1. Use the following predicates to express the sentences below in predicate calculus:

        male(x) - means that the object denoted by x is male.

        female(x) - means that the object denoted by x is female.

        vegetarian(x) - means that x is a vegetarian.

        butcher(x) - means that x is a butcher

        likes(x,y) - means that x likes y

    a. No man is both a butcher and a vegetarian.

    b. All men except butchers like vegetarians.

    c. The only vegetarian butchers are women.

    d. No man likes a woman who is a vegetarian.

    e. No woman likes a man who does not like all vegetarians.

2. Translate the following predicate calculus to colloquial English:

    a. ( $ x" t Person(x) Ù Time(t) Ù Canfool(x, t) )

    b. Ø (" x Glitters(x) Þ Gold(x))

3. Convert the following to clause form:

a. (" x $ y P(x,y) Þ Q(x,y))

b. (Ø " x $ y P(x,y) Þ Q(x,y))

c. (Ø " x P(x)) Þ ($ x P(x))

d. (" x " y (P(x,y) Ú Q(x,y) ) Þ R(x,y))

e. ($ x " y $ z (P(x) Ù S(x)) Þ (Q(y) Þ R(z)))

f. ( " x [ a(x) Ù b(x) ] Þ [ c(x,i) Ù ( $ y ( $ z c(y,z) ) Þ d(x,y) ) ] )

4.  Prove (PÞ (QÞ R)) Þ ((PÞ Q) Þ (PÞ R)) using resolution refutation.

5.  Prove (" x P(x) Þ Q(x)) Þ ((" x P(x)) Þ (" x Q(x))) using resolution refutation.

6. The law says that it is a crime to sell an unregistered gun. Ralph has several unregistered guns, and all of them were purchased from Lefty. First, use predicate calculus to represent the knowledge and data in these statements, then, convert to clause form and using resolution refutation, develop a proof that Lefty is a criminal.