. Å , 1.
x Î M Î M, x = e x = e ( e ).
, e =0, x x.
, e = 0 (-), x x.
, e =1, x x 1. x 1 x ¹0, M. , , ( , ) , . x ¹0 y ¹0, z = x × y ¹0, ( ).
|
|
, . e E, X X 1. , X , .. det X ¹0. M n. ? , det X ¹0 det Y ¹0, Z = X × Y, , det Z ¹0? , , , .. det(X × Y)=det X ×det Y.
e =1. x ÎN ,
(x, )=1 , (x, y)³max(x, y).
, , f (x) f (x), . , .. , 0. , f (x)= x +2, .
, f (x) , . f (x)= x 2. , , -, , -, . , , , .. f (x)= ax + b. , y = ax + b, . , f (x)= ax + b , x y . , a ¹0. a ¹0 ( a =0 ).
. , f (x)= a 1× x + b 1 g (x)= a 2× x + b 2 , f g
f (g (x))= a 1×(a 2× x + b 2)+ b 1= a 1× a 2× x +(a 1× b 2+ b 1),
.. .
, Ù ( e =1), x . ( e =0), x . XOR ( e =0), x : x XOR x =0.
M ={ a, b, c }, 1.4, a, b c, c b, b c = c b = a, a a.
, A , A D A =Æ, D Æ.
: x Å y = x + y 1. : x? , x Å = e. x Å = x + 1, e =1. , x + 1=1, =2 x. , Å .
( ) x + y + z x × y × z, , , , .. "" . , ( ) .
, x, y, z Î M (x y) z = x (y z). x y z , x 1 x 2¼ xn.
, ? : , ? ( ) , " ", , " " ( ) .
, " ".
, "" . , , , ¼
, .
, .
. , . i, j, k. , , (i ´ j)´ j = k ´ j = i. - i ´(j ´ j)= i ´ 0=0.
, .
. , f, g, h (f g) h f (g h) f (g (h (x))).
, XOR, . x, y, z Î{0,1} ( 23=8 ), .
M ={ a, b, c }, 1.4, 33=27 , . .
, . " ", , , ( ).
: x Å y = x + y 1. : ? (x Å y)Å z = x Å(y Å z) x, y, z. (x Å y) z =(x + y 1)+ z 1= x + y + z 2, x Å(y Å z)= x +(y + z 1)1= x + y + z 2. , .
. , , : 3- V R . " " , .. RÈV , ×. . , , . , , , . , , . , . , "-" , .. , (x × y)× z = x ×(y × z), , .
, , x × y × z, .
, , , x y , z , , z, x × y, .
, , x y , z . x × y, z, .. . x y × z, y .
, , , , z, , x. , x z , , "-" .
. .
M G ( G =(M, )),
, .. (x y) z = x (y z) x, y, z Î M;
M e , x Î M x e = x e x = x;
x Î M Î M , x = e x = e.
. x , .
, G =(M, ) ( ).
, .
a x = b y a = b, (*)
a b () , x y .
, (*) .
( ) , .. , a. (a x), b. , ( a) x. a = e, e x = x. x = b.
(*) .
: (M, ) (*) , .
. , . 1.4 .
, , .
1. (Z, +) . "" - , (additio - , ). (Q, +), (R, +) (C, +). 0, x x. , . "" (+) "" , (. ). , .
2. (Q*, ×) , . Q*={ , m ÎZ, m ¹0, n ÎN}. "" - , (multiplicatio - ). , (R*, ×) , (C*, ×). 1, x x 1. 0 : 01.
3. (Q+, ×) . Q+={ , m, n ÎN}. (R+, ×). 1, x x 1.
4. (R m ´ n, +) m n . , , X X ( ).
5. (Mn, ×) n. E, X X 1.
6. (Un, ×) n, 1 ( ). E. Un - X det X =1, det X 1=1.
7. (Z[ x ], +), (Q[ x ], +), R[ x ], +), (C[ x ], +) . , f (x) f (x).
8. (L [ x ], ) f (x)= ax + b a ¹0 . a b Q, R, C.
9. (V, +) . - 0, x x.
10. (Z n, + n) n. 0, x n x.
