F
T
pUnit 5 Answers
1. The equivalences follow by showing that the appropriate pairs of columns of the following table agree.
| p | p
F | p
T | p
p |
| T | F | T | T |
| F | F | T | F |
2.
| p q r | q
r | p
(q r)
|
p
q | p
r |
(p
q)
(p r)
|
| T T T | T | T | T | T | T |
| T T F | T | T | T | F | T |
| T F T | T | T | F | T | T |
| T F F | F | F | F | F | F |
| F T T | T | F | F | F | F |
| F T F | T | F | F | F | F |
| F F T | T | F | F | F | F |
| F F F | F | F | F | F | F |
3. a) If the hypothesis p is true, by the definition of
disjunction, the conclusion p
q is also true.
(p
q) is true.
Altenatively, p
(p
q)
(
p
(p q))
(( p
p )
q)
(T
q)
T
b) If the hypothesis p
q is true,
then both p and q are true so that the conclusion p
q
is also true. If the hypothesis is false, then "if-then" statement is always true.
This can also be proven similarly to the alternative proof for a).
c) If the hypothesis
(p q)
is true, then p
q is false, so that p is true and q is false. Hence,
the conclusion
q is true.If the hypothesis is false, then "if-then" statement is always true.
This can also be proven similarly to the alternative proof for a).
4. a) If p is true, then p
(p
q) is true
since the first proposition in the disjunction is true. On the other hand, if pis
false, then p q is also false, so p (p q) is false. Since p and p
(p
q)
always have the same truth value, they are equivalent.
This can also be proven similarly to b).
b) [ p
(p
q) ]
[ (p
F )
(p
q) ]
[ (p
( F
q) ]
[ p
F ]
p
This can also be proven similarly to a).
5. a) (p
q
r)
b) (p
q
r)
s
c) (p
T)
(q
F)
6.
(p
q
r )
( p
q
r )
( p
q
r )