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, .
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. , ( , ) , - (, , ). ()
, , .
. (.. " "), , ( " " " "). "" .
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, . , , , ( , ). , (, ), () () .
. - , , .. : xn=xn-1+xn-2, n>2, x1=1, x2=1 . : 1, 1, 2, 5, 29,..., , , . , ( ) , . : xn=(xn-1)2+(xn-2)2, n=3, 4,.... " " , , , , .
:
;
() ( ), ;
( "-"), .
, , , , ( ). - " " ( ).
. , , , , , , . , , , , , - - . , , ("") -.
, , , .
. - HDI (Human Development Index - , ), 4 , :
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1. (25-85 );
2. (0-100 %);
3. (0-15 );
4. (200-40000 $).
HDI, , . () , , , , HDI. , HDI (, ) , - , , , , , . HDI, , ( ) , , 1994 34 ( 200 ), 1996 - 57- ; ( - ).
.
. - - , , .
, , . ( ) , , , . ( - ) .
(, , ) , . , . , - .
( , ) "" " ".
r, , - , ( ) . r n- , n X. (,), (2-) , - X×Y. r , y Y , r(x,y) r(X,Y).
. : 1 - , 2 - -, 3 - , 4 - , 5 - . " " : i r j, i j . R ( r i j , i j, - ):
R = r r r r
r r r
r
, " " x X , " " . , , r, ={: r} ={|r}.
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() r=r1 o r2. r1 r2, , - r, :
r 1) ; 2) ; 3) mp; 4) ; 5) s, , , ,
1.
2.
3.
4.
5.
. "=" - ( x=x), ( x=y => y=x), ( x=>y, y=>z => x=>z). " " - , , (). " " - , , ().
r , (.. ) r(), , , .
r() - , , x, y X, , .
( ) , - .
. N - . r(x,y): "x y" N, , . r(x,y): "x y" R, - . r(x,y): "x<y" R ( ). "x y" - , , ( ).
.
, ( ) . , : () - X, {,} ( ).
, () S = <A, R>, = {}, R = {r), r - , A - . . - . - , , . ("") .
Y - ( ) , r C, S = <A,Y,R> , . ( ) , , .
, , . r() - , X, r(), x .
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. . x,y X r(x,y), , . X r(x,y), x,y X, Xx, Xy - x, y , Xx=Xy.
. x, y, x = y+ka, x, y, k, a Z, x y a x = y (mod a). :
1. x = x (mod a), k=0 ();
2. x = y (mod a) => x = y+ka => y = x+(-k)a => y = x (mod a) ();
3. x = y(mod a), y = z(mod a) => x = y+ka, y = z+ma => x = z+(k+m)a => x=z(mod a) ().
Z k :
X0={x: x=ka, k, a Z},
X1={x: x=1+ka, k, a Z},
X2={x: x=2+ka, k, a Z},
...
Xk-1 = {x: x=k-1+ka, k, a Z}.
, k=2 Z X0 - X1 - ; k=3 - Z X0 - 3, X1 - 3 1, 2 - 3 2.
, , , . ( , ) .
"" - (, ).
X Y X ( , ) I. , Y ( , ) I, I X Y. . .
. , , . , - , ( , , ). - , ; , , . "" - .
S - r X×Y:
r - S-1 Y×X
. ( ) :
1. r1(x1, x2,..., xn), r2(x1, x2,..., xn), X, r3(X)=r1 r2 - X, r1 r2;
2. - r3(X)=r1 r2 - - X, r1 r2;
3. r1() k, .. r1=r1(x1, x2,..., xk), x1, x2,..., xk X ( k ), - r2 m<k, .. () ;
4. r1(x1, x2,..., xk), r2(x1, x2,..., xk) - r3=r1 - r2, X, r1, r2;
5. r2(x1, x2,..., xk) r1(xn+1, xn+2,..., xn+m) - r3=r1×r2, X, r1, r2; n r3 , r1, m r2;
6. (, ) q , r; q - .
.
"" , . - , .
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A=<X, f> X, f ( ), - .
f n-, n ( - ).
F={f} A , X={x} - .
, , , , , - , () :
1. - + = +, = ;
2. - (+)+z = +(+z), (xy)z = x(yz);
3. - + = , xx = x;
4. - (x+y)z = xz+yz, xy+z = (x+z)(y+z);
5. ( ) - ;
6. - x(x+y) = x, x+xy = x;
7. - x+y = xy, xy = x+y
8.: x(y+y) = x, x+yy = x.
9. ( " -1" "-0"), 0 = 1, 1 = 0, x+x = 1, xx = 0.
- A=<X, f> f.
- , . .
. ={x1, x2,..., xn} - . S(X). ( ) - : S(X). , , .
- ( : ==), f , .. ( ) xfa=b, afx=b.
. ={x1, x2,..., xn} - . i, i=1, 2,..., n xi-1, . () , : xixi-1= , xi-1xi= . - ( ) : . , .
- : () , () - ( ), .
- , .
. , , - , .
( r) X Y - - f: X Y, , x1 X x2 X r , y1=f(x1) y2=f(x2) r .
, () , () , .
, , ( ) () ( ), .
1. , ?
2. , , ?
3. ? ? ?
1. ( ), . .
2. , : ) , , ; ) , , ; ) , , ; ) , , ; ) .
3. , . ?