m: U [0, 1]. U , :
m = {(x1, m(x1)), , (xn, m(xn))}.
m = m(x1)/x1 + + m(xn)/xn
:
x1 | x2 | xn | |
m(x1) | m(x2) | m(xn) |
R m(x) . , , m(a) = 1. , , :
. .
a- m.
1. , , .
1)
2) a < b, ,
3) .
2 ( ). .
.
, ; m1 m2, , : .
. :
();
();
();
( );
( );
( );
( );
();
().
m a-, . :
();
();
();
:
.
. , 0. , :
, a- . , m1 m2 R .
{0, 1} &, Ú, Ø, [0, 1].
:
, a, b, c Î [0, 1] :
1) a Ù 1 = a (1 );
2) a £ b, a Ù c £ b Ù c ();
3) a Ù b = b Ù a ();
4) (a Ù b) Ù = a Ù (b Ù c) ().
, , 0 £ 0Ùx £ 0Ù1 = 0, : 0 Ù x = 0.
:
1) a Ç b = min (a, b) ();
2) a * b = max(0, a + b 1) ();
3) a × b = ab ( ).
.
, a, b, c Î [0, 1] :
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|
1) 0 Ú a = a (0 );
2) a £ b, a Ú c £ b Ú c ();
3) a Ú b = b Ú a ();
4) (a Ú b) Ú = a Ú (b Ú c) ().
:
1) a È b = max(a, b) ();
2) a Ú b = min(a + b, 1) ();
3) a Ú b = a + b ab ( ).
g: [0,1] [0,1] , , , :
1) g(0) = 1; g(1) = 0;
2) a £ b, g(a) ³ g(b).
:
1) ();
2) ( );
3) ( );
4) , -1 < l < ¥ ().
Ù Ú g-,
.
, :
- ( ).
Ù . , Ù, , x Î [0, 1] :
x £ (a b), x Ù a £ b.
:
a b = sup {x Î [0, 1]: x Ù a £ b }.
1) C :
a b = min {1 a + b, 1}.
2) C ø:
3) C a×b :
. , - :Øa Ú b, a Ú b = max(a,b):
a b = max(1-a, b).
, Øa Ú b , a Ú b = a + b ab :
a b = 1 a + ab.
:
a b = max(1 a, min(a, b)).
, Øa = a 0.
, U1, U2, , Un R Í U1 ´ U2 ´´ Un. ( ), .
U1, U2, , Un . U1, U2, , Un . - . r, s Î F (X ´ Y). :
(r È s)(x, y) = max (r(x, y), s(x, y)), (r Ç s)(x, y) = min (r(x, y), s(x, y)).
max min, , , .
F (X ´ Y) X Y X ´ Y. , r Í s , r(x, y) £ s(x, y) x Î X y Î Y.
r Î F (X ´ Y) s Î F (Y ´ Z). rs Î F (Y ´ Z) . :
1) (rs)t = r(st),
2) ,
, ,.
|
|
, .
r Î F (X ´ Y) , . , :
1) ();
2) ();
3) ().
2 , .
, , :
.
.
5.4.
0 () 1 () Ù, Ú, Ø :
1) i = 1, 2, ;
2) 0 1 ;
3) g f , (f Ù g) (f Ú g) ;
4) f , Øf .
F.
:
(F1) Ø0 = 1,
(F2) A Ù 1 = A, A Ú 1 = 1, A Ù 0 = 0, A Ú 0 = A,
(F3) Ø(A Ù B) = ØA Ú ØB, Ø(A Ú B) = ØA Ù ØB,
(F4) A Ù (B Ú C) = (A Ù B) Ú (A Ù C), A Ú (B Ù C) = (A Ú B) Ù (A Ú C),
(F5) ØØA = A,
A, B, C Î F.
, ,
t(0) = 0, t(1) =1, t(f Ù g) = min (t(f), t(g)),
t(f Ú g) = max (t(f), t(g)), t(Øf) = 1 t(f).
.
f Î F , t t(f) ³ 0.5. f Î F , t : t(f) £ 0.5.
, , .
1. f Î F ,
f K. f Î F , K.
. , .
: , .
f Î F (), . ().
, 0 1.
fÞg F, , t(f) £ t(g).
f , i Î w. , , , . : . :
. , :
.
. ,
t(x) = 0.3, t() = 0.1, t() = 0.2. t() = 0.3, t() = 0.7. , t() = 0.3. , , , , . , .
2. , . :
1) , ;
2) , .
, , , , .
5.5.
|
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. , :
(Modus Ponens), (Modus Tollens).
.
U , A Í U , , . : X A , X U, , . : X A, , U = w, A , : X X Î A.
, X, U, A, . : X A. X , A . , , , .
, : X A , , , , :
1) X A Y B (X, Y) A Ç B, , , A Ç B U ´ V ;
2) X A Y B (X, Y) A È B, ;
3) X A, Y B (X, Y) A B, ;
4) X A X A, .
, :
.
, a Ù b = min(a, b), a b . .
Modus Ponens . . A, B, A¢ : , , . :
X A, Y B
,
B¢ , : .
B¢ , , B¢ A¢, ( sup, ).
A, B, B¢, , , , Modus Tollens :
X A, Y B
.
:
,
. , , .
4. ......................................................................................................................................... 26
4.1. ......................................................................................................................................................... 26
4.2. .................................................................................................................................................. 29
4.3. ............................................................................................................................... 32
4.4. .................................................................................................................................... 34
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4.5. ........................................................................................................... 36
4.6. ........................................................................................................................................................ 38
4.7. ......................................................................................................................................... 39
5. .......................................................................................................................................................... 41
5.1. ........................................................................................................................................................ 41
5.2. ...................................................................................................................................................... 43
5.3. ....................................................................................................................................................... 45
5.4. ............................................................................................................................. 46
5.5. ....................................................................................................................................... 47