Problem-22
Complete the truth table for each of these compound propositions.
a) p→(¬q∨r)
b) ¬p→(r→q)
c) (p→q)∧(¬p→r)
d) (p↔q)∨(¬q↔r)
e) (¬p↔¬q)↔(q↔r)
Solution
a) p→(¬q∨r)
|
p |
q |
r |
¬q |
¬q∨r |
p→(¬q∨r) |
|
T |
T |
T |
F |
T |
T |
|
T |
T |
F |
F |
F |
F |
|
T |
F |
T |
T |
T |
T |
|
T |
F |
F |
T |
T |
T |
|
F |
T |
T |
F |
T |
T |
|
F |
T |
F |
F |
F |
T |
|
F |
F |
T |
T |
T |
T |
|
F |
F |
F |
T |
T |
T |
b) ¬p→(r→q)
|
p |
q |
r |
¬p |
r→q |
¬p→(r→q) |
|
T |
T |
T |
F |
T |
T |
|
T |
T |
F |
F |
T |
T |
|
T |
F |
T |
F |
F |
T |
|
T |
F |
F |
F |
T |
T |
|
F |
T |
T |
T |
T |
T |
|
F |
T |
F |
T |
T |
T |
|
F |
F |
T |
T |
F |
F |
|
F |
F |
F |
T |
T |
T |
c) (p→q)∧(¬p→r)
|
p |
q |
r |
¬p | p→q | ¬p→r |
(p→q)∧(¬p→r) |
|
T |
T |
T |
F | T | T |
T |
|
T |
T |
F |
F | T | T |
T |
|
T |
F |
T |
F | F | T |
F |
|
T |
F |
F |
F | F | T |
F |
|
F |
T |
T |
T | T | T |
T |
|
F |
T |
F |
T | T | F |
F |
|
F |
F |
T |
T | T | T |
T |
|
F |
F |
F |
T | T | F |
F |
d) (p↔q)∨(¬q↔r)
|
p |
q |
r |
¬q | p↔q | ¬q↔r |
(p↔q)∨(¬q↔r) |
|
T |
T |
T |
F | T | F |
T |
|
T |
T |
F |
F | T | T |
T |
|
T |
F |
T |
T | F | T |
T |
|
T |
F |
F |
T | F | F |
F |
|
F |
T |
T |
F | F | F |
F |
|
F |
T |
F |
F | F | T |
T |
|
F |
F |
T |
T | T | T |
T |
|
F |
F |
F |
T | T | F |
T |
e) (¬p↔¬q)↔(q↔r)
|
p |
q |
r |
¬p | ¬q | ¬p↔¬q | q↔r |
(¬p↔¬q)↔(q↔r) |
|
T |
T |
T |
F | F | T | T |
T |
|
T |
T |
F |
F | F | T | F |
F |
|
T |
F |
T |
F | T | F | F |
T |
|
T |
F |
F |
F | T | F | T |
F |
|
F |
T |
T |
T | F | F | T |
F |
|
F |
T |
F |
T | F | F | F |
T |
|
F |
F |
T |
T | T | T | F |
F |
|
F |
F |
F |
T | T | T | T |
T |