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LPS 30
HOMEWORK 2
(50 POINTS )
Problem 1. (20 points) Consider the following dictionary:
● Let s mean“The sky is rarely cloudy.”
● Let d mean“The dog is sleeping inside.”
● Let c mean“My car is broken.”
Using the dictionary, translate these propositions into the language of propositional logic.
(1) There sky is rarely cloudy but the dog is sleeping inside.
(2) The dog is sleeping inside only if my care is broken.
Using the dictionary, translate the following formulae of propositional logic into English:
(1) (s&c) 5 d.
(2) c& ~ d.
Problem 2. (30 points) Complete the truth tables for the following propostions and determine whether they are tautologies, contradictions or contingencies:
(1) (p&q) V q
| p | q | ( p & q ) V q |
| 0 | 0 |
|
| 0 | 1 |
|
| 1 | 0 |
|
| 1 | 1 |
|
(2) (q = q) = (p =~ p)
| p | q | ( q = q ) = ( p = ~ p ) |
| 0 | 0 |
|
| 0 | 1 |
|
| 1 | 0 |
|
| 1 | 1 |
|
(3) (p&((p V q) 5 r)) 5 r
| p | q | r | ( p & (( p V q ) 5 r )) 5 r |
| 0 | 0 | 0 |
|
| 0 | 0 | 1 |
|
| 0 | 1 | 0 |
|
| 0 | 1 | 1 |
|
| 1 | 0 | 0 |
|
| 1 | 0 | 1 |
|
| 1 | 1 | 0 |
|
| 1 | 1 | 1 |
|