Prove the following logic rules from other ones. (The only rule you can\'t use i
ID: 3824800 • Letter: P
Question
Prove the following logic rules from other ones. (The only rule you can't use is the one you are trying to prove). Don't forget to prove both directions of doubleheadarrow rules. a. Double negation: P doubleheadarrow P b. Demorgan's Law: (A logicalor B) doubleheadarrow A logicaland B c. Associativity of V: X logicalor (Y logicalor Z) doubleheadarrow (X logicalor Y) logicalor Z d. Distribution/Factorization: (P logicaland Q) logicalor R doubleheadarrow (P logicalor R) logicaland (Q logicalor R) e. Modus ponens: (A logicaland (A rightarrow B) rightarrow B f. commutativity of V: (X logicalor Y) doubleheadarrow (Y logicalor X) g. Transitivity of rightarrow: ((P rightarrow Q) logicaland (Q rightarrow R)) rightarrow (P rightarrow R)Explanation / Answer
A
B
A<->B
F
F
T
F
T
F
T
F
F
T
T
T
p
¬p
¬(¬p)
¬(¬p)<->p
T
F
T
T
F
T
F
T
¬(¬p)<->p is a tautology
So ¬(¬p)<->p
A
B
A V B
¬( A V B )
¬A ^ ¬B
F
F
T
T
F
T
T
T
F
T
T
F
T
F
F
T
T
F
F
T
T
F
F
T
T
T
F
F
T
F
F
T
¬( A V B )<-> ¬A ^ ¬Bis a tautology
So
c)X V (y V z) <-> (x V y) V z
x
y
z
y V z
X V y
X V (y V z)
(x V y) V z
X V (y V z) <-> (x V y) V z
F
F
F
F
F
F
F
T
F
F
T
T
F
T
T
T
F
T
F
T
T
T
T
T
F
T
T
T
T
T
T
T
T
F
F
F
T
T
T
T
T
F
T
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
X V (y V z) <-> (x V y) V z is a tautology
So X V (y V z) <-> (x V y) V z
D)
( p ^ q ) V r <-> (p V r ) ^ (q V r)
p
q
r
P ^ q
p V r
q V r
( p ^ q ) V r
(p V r ) ^ (q V r)
( p ^ q ) V r<->(p v r) ^ (q v r)
F
F
F
F
F
F
F
F
T
F
F
T
F
T
T
T
T
T
F
T
F
F
F
T
F
F
T
F
T
T
F
T
T
T
T
T
T
F
F
F
T
F
F
F
T
T
F
T
F
T
T
T
T
T
T
T
F
T
T
T
T
T
T
T
T
T
T
T
T
T
T
T
So ( p ^ q ) V r <-> (p V r ) ^ (q V r) is a tautology
So ( p ^ q ) V r <-> (p V r ) ^ (q V r)
e)
A ^ ( A -> B) < - > B
A
B
A -> B
A ^ ( A -> B)
A ^ ( A -> B) - > B
F
F
T
F
T
F
T
T
F
T
T
F
F
F
T
T
T
T
T
T
A ^ ( A -> B) < - > B is a tautology
So A ^ ( A -> B) < - > B
g) (x V y ) <-> ( y V x)
X
Y
X V y
Y V x
(x v y ) <-> (y v x)
F
F
F
F
T
F
T
T
T
T
T
F
T
T
T
T
T
T
T
T
(x V y ) <-> ( y V x) is a tautology
So ) (x V y ) <-> ( y V x)
A
B
A<->B
F
F
T
F
T
F
T
F
F
T
T
T
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.