Preparation for Second Midterm

1. What is the negation of each of the following sentences?

   1. for-every x, exists y: son(x) = y
   2. for-every x, for-every y: son(x) = y <=> father(y) = x
   

   In your answers, any "not" may only appear after the last appearance of a universal or existential quantifier.

2. Textbook exercise 8.8 (in both second and third edition).

3. Consider the technique of propositionalization. For each of the following two knowledge bases, decide if propositionalization can be applied successfully. If not, why not?

   KB 1:
   
   for-every x: king(x) and greedy(x) => evil(x)
   king(John)
   greedy(John)
   brother(Richard, John)
   
   
   KB 2:
   for-every x: king(x) and greedy(x) => evil(x)
   king(John)
   greedy(John)
   brother(Richard, John)
   king(father(John))

4. Textbook exercise 9.19, parts (a), (b), (c) (second edition), exercise 9.24, parts (a), (b), (c) (third edition).