1. ? , . .
) ((a) Ú ($ x P(x)); ) (a Ú ($ x P(x)); ) ((" x P(x, y)) Ú Q(x)); ) (P(x) Ù 
(y));
) (P(x) Ú Q(y, z)); ) ($ t Q(x) (" z P(z, t))); ) (($ x P(x, y)) (" y P(x, y)));
) (" x 
); ) (a ($ y R(x))); ) 
; ) 
.
2. , , :
) ((a Ú b) ($ x P(x, y))); ) ( 
(" x P(x, y))); ) P(x) Ú ($ x P(x));
) (((" P(x, y)) Ú P(x, x)) ($ y (P(x, y) Ù Q(x, y))))); ) ($ x (P(x) Ú (a b)) Ù P(x));
) (" x ($ y (P(z, y) Ù (" z Q(z, x)))) R(x, z)); ) ( 
Ú (" z P(x, z))).
3. ? , ? , ? , , ? ?
4. , :
) $ y P x, y Ú S x; ) a Ú b c; ) a Ù " y P x; ) a Ù " y R y Q x, y;
) 
a $ x P x, y Q x; ) R x " z R z Ú P x, x.
5. :
) x, y z, y ³ z;
) x > y, z z×x > z×y;
) x, y , x ³ y, z t z 
t;
) 5 < 2, , , ;
) [0; 1] , 10;
) , , , ;
) , , ;
) 0 < 
³ 0;
) , , , , ;
6. :
) ;
) ;
) , ;
) , , ;
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) , , ;
) , .
1. :
) ($ x P(x, y)), J = (M = N, a = 1, P(x, y) = (x > y));
) (a ((" x P(x)) P(y))), J = (M = {0}, a = 5, P(x, y) = (x £ y));
) ($ x P(x, y)), J = (M = N, P(x) = (x > 1));
) (a ((" x P(x)) P(y))), J = (M = {0}, a = 1, P(x) = (x > 0));
) (($ x P(x, y)) (" y (P(x, y) Ú Q(y)))), J = (M = {1}, P(x, y)=(x | y), Q(y)=(y = 1));
) ($ x P(x, y)), J = (M = R, P(x, y) = (
> 2));
) (P(x) Q(x, y)), J = (M = R, x = 1, y = 1+i, P(x) = (x > 2), Q(x, y)=(y = 1));
) (P(x, y) P(y, x)), J = (M = R, x = 1, y = 1, P(x, y) = (x > y), P(y, x)=(y = 1));
) (" x (P(x, y) ($ y (P(y, y) Ú P(y, x))))), J = (M = R, x = 1, y = 0, P(x, y) = (x | y)).
2. :
) (" x Q(x, y)), J = (M = N, y = 1, Q(x, y) = (x ³ y));
) (" x Q(x, y)), J = (M = {0, 1}, y = 0, Q(x, y) = (x > y));
) (" x (Q(x, y) P(x)), J = (M = N, y = 2, Q(x, y) = (x = y), P(x) = (x ¹ 1));
) (" x (Q(x, y) P(x)), J = (M = N, y = 2, Q(x, y) = (x = y), P(x) = (x ³ 1));
) (b (($ x R(x)) R(y))), J = (M = {0, 1}, b = 0, y = 1, R(x) = (x ¹ 0));
) (b (($ x R(x)) R(y))), J = (M = {0, 1}, b = 1, y = 1, R(x) = (x > 1));
) (($ x R(x, y)) R(y, x))), J = (M = {0, 1}, y = 1,,x = 1, R(x, y) = (x > y));
) (($ x R(x, y)) R(y, x)), J = (M = {0, 1}, y = 0, x = 1, R(x, y) = (x > y));
) (P(x) ($ y P(y))), J = (M = R, x = 0, P(x) = (x > 0));
) (P(x) ($ y P(y))), J = (M = R, x = 1, P(x) = (x > 0)).
3. : ) (P(x) Ú Q(x)); ) (Q(x) Q(y)); ) (a Ù ($ x R(x, y)));
) (a ($ x P(x, y))); ) (($ x Q(x)) Ù (" y (Q(y) Ú Q(x)))); ) ((" y R(y)) R(x));
) ($ x (" y (P(x, y) Q(x)))); ) ((" x (" y S(x, y))) S(z, z)); ) (P(x) 
);
) 
; ) ((" x R(x)) R(y)); ) (P(x) (a P(x))).
4. ?
) (P(x) P(y)); ) (($ x R(x)) Ú 
); ) ((" x S(x)) ($ y S(y))); ) (a Ú P(x));
) (a Ú (P(x) Ú 
)); ) ((" x (" y Q(x, y))) (" x Q(x, x))); ) (R(x) (" x R(x)));
) (" x ((P(x) (x)) 
)); ) (
Ù A(x));
) 
; ) (($ x (P(x) Q(x))) ((" x P(x)) ($ x Q(x))).
5. ?
) (P(x) Ù Q(x)); ) (Q(x) Q(y)); ) (" x (R(x) Ú S(x))); ) ($ x (P(x, y) Ú R(x))); ) (" x (($ y P(x, y)) Ú R(z))); ) 
; ) 
($ x P(x));
) ((" x (P(x) a)) Ú ($ x (P(x) ))); ) ((" z T(z)) Ú (a Ù 
).
6. , :
) $ x P(x) Ú ; ) R(x) " y P(y) R(x); ) " x S(x) S(y); ) a Ú P(x); ) " x S(x) 
; ) $ y Q(y, x) Q(x, y) Ú " x Q(x, x) Ù 
.
7. , ) ; ) .
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8. , , ?