11. . 0 x × n 0 0 x, x × n 0 = 1 . (Z n)*={1,¼, n 1}. . n =6, (Z6)*={1,2,3,4,5}. 2×6 3 = 0 , (Z6)* ×6. 2× 3 6, .. n =6. -, n , .. ("" n = n ×1, , ). ((Z p)*, × p). , , . p =5 ( 1.3). 2 3, 1. 5, 21=3. 31=2, 41=4.
12. (Z, Å) Å, x Å y = x + y 1. , e =1 , x, =2 x. (Q, Å), (R, Å), (C, Å). , .
13. : Q ( R, C) x Ä y = x + y x × y. (Q, Ä), (R, Ä), (C, Ä) ? , - .
. G =(M, ) . G '=(M ', ), M 'Í M, G, G 'Í G. , . , , : , e. , ( ) .
. (Z, +) (Q, +), (R, +), (C, +). 0, x x.
:
(Z, +)Ì(Q, +)Ì(R, +)Ì(C, +), , .
: (Q+, ×) (Q*, ×) , ,
(Q+, ×)Ì(Q*, ×).
(Un, ×) (Mn,×).
(2Z, +) (Z, +).
, . .. G =(M, ) - M 'Í M, - à ( ) G '=(M ', à), G, M 'Í M.
, n +, n ×, .
M ' , , (M ', ) (M, )?
. , , , x Î M ', M '.
, , (M, ) M, , M '.
1. Z 2Z 2Z+1. 2Z , (Z, +) 0 , x Î2Z, , .. x Î2Z. , , (2Z, +) . 2Z+1 , (2Z+1, +) .
2. Z N. , (N, +) (Z, +), 0 .
Z0={0,1,2,¼}. : 0ÎZ0, (Z0, +) (Z, +), x ÎZ0 x ÏZ0.
(N, +) (Z0, +) . , , . , (Z0, +) , . , n , .
3. (Mn, ×) n Tn , . (, , 0.) ( ), , .. , , A Î Tn, , .. A 1Î Tn. 2- . , p ¹0 r ¹0. . , , (Tn, ×) (Mn, ×).
. G 1=(M 1, ) G 2=(M 2, à) ( ) . G 1 G 2 ,
j: M 1 M 2,
x, y Î M 1 j(x y)= j(x) à j(y).
: G 1 @ G 2.
. G 1=(R+, ×) , G 2=(R, +) . j: R+R , , , , .. j(x)=ln x. , ex. , ln (x × y)= ln (x)+ln (y). , , .. (R+, ×) @ (R, +).
: (Z, +) @ (2Z, +).
j: Z2Z j(x)=2 x. , .
. M ={ a, b, c }, 1.4. - (, a, ) " " ( ). "" .
. (M, ) 3, .. (Z3, +3), Z3 ={0,1,2}. j: M Z3, j(a)=0, j(b)=1, j(c)=2., , j(b c)= j(b) +3 j(c). , 1.4, b c = a j(a)=0. 1 +3 2 =0. 1.4. , . : (M, ) , .
M , (+) (×). K =(M, +, ×). .
(M, +) , . 0, , x, x.
: , . , , , .
( ): x × (y + z)=(x × y)+(x × z) (y + z) × x =(y × x)+(z × x) ( , ).
(), x y = x +( y).
.
1. (Z, +, ×), (Q, +, ×), (R, +, ×), (C, +, ×) . .
2. (Mn, +, ×) n ( ) . , . .
3. (Tn, +, ×) n ( ) . . Tn ( ) ( ).
4. (Z n, + n, × n) n. .
5. Z[ x ], +, ×), (Q[ x ], +, ×), (R[ x ], +, ×), (C[ x ], +, ×) . . .
6. (V, +, ´) . .
7. . U M =2 U. D, Ç. (2 U, D, Ç) , . Æ.
8. (Z, Å, Ä) : x Å y = x + y 1 x Ä y = x + y x × y . (Q, Å, Ä), (R, Å, Ä), (C, Å, Ä). .
9. , (L [ x ], +, ), L [ x ] f (x)= ax + b a b, , () . , .
, .. f 1(<