Extra SL Derivation Problems: Solutions

 

1. DERIVE: ~J

1. ( ~C v ~R ) & ( R L ) PR
2. C ( C v L ) PR
3. J R PR
4.
| J
 AS
5.
| R
 3, 4 E
6.
| R L
 1 &E
7.
| L
 5, 6 E
8.
| C v L
 7 vI
9.
| C
 2, 8 E
10.
| ~C v ~R
 1 &E
11.
| ~~R
 5 DN
12.
| ~C
 10, 11 vE
13. ~J 4-12 ~I

 

2. DERIVE: S

1 ( P v Q ) ( R S ) PR
2. (~S v T ) ( P & R ) PR
3.
| ~S
 AS
4.
| ~S v T
 3 vI
5.
| P & R
 2, 4 E
6.
| P
 5 &E
7.
| P v Q
 6 vI
8.
| R S
 1, 7 E
9.
| R
 5 &E
10.
| ~R
 3, 8 MT
11. S 3-10 ~E

 

3. DERIVE: A & C

1. A C PR
2. A v C PR
3.
| ~A
 AS
4.
| C
 2, 3 vE
5.
| A
 1, 4 E
6.
| ~A
 3 R
7. A 3-6 ~E
8. C 1, 7 E
9. A & C 7, 8 &I

 

4. DERIVE: E R

1. ( A & R ) v ( N & A ) PR
2. ( A ~M ) & ( N M ) PR
3.
| E
 AS
4.
| ( A & R ) v ( A & N )
 1 COM
5.
| A & ( R v N )
 4 DIST
6.
| A
 5 &E
7.
| A ~M
 2 &E
8.
| ~M
 6,7 E
9.
| N M
 2 &E
10.
| ~N
 8,9 MT
11.
| R v N
 5 &E
12.
| R
 10, 11 vE
13. E R 3-12 I

 

5. DERIVE: ( P Q ) v ( ~Q P )

1.
| ~[ ( P Q ) v ( ~Q P ) ]
 AS
2.
| ~( P Q ) & ~( ~Q P )
 1 DeM
3.
| ~( P Q )
 2 &E
4.
| ~( ~Q P )
 2 &E
5.
| ~( ~P v Q )
 3 IMP
6.
| ~~P & ~Q
 5 DeM
7.
| ~( ~~Q v P )
 4 IMP
8.
| ~~~Q & ~P
 7 DeM
9.
| ~~P
 6 &E
10.
| ~P
 8 &E
11. ( P Q ) v ( ~Q P ) 1-10 ~E

 

6. DERIVE: A ( B v C )

1. ( A B ) & ( A C ) PR
2. A B 1 &E
3. A C 1 &E
4.
| A
 AS
5.
| B
 2, 4 E
6.
| B v C
 5 vI
7. A ( B v C ) 4-6 I
8.
| B v C  
AS
9.
| | ~B
AS
10.
| | C
8, 9 vE
11.
| | A
3, 10 E
12.
| | B
2, 11 E
13.
| | ~B
9 R
14.
| B
9-13 ~E
15.
| A
2, 14 E
16. ( B v C ) A 8-15 I
17. A ( B v C ) 7, 16 I

 

7. DERIVE: ( A & ~B ) v A

1. C B PR
2. ( ~C A ) v E PR
3. D & ~E PR
4. B ( A & ~B ) PR
5. ~E 3 &E
6. ~C A 2, 5 vE
7. C ( A & ~B ) 1, 4 HS
8. ~A ~~C 6 TRANS
9. ~A C 8 DN
10. ~A ( A & ~B ) 7, 9 HS
11. ~~A v ( A & ~B ) 10 IMP
12. A v ( A & ~B ) 11 DN
13. ( A & ~B ) v A 12 COM

 

8. DERIVE: P [ ( Q & ~Q ) R ]

1.
| P  
AS
2.
| | Q & ~Q
AS
3.
| | Q
2 &E
4.
| | ~Q
2 &E
5.
| | Q v R
3 vI
6.
| | R
4, 5 vE
7.
| ( Q & ~Q ) R  
2-6 I
8. P [ ( Q & ~Q ) R ] 1-7 I

 

9. DERIVE: I ~H

1. ~H PR
2. ( ~J v H ) H PR
3. I v ( J H ) PR
4. ~( ~J v H ) 1, 2 MT
5. ~( J H ) 4 IMP
6. I 3, 5 vE
7. I & ~H 1, 6 &I
8. ( I & ~H ) v ( ~I & ~~H ) 7 vI
9. I ~H 8 EQUIV

 

