1. Two sentences in propositional calculus can be shown to be equivalent by proving that one entails the other and vice-versa. Show that (P ? Q) ? P ? Q by doing the following steps:

a. Prove by contradiction using resolution (P ? Q) ? P ? Q

b. Prove by contradiction using resolution P ? Q ? (P ? Q)

2. Write each of the following sentences, in predicate calculus. Use the same predicates across the sentences. Use predicates such as House(x), Pet(x), In(x,y), Live(x,y), Big(x), and Cost(x,y).

a. All houses have at least one bathroom.

b. There is a house in Minneapolis which costs more than any other house.

c. There is only one house in Minneapolis which is in the historical register.

d. There is a pet in each house.

e. Rich people have big houses.

f. All big houses are expensive.

g. All expensive houses are big.

h. A house is expensive if it is big.

i. Any small house costs less than any big house.

j. There is only one house with a garden.

3. Transform to CNF the following knowledge base, where upper case arguments are constants, lower case arguments are variables. Do not forget to standardize variables apart.

a. ? x G(x) ? H(x)

b. ? x H(x) ? I(x)

c. ? x H(x) ? J(x,D)

d. ? x I(x) ? J(C,x)

e. ? x I(x) ? J(x,y)

Prove by resolution with refutation " G(B)".

4. You are given the following knowledge base:

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

b. Everyone, except butchers, likes vegetarians.

c. John is a vegetarian.

Write the sentences in predicate calculus, transform them to CNF, and use resolution to answer the question "Is there someone that John likes?".