Practice With Long Truth-Tables





INSTRUCTIONS: Construct a long truth-table for each of the arguments, and then determine whether the argument is valid. Answers follow the exercises.
 

1.
P1 P & Q
P2 P v Q
C P -> Q

2.
P1 P -> Q
P2 ~P
C ~Q

3.
P1 Q v P
P2 P -> (Q -> P)
C P <-> Q

4.
P1 Q
C Q v Q

5.
P1 (P & Q) -> (P v Q)
P2 ~P
P3 ~Q
C ~ (P -> Q)

6.
P1 P -> (Q -> R)
P2 ~(Q v R)
C ~P

7.
P1 ~(P <-> Q)
P2 ~(P -> (Q -> P))
C R

8.
P1 P -> R
P2 ~R v (P v R)
P3 P -> ~(P v R)
C ~P

9.
P1 P -> (P <-> Q)
P2 Q & (P v Q)
C ~~P

10.
P1 G -> S
P2 S v G
C G & S



ANSWERS:

1. valid


 
P1 P2 C
P Q P & Q P v Q P -> Q
T T  T  T T
T F F T F
F T F T T
F F F F T

 
2. invalid (third row)
P1
P2
C
P
Q
P -> Q
~P
~Q
T
T
T
F
F
T
F
F
F
T
F
T
T
T
F
F
F
T
T
T

 
3. invalid (rows 2 and 3)
P1
P2
C
P
Q
Q v P
P -> (Q -> P)
P <-> Q
T
T
T
T       T
T
T
F
T
T       T
F
F
T
T
 T        F
F
F
F
F
 T       T
T

 
4. valid
P1
C
Q
Q v Q
T
T
F
F

 
5. invalid (row 4)
P1
P2
P3
C
P
Q
(P & Q) -> 
(P v Q)
~P
~Q
~(P -> Q)
T
T
T
F
F
F
T
F
T
F
T
T
F
T
T
T
F
F
F
F
T
T
T
F

 
6. invalid (row 4)
P1
P2
C
P
Q
R
P -> (Q -> R)
~(Q v R)
~P
T
T
T
T
F
F
T
T
F
F
F
F
T
F
T
T
F
F
T
F
F
T
T
F
F
T
T
T
F
T
F
T
F
T
F
T
F
F
T
T
F
T
F
F
F
T
T
T

 
7. valid
C
P1
P2
P
Q
R
~(P <-> Q)
~(P -> (Q ->P))
T
T
T
F
F
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
F
T
T
T
F
F
T
F
T
F
F
F
T
F
F
F
F
F
F
F

 
8. valid
P1
P2
P3
C
P
R
P -> R
~R v (P v R)
P -> ~(P v R)
~P
T
T
T
T
 F
F
T
F
F
T
 F
F
F
T
T
T
T
T
F
F
T
T
T
T

 
9. valid
P1
P2
C
P
Q
P -> (P <-> Q)
Q & (P v Q)
~~P
T
T
T
T
T
T
F
F
F
T
F
T
T
F
F
F
F
T
F
F

 
10. invalid (row 3)
P1
P2
C
G
S
G -> S 
S v G
G & S
T
T
T
T
T
T
F
F
T
F
F
T
T
T
F
F
F
T
F
F