10. DERIVE: K

1. H ~( I ~K) PR
2. ~( H v I ) PR
3. ~H & ~I 2 DeM
4.
| ~K  
AS
5.
| | I ~K
AS
6.
| | I
4, 5 E
7.
| | ~I
3 &E
8.
| ~( I ~K)  
5-7 ~I
9.
| H  
1, 8 E
10.
| ~H  
3 &E
11. K 4-10 ~E

 

11. DERIVE: [ ~A ( ~B C ) ] [ ( A v B ) v ( ~~B v C ) ]

1.
| ~A ( ~B C )
 AS
2.
| ~~A v ( ~B C )
 1 IMP
3.
| ~~A v ( ~~B v C )
 2 IMP
4.
| A v ( ~~B v C )
 3 DN
5.
| A v [ ( ~~B v ~~B ) v C ]
 4 IDEM
6.
| A v [ ~~B v ( ~~B v C ) ]
 5 ASSOC
7.
| A v [ B v ( ~~B v C ) ]
 6 DN
8.
| ( A v B ) v ( ~~B v C )
 7 ASSOC
9. [ ~A ( ~B C ) ] [ ( A v B ) v ( ~~B v C ) ]  1-8 I

 

12. DERIVE: ~X Y

1. ~( X Y ) PR
2. ~[ ( X & Y ) v (~X & ~Y ) ] 1 EQUIV
3. ~( X & Y ) & ~( ~X & ~Y ) 2 DeM
4. (~X v ~Y ) & ~( ~X & ~Y ) 3 DeM
5. ( ~Y v ~X ) & ~( ~X & ~Y ) 4 COM
6. ( Y ~X ) & ~( ~X & ~Y ) 5 IMP
7. ( Y ~X ) & ( ~~X v ~~Y ) 6 DeM
8. ( Y ~X ) & ( ~X ~~Y ) 7 IMP
9. ( Y ~X ) & ( ~X Y ) 8 DN
10. ( ~X Y ) & ( Y ~X ) 9 COM
11. ~X Y 10 EQUIV

 

13. DERIVE: H J

1. ( H & T ) J PR
2. ( M  D ) & ( ~D M ) PR
3. ~T ( ~D & M ) PR
4.
| H  
AS
5.
| | ~T
AS
6.
| | ~D & M
3, 5 E
7.
| | M  D
2 &E
8.
| | M
6 &E
9.
| | D
7, 8 E
10.
| | ~D
6 &E
11.
| T  
5-10 ~E
12.
| H & T  
4, 11 &I
13.
| J  
1, 12 E
14. H J 4-13 I

 

14. DERIVE: ~D

1. ~( A B) PR
2. ~( B C ) PR
3. ~( ~A v B ) 1 IMP
4. ~~A & ~B 3 DeM
5. ~( ~B v C ) 2 IMP
6. ~~B & ~C 5 DeM
7. ~B 4 &E
8. ~~B 6 &E
9. ~B v ~D 7 vI
10. ~D 8,9 vE

 

15. DERIVE: A S

1. A [ ( ~N v N ) ( S v T ) ] PR
2. T ~( F v ~F ) PR
3.
| A  
AS
4.
| ( ~N v N ) ( S v T )  
1, 3 E
5.
| | N
AS
6.
| | N
5 R
7.
| N N  
5-6 I
8.
| ~N v N  
7 IMP
9.
| S v T  
4, 8 E
10.
| | ~S
AS
11.
| | T
9, 10 vE
12.
| | ~( F v ~F )
2, 11 E
13.
| | ~F & ~~F
12 DeM
14.
| | ~F
13 &E
15.
| | ~~F
13 &E
16.
| S  
10-15 ~E
17. A S 3-16 I

 

16. DERIVE: O

1. ( N & O ) v [ ( ~P v O ) & N ] PR
2. ~( ~O ~P ) v ~N PR
3. ( N & O ) v [ N & ( ~P v O ) ] 1 COM
4. N & [ O v ( ~P v O ) ] 3 DIST
5. N 4 &E
6. O v ( ~P v O ) 4 &E
7. ~~N 5 DN
8. ~( ~O ~P ) 2, 7 vE
9. ~( P O ) 8 TRANS
10. ~( ~P v O ) 9 IMP
11. O 6, 10 vE

 

17. DERIVE: ~L & ~M

1. M & L PR
2. [ L & ( M & ~S ) ] K PR
3. ~K v ~S PR
4. ~( K ~S ) PR
5. K ~S 3 IMP
6.
| ~S
 AS
7.
| M
 1 &E
8.
| M & ~S
 6, 7 &I
9.
| L
 1 &E
10.
| L & ( M & ~S )
 8, 9 & I
11.
| K
 2, 10 E
12. ~S K 6-11I
13. K ~S 5, 12 I
14. ( K ~S ) v ( ~L & ~M ) 13 vI
15. ~L & ~M 4, 14 vE

