M a shqlar : 1 misol

M a sh q l a r :

1 - misol. $x R ( x, u ) – formula bajariluvchi formuladir. Haqiqatdan, R ( x, u ) – natural sonlar to`plamida aniqangan “ u ∶ x ” predikat bo‘lsin, u holda

R ( 1, u ) = 1. Demak, $x R ( x, u ) = 1.

2 - misol . $x $u R ( x, u ) – predikat bajariluvchi predikatdir. Ha=iqatdan, R ( x, u ) – predikat natural sonlar to‘plamida aniqangan « x > u » predikati bo‘lsin, u holda

R ( 5, 1 ) = 1. Demak, $x $u R ( x, u ) = 1.

3 - misol. R ( x ) Ú ù R ( x ) – predikat umumqiymatli predikatdir.

4 - misol. R ( x ) Ù ù R ( x ) – predikat aynan yolg‘on predikatdir.

1. 
   Natural sonlar to`plamida qaralayotgan quyidagi predikatli formulalarning qaysilari bajariluvchi ( aynan rost, aynan yolg‘on ) ekanligini aniqlang :
   

1. 
   $x "u (ù ( u > x ) Û $x ( ù $u ( u > x ));
   
2. 
   $x "u (( u > x ) Ú ù ( u > 0 )) Û $x "u ( u > 0 Þ u > x);
   

3).ù ( "x $u $z ( x < y Ù z² > y ) Û "x $y ( x < y Ù $z ( z² > y )).
1.Yechish: $x "u (ù ( u > x ) Û $x ( ù $u ( u > x )) ≡ $x "u (ù ( u > x ) Û $x ("u ù ( u > x )) ≡ 1 demak berilgan formula aynan rost formula eran.

1. 
   quyidagi formulalarning bajariluvchi ekanligini isbotlang :
   

1. 
   $x $y ( A ( x ) Ù ù A ( y ));
   
2. 
   $x "u ( V ( x, u ) Þ "z C ( x, y, z ));
   
3. 
   A ( x ) Þ "y A ( y );
   
4. 
   "x ( A ( x ) Ú B ( x )) Þ ( "x A ( x ) Ú "x B ( x )).
   

4. quyidagi formulalarning umumqiymatli ekanligini isbotlang :

1. 
   $x "u V ( x, u ) Þ "u $x V ( x, u );
   
2. 
   "x ( A ( x ) Þ V ( x )) Þ ( "x A ( x ) Þ "x V ( x ));
   
3. 
   "x ( A ( x ) Þ V ( x )) Þ ( $x A ( x ) Þ $x V ( x ));
   
4. 
   $x ( A ( x ) Þ V ( x )) Û ( "x A ( x ) Þ "x V ( x )).
   

5.Teng kuchli almashtirishlar yordamida quyidagi formulalarni keltirilgan normal formaga aylantiring :

1. 
   $x ( A ( x ) Þ "u ( V ( u ))).
   
2. 
   ù ( "x A ( x ) Þ $u V ( u )).
   
3. 
   $x ( "u A ( u ) Þ V ( x )) Ù ù ( "u $x ( V ( x ) Þ A ( u ))).
   
4. 
   ù ("x A ( x ) Ú $x( V ( x ) Þ S ( x ))).
   

6.Teng kuchli almashtirishlar yordamida quyidagi formulalarni preneksli normal formalarga aylantiring :

1. 
   "u A ( x, u ) Þ V ( x, x ).
   
2. 
   "u A ( u, z ) Þ $x B ( x, t, z ).
   
3. 
   $x A ( x, y, z ) Þ ù ( "x B ( x, y )).
   
4. 
   "x ( A ( x ) Þ B ( x )) Þ ( $x A ( x ) Þ $y B ( y )).