1. Decide if the following sentences are valid, unsatisfiable, or neither. To do it, use the truth tables and equivalency rules in Chapter 7.

Big Big

Big Heavy

(Big Heavy) ( Big Heavy)

Big Heavy Heavy

(Big Dense) Heavy) (Big Dense) (Heavy Dense)

(Big Dense) ((Big Heavy) Dense)

Small Cute (Small Cute)

(Small Cute) Cute

((Rain Wet) (Wet Cold)) (Rain Cold)

( Rain Wet ) ( Wet Cold ) ( Rain Cold )

2. For each of the following formulas, state briefly if it is a correct representation in propositional calculus of the sentence "If Bill works and his father stays at home, then his mother is happy" or not and explain why. The propositions used in the sentences should have an obvious interpretation.

BillWork DadHome MomHappy

(BillWork DadHome) MomHappy

(BillWork DadHome) MomHappy

MomHappy (BillWork DanHome)

BillWork ( MomHappy DadHome)

3. Convert the following set of propositional clauses to CNF

Cold Dry Pleasant

January Winter Wet

Winter Dry

Wet Cold

January

and prove by resolution with refutation that it is Pleasant.