 

18. DERIVE: ~( K L ) v {( M L ) [ L v ( ~K & ~M )]}

1.
| K L  
AS
2.
|  | M L
AS
3.
| | | ~L
AS
4.
| | | ~M
2, 3 MT
5.
| | | ~K
1, 3 MT
6.
| | | ~K & ~M
4, 5 &I
7.  
| | ~L ( ~K & ~M )
3-6 I
8.  
| | ~~L v ( ~K & ~M )
7 IMP
9.  
| | L v ( ~K & ~M )
8 DN
10.
| ( M L ) [ L v ( ~K & ~M )]  
2-9 I
11.  ( K L ) {( M L ) [ L v ( ~K & ~M )]} 1-10 I
12. ~( K L ) v {( M L ) [ L v ( ~K & ~M )]} 11 IMP

 

19. DERIVE: ~A [ B ( G v D ) ]

1. ( A v ~B ) v ~C PR
2. ( D v G ) v C PR
3.  A v ( ~B v ~C ) 1 ASSOC
4.
| ~A  
AS
5.
| ~B v ~C  
3, 4 vE
6.
| | B
 AS
7.
| | ~~B
6 DN
8.
| | ~C
5, 7 vE
9.
| | D v G
2, 8 vE
10.
| | G v D
9 COM
11.
| B ( G v D )  
6-10 I
12. ~A [ B ( G v D ) ] 4-11 I

 

20. DERIVE: ~R

1. ( O R ) S PR
2. ( P R ) ~S PR
3.
| R
 AS
4.
| ~O v R
 3 vI
5.
| O R
 4 IMP
6.
| S
 1, 5 E
7.
| ~P v R
 3 vI
8.
| P R
 7 IMP
9.
| ~S
 2, 8 E
10. ~R 3-9 ~I

 

21. DERIVE: ~T v S

1.  ( T & Q ) R PR
2. ( Q R ) ( ~T v S ) PR
3.  T ( Q R ) 1 EXP
4.  T ( ~T v S ) 2, 3 HS
5.  ~T v ( ~T v S ) 4 IMP
6.  ( ~T v ~T ) v S 5 ASSOC
7. ~T v S 6 IDEM

 

22. DERIVE: ~( I v D ) ~A

1. ~( A v G ) v P PR
2. ( P & ~I ) ( O & D ) PR
3.
| A
 AS
4.
| A v G
 3 vI
5.
| ~~( A v G )
 4 DN
6.
| P
 1, 5 vE
7.
| P [ ~I ( O & D ) ]
 2 EXP
8.
| ~I ( O & D )
 6, 7 E
9.
| ~~I v ( O & D )
 8 IMP
10.
| I v ( O & D )
 9 DN
11.
| ( I v O ) & ( I v D )
 10 DIST
12.
| I v D
 11 & E
13. A ( I v D ) 3-12 I
14. ~( I v D ) ~A 13 TRANS

 

23. DERIVE: ~A ~( K C )

1. A v ~( ~K v C ) PR 
2. ( K C ) v ~A PR
3. ~~A v ~( ~K v C ) 1 DN
4. ~A ~( ~K v C ) 3 IMP
5. ~A ~( K C ) 4 IMP
6. ~~( K C ) v ~A 2 DN
7. ~( K C ) ~A 6 IMP
8. ~A ~( K C ) 5, 7 I

 

24. DERIVE ~C v ~E

1. B ( E F ) PR
2. A ( C D ) PR
3. A v B PR
4. ~D & ~F PR
5.
| ~( ~C v ~E )  
AS
6.
| ~~( C & E )  
5 DeM
7.
| C & E  
6 DN
8.
| | A
AS
9.
| | C D
2, 8 E
10.
| | C
7 & E
11.
| | D
9, 10 E
12.
| | ~D
4 & E
13.
| ~A  
8-12 ~I 
14.
| B  
3, 13 vE
15.
| E F  
1, 14 E
16.
| E  
7 &E
17.
| F  
15, 16 E
18.
| ~F  
4 &E
19. ~C v ~E 5-18 ~E

 

25. DERIVE: W U

1. ( U ~W ) ~X PR
2. Y v ( W & Z ) PR
3. ( Y v Z ) X PR
4. ( Y v W ) & ( Y v Z ) 2 DIST
5. Y v Z 4 & E
6.
| W  
AS
7.
| | ~U
AS
8.
| | ~~W
6 DN
9.
| | ~U & ~~W
7, 8 &I
10.
| | ( U & ~W ) v ( ~U & ~~W )
9 vI
11.
| | U ~W
10 EQUIV
12.
| | X
3, 5 E
13.
| | ~X
1, 11 E
14.
| U  
7-13 ~E
15. W U 6-14 I

 

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