ME290M, Spring 1999, Week 4a
Week 4b
Solution to Exercises in
Week 4b
3 Exercises on Interpretation
For each of the following sentences, give interpretations to the symbols so that the sentence makes sense and represents the world accurately (i.e., so that you believe the sentence to be true).
- (> 3 7)
3 is greater than 7
- (SHORTER ALICE (ETCHEVERRY-HALL))
Alice is shorter than Etcheverry Hall.
- (Ductile+ COPPER BRASS)
Copper is more.ductile than brass.
- (EXIST something (AND (GOLD something) (NOT (GLITTER something))))
There exists something that is gold and does not glitter. Or as Tolkein said: All that is gold does not glitter.
- (EXIST something (AND (WANDER something) (NOT (LOST something))))
There exists something that wanders but is not lost. Or as Tolkein said: Not all those who wander are lost.
- (IF (REPTILE x)(ANIMAL x))
If something is a reptile than it is an animal.
- (IF (R a b c) (R b c a))
If relation R holds for a, b and c, then it holds for b, c and a.
- (IF (NOT (P)) (NOT (Q)))
If P is not true, then Q is not true.
4 Exercises on Satisfaction and Validity
Say whether each of the following sentences is valid, satisfiable or unsatisfiable.
- (IF P (NOT P))
Satisfiable only if P is false.
- (IF (NOT P) P)
Satisfiable only if P is true.
- (IF P P)
Valid, or tautological.
- (AND (IF P (NOT P))
(IF (NOT P) P))
Unsatisfiable or inconsistent.
- (AND (NOT P) P)
Unsatisfiable or inconsistent.
- (OR (NOT P) P)
Valid, or tautological.
- (IF P (IF Q P))
Valid, or tautological.
One way to figure this out is to use the fact that
(IF A B) is the same as (OR (NOT A) B). Thus
(IF P (IF Q P)) is the same as (OR (NOT P) (NOT Q) P),
which is clearly tautological.
Or using a truth table:
|
P | Q | (IF P Q) | (IF P (IF Q P))
|
|
T | T | T | T |
|
T | F | T | T |
|
F | T | F | T |
|
F | F | T | T |
- (OR (NOT P) (IF Q P))
Valid, or tautological.
Again converting to a disjunction it is obviously tautological.
(OR (NOT P) (IF Q P)) is the same as (OR (NOT P) (NOT Q) P).
Or using a truth table:
|
P | (NOT P) | Q | (IF Q P) | (OR (NOT P) (IF Q P))
|
|
T | F | T | T | T |
|
T | F | F | T | T |
|
F | T | T | F | T |
|
F | T | F | T | T |
Last updated: 20 February 99
Send Comments to: Alice Agogino, aagogino@me.berkeley.edu
Copyright © 1999 Alice Agogino;
All Rights Reserved.
