. .
1. . :
:
.
2. 1 ( ). .
3. , , . . .
4. , , , ..
, □, .
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2) .
3) .
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1) ; ;
2)
: 1. ;
2. ;
3. ;
4. ;
5. ;
3) 6. (2, 4)
7. (3, 5)
8. (1, 6)
9. .
, , . . , . .
5.
5.1.
, .
, , : . .
1. n , 1´ 2´´ n , n x 1, 2,..., n, 1, 2,..., n, 1, 2,..., n .
( 1, 2,..., n) : 1´ 2´´ n {0,1}. , ( 1, 2,..., n) 1´´ n {0,1}, 0, ( 1, 2,..., n) 1, . 1, 2,..., n , 1, 2, , n, . , . .
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1. " " , , : Z {0, 1}. Z, {0, 1}.
2. " < " " " , R ´ R Z ´ Z .
2. ( 1, 2,..., n), 1´ 2´´ n, ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) .
: P + ={( 1, 2,..., n): l ( ( 1, 2,..., n)) =1}.
.
1. (, ):" ", ´ ,
= {1, 2, 3, 4, 5, 6}, + = {(1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (4, 2), (6, 2), (6,3)}.
2. (,y): " x < ", 1´ 2, 1 = {1, 2, 3}, 2 = {2, 4, 6}, P+ = {(1,2), (1,4), (1,6), (2,4), (2,6), (3,4), (3,6)}.
3. ( 1, 2,..., n) Q ( 1, 2,..., n), 1´ 2´´ n, , ( 1, 2,..., n) 1´ 2´´ n, , += Q+. Q ( 1, 2,..., n), 1´ 2´´ n ( 1, 2,..., n), , 1, 2,..., n 1, 2,..., n, ( 1, 2,..., n), , .
5.2.
4. ( 1, 2,..., n), 1´ 2´´ n, ( 1, 2,..., n), , ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) , ( 1, 2,..., n) .
, (): " 3", = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} : " 3". + = {0, 3, 6, 9}, ()+ = {1, 2, 4, 5, 7, 8}.
+ , (M ).
5. n ( 1, 2,..., n) Q ( 1, 2,..., n), 1´ 2´´ n, n ( 1, 2,..., n) Q ( 1, 2,..., n), ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) Q ( 1, 2,..., n) .
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6. n ( 1, 2,..., n) Q ( 1, 2,..., n), 1´ 2´´ n, n ( 1, 2,..., n) Ú Q ( 1, 2,..., n), ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) Q ( 1, 2,..., n) .
7. n ( 1, 2,..., n) Q ( 1, 2,..., n), 1´ 2´´ n, n ( 1, 2,..., n) Q ( 1, 2,..., n), ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) , Q ( 1, 2,..., n) . .
8. n ( 1, 2,..., n) Q ( 1, 2,..., n), 1´ 2´´ n, n ( 1, 2,..., n) Q ( 1, 2,..., n), ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) Q ( 1, 2,..., n) , . .
( 1, 2,..., n) Q (y 1, y 2,..., ym) , , , , (n + m k) , k , .
, , , : () (). () , , () P (x) (), " " " ".
1. () " 7", Z. : " Z 7" " Z, 7". , () () () , , . .
. ( 1, 2,..., n) n , Q ( 2,..., n)= ( 1, 2,..., n), R ( 2,..., n)= ( 1, 2,..., n), 2, 2,..., n, (n 1) .
, , , . , , .
2. (, y) , " 2 + 2 1" R ´ R. (, y) , 1, . Q (y)= x P (x, y), R (x)= (, ), S = x y P (x, y). Q (y) , R. . , Q , Q+ = Æ. R (x) , , . [1, 1]. , R. S , . S , , . . , , [1, 1], . S .
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5.3.
, , . n ( 1, 2,..., n) Q ( 1, 2,..., n) 1´ 2´´ n, :
1) , ;
2) Q + Q+, ;
3) Q + Q+, ;
4) Q ;
5) Q .
. ():" " Q (x): " ", = {1, 2, 3, 4, 5, 6, 7, 8}. : + = {2, 4, 6, 8}, Q+ = {3, 6}. , (1)(5)
) {1, 3, 5, 7}, {l, 2, 4, 5, 7, 8};
) {3, 6} = {6};
) = {2, 4, 6, 8} {3,6}= {2, 3, 4, 6, 8};
) ={1, 3, 5, 7} {3,6} ={1, 3, 5, 6, 7};
) ={1, 3, 5, 6, 7} {1, 2, 4, 5, 6, 7, 8}= {1,5,6,7}.
5.4.
9. ( 1, 2,... n), 1´ 2´´ n, , ( 1, 2,..., n) 1´ 2´´ n ( 1, 2,..., n) , ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) . ( 1, 2,... n), 1´ 2´´ n, , ( 1, 2,..., n) 1´ 2´´ n, ( 1, 2,..., n) , ( 1, 2,..., n) 1´ 2´´ n ( 1, 2,..., n) .
, , . " x P (x) , () . . $ () , () .
5.5. .
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. , :
: , , z, xi, yi, zi ();
: , Q, R, i, Qi, Ri,
(i N);
n () : (,...,), Q (,...,), R (,...,), i (,...,), Qi (,...,), Ri (,...,) (i N) ;
: ;
: ", $;
: (,) ;, .
10. ( ).
1. .
2. n ( 1, 2,..., n) .
3. F, G , , , (), (F Ù G), (F Ú G), (F G), (F G) .
4. F , , F, " x F (x) $ x F (x) .
5. .
, , , , , .
, .
11. F G , , , , , .
. , , :
1. : , .
2. :
;
.
3. Q , :
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4. :
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5. :
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5.6.
12. F () , , , , () . ( ) , , , ( ) . (), , , ( ) .
F G (F G).
1. , x y , N = .
, x y, x, . , x, y. . , .
2. , .
A , . , , . , - y:x, A , , , . , y<x, A , , . , A .
3. , .
, A M, :
.
A .
4. , .
, , , , A.
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13. F', F, , F' , , . () F , , .
, " $ (), F .
.
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.
, . , , . , . 1936 . . , . , , .
6. ()
, .
. , -. , ( ):
(P1) ;
(P2) ,
t x.
modus pones (MP) :
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F.
.
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. :
1. .., .. . . : . 3 ., . .: , 2008. 288 .
2. : , , / .. . 7- . .: , 2005. 268 .: .
3. .. : . . .. / ... 2- .,. .: , 2006. 304 .
4. .. : . . .. / ... .: , 2004. 448 .
5. .., .., .. . 2- . .: , 2002. 128 .
6. . . I: . / . . . -; . ... , 2005. 48 .
1. ... 3
1.1. 3
1.2. 5
1.3. . .. 6
1.4. .....11
1.5. 18
1.6. 21
2. .25
2.1. 25
2.2.
.... 30
2.3.
- .................. 33
3. ... 35
4. Ʌ.41
5. .... 43
4.
5.
5.1. .. 43
5.2. .... 45
5.3. - ..48
5.4. ... 50
5.5. .
. 50
5.6. ⅅ 52
6. . 55
.. 57