SOLVED LOGIC PROBLEMS
1  (Ex)(Ey)(Px > Qy) / (x)Px > (Ex)Qx                                         
                                                         
2  a(Ey)(Pa > Qy)     A                                                      
                                                        
3   bPa > Qb         A                                                      
                                                       
4     (x)Px         A                                                      
                                                      
5     Pa > Qb       3,R                                                    
6     Pa            4,UE                                                   
7     Qb            5,6,>E                                                 
8     (Ex)Qx        7,EI                                                   
9    (x)Px > (Ex)Qx     4-8,>I                                              
10  (x)Px > (Ex)Qx    2,3-9,EE                                               
11 (x)Px > (Ex)Qx      1,2-10,EE                                              
                                                                 SET E PROB M  
1  (Ex)(y)(Px > Qy) / (x)Px > (x)Qx                                           
                                                         
2  a(y)(Pa > Qy)      A                                                      
                                                        
3   Pa > Qb           2,UE                                                   
4    (x)Px           A                                                      
                                                       
5    Pa              4,UE                                                   
6    Pa > Qb         3,R                                                    
7    Qb              6,5,>E                                                 
8    (x)Qx           7,UI                                                   
9   (x)Px > (x)Qx     4-8,>I                                                 
10 (x)Px > (x)Qx       1,2-9,EE                                               
                                                                 SET E PROB N  
1  (x)(Ey)(Px > Qy) / (Ex)Px > (Ex)Qx                                         
                                                         
2   (Ex)Px            A                                                      
                                                        
3   aPa              A                                                      
                                                       
4    (x)(Ey)(Px > Qy)     1,R                                               
5    (Ey)(Pa > Qy)     4,UE                                                 
6    bPa > Qb       A                                                      
                                                      
7     Pa            3,R                                                    
8     Qb            6,7,>E                                                 
9     (Ex)Qx        8,EI                                                   
10   (Ex)Qx          5,6-9,EE                                               
11  (Ex)Qx            2,3-10,EE                                              
12 (Ex)Px > (Ex)Qx     2-11,>I                                                
                                                                 SET E PROB O  
1  (x)Px > (x)Qx / (Ex)(y)(Px > Qy)   16   ~(y)(Pa > Qy)     6,R           
                 17  ~~Pa              15,16,(7-16),~I 
2   ~(Ex)(y)(Px > Qy)     A          18  Pa                17,~E           
                19  (x)Px             18,UI           
3    (y)(Pa > Qy)    A              20  (x)Px > (x)Qx     1,R             
               21  (x)Qx             19,20,>E        
4    (Ex)(y)(Px > Qy)     3,EI      22  Qb                21,UE           
5    ~(Ex)(y)(Px > Qy)     2,R      23   Pa              A               
6   ~(y)(Pa > Qy)     4,5,(3-5),~I                   
7    ~Pa             A              24   Qb              22,R            
               25  Pa > Qb           23-24,>I        
8     Pa            A              26  (y)(Pa > Qy)      25,UI           
              27  (Ex)(y)(Px > Qy)     26,EI        
9      ~Qb         A              28 ~~(Ex)(y)(Px > Qy)     2,27,(2-27),~
             29 (Ex)(y)(Px > Qy)    28,~E           
10     ~Pa         7,R                                                    
11     Pa          8,R                                                    
12    ~~Qb          10,11,(9-11),~I                                        
13    Qb            12,~E                                                  
14   Pa > Qb         8-13,>I                                                
15   (y)(Pa > Qy)    14,UI                                                  
                                                                 SET E PROB P  
1  (x)Px > (x)Qx / (Ex)(y)(Px > Qy)                                           
                                                         
2   ~(Ex)(y)(Px > Qy)     A                                                  
                                                        
3   (x)~(y)(Px > Qy)     2,~X                                                
4   ~(y)(Pa > Qy)     3,UE                                                   
5   (Ey)~(Pa > Qy)    4,~U                                                   
6   b~(Pa > Qb)      A                                                      
                                                       
7    Pa & ~Qb        6,C                                                    
8    Pa              7,&E                                                   
9   Pa                5,6-8,EE                                               
10  (x)Px             9,UI                                                   
11  (x)Px > (x)Qx     1,R                                                    
12  (x)Qx             10,11,>E                                               
13  Qb                12,UE                                                  
14  Pa > Qb           13,W                                                   
15  (y)(Pa > Qy)      14,UI                                                  
16  (Ex)(y)(Px > Qy)     15,EI                                               
17 (Ex)(y)(Px > Qy)    2,16,(2-16),RD                                         
                                                                 SET E PROB P  
1  (x)(y)(Px v Qy) / (x)Px v (y)Qy                                            
                                                         
2  (y)(Pa v Qy)        1,UE                                                   
3  Pa v Qb             2,UE                                                   
4   ~((x)Px v (y)Qy)     A                                                   
                                                        
5   ~(x)Px & ~(y)Qy     4,DM                                                 
6   ~(x)Px            5,&E                                                   
7   ~(y)Qy            5,&E                                                   
8   (Ex)~Px           6,~U                                                   
9   (Ey)~Qy           7,~U                                                   
10  a~Pa             A                                                      
                                                       
11   Pa v Qb         3,R                                                    
12   Qb              10,11,vE                                               
13  Qb                8,10-12,EE                                             
14  (y)Qy             13,UI                                                  
15  (x)Px v (y)Qy     14,vI                                                  
                                                                 SET E PROB Q  
1  (x)(y)(Px v Qy) / (x)Px v (y)Qy                                            
                                                         
2  (y)(Pa v Qy)        1,UE                                                   
3  Pa v Qb             2,UE                                                   
4   ~((x)Px v (y)Qy)     A                                                   
                                                        
5   ~(x)Px & ~(y)Qy     4,DM                                                 
6   ~(x)Px            5,&E                                                   
7   (Ex)~Px           6,~U                                                   
8   a~Pa             A                                                      
                                                       
9    Pa v Qb         3,R                                                    
10   Qb              8,9,vE                                                 
11  Qb                7,8-10,EE                                              
12  (y)Qy             11,UI                                                  
13  (x)Px v (y)Qy     12,vI                                                  
14 (x)Px v (y)Qy       4,13,(4-13),RD                                         
                                                                 SET E PROB Q  
1  (Ex)(y)Jxy          P              18  (Ez)Gz            8,9-17,EE       
2  (Ey)(Ez)(Hzy & ~Py)     P          19 (Ez)Gz              2,4-18,EE       
3  (z)(w)((Jzw&~Pw)>Gz) / (Ez)Gz                                              
                                                         
4  c(Ez)(Hzc & ~Pc)     A                                                    
                                                        
5   dHdc & ~Pc       A                                                      
                                                       
6    ~Pc             5,&E                                                   
7   ~Pc               4,5-6,EE                                               
8   (Ex)(y)Jxy        1,R                                                    
9   a(y)Jay          A                                                      
                                                       
10   Jac             9,UE                                                   
11   ~Pc             7,R                                                    
12   Jac & ~Pc       10,11,&I                                               
13   (z)(w)((Jzw&~Pw)>Gz)     3,R                                           
14   (w)((Jaw & ~Pw) > Ga)     13,UE                                        
15   (Jac & ~Pc) > Ga     14,UE                                             
16   Ga              12,15,>E                                               
17   (Ez)Gz          16,EI                                                  
                                                                 SET E PROB R  
