1. In transforming a formula into an equivalent CNF, you can use the absorption laws to eliminate conjunctions within disjuntions, i.e. expressions such as p ˅ (q ˄ r) and (p ˄ q) ˅ r. True False 2. Let p, q, and r be propositional variables. Which of the following expressions would NOT be formulas in conjunctive normal form? (p ˄ ¬q) ˅ (¬r ˄ q ˄ p) ¬q p ˅ q ˅ ¬p p ˄ ¬p 3. Consider the propositional logic formula (p ˄ q) ˅ (r ˅ ¬s) From the options below, which one is the equivalent CNF? To determine the correct answer, transform the formula above into CNF using the steps learnt in this module. (p ˅ r ˅ s) ˄ (q ˅ r ˅ ¬s) (¬p ∨ r ∨ s) ∧ (¬q ∨ ¬r ∨ ¬s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ s) (p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ ¬s)
1. In transforming a formula into an equivalent CNF, you can use the absorption laws to eliminate conjunctions within disjuntions, i.e. expressions such as p ˅ (q ˄ r) and (p ˄ q) ˅ r.
True
False
2. Let p, q, and r be propositional variables. Which of the following expressions would NOT be formulas in conjunctive normal form?
(p ˄ ¬q) ˅ (¬r ˄ q ˄ p)
¬q
p ˅ q ˅ ¬p
p ˄ ¬p
3. Consider the propositional logic formula
(p ˄ q) ˅ (r ˅ ¬s)
From the options below, which one is the equivalent CNF? To determine the correct answer, transform the formula above into CNF using the steps learnt in this module.
(p ˅ r ˅ s) ˄ (q ˅ r ˅ ¬s)
(¬p ∨ r ∨ s) ∧ (¬q ∨ ¬r ∨ ¬s)
(p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ s)
(p ˅ r ˅ ¬s) ˄ (q ˅ r ˅ ¬s)
4. Consider the propositional logic formula
r → ¬(¬p → s)
From the options below, which one is the equivalent CNF? To determine the correct answer, transform the formula above into CNF using the steps learnt in this module.
(¬r ˅ ¬p) ˄ (¬r ˅ ¬s)
(r ˅ p) ˄ (r ˅ s)
¬r ˅ ¬(p ˅ s)
r ˅ (¬p ˄ ¬s)
5. Which of the following are correct statements? Select all that apply.
The resolvent of p and ¬p is ∅.
Let C1, C2 be clauses. The set of clauses {C1, C2, T} and {C1, C2} are logically equivalent.
The empty set of clauses is unsatisfiable.
A set of clauses is valid if and only if every clause in the set is true in every interpretation
6. Which formulas are in 3CNF? Select all that apply.
p
(¬p ˅ q ˅ ¬r) ˄ (p ˅ ¬q ˅ r)
p ˄ q ˄ r
(¬p) ˄ (¬q ˅ r)
7. Let p, q, r, and s be propositional variables. Consider the formula
(p ˅ q) ˄ (r ˅ s)
If you applied the
(p ˅ q ˅ r) ˄ (q ˅ r ˅ s)
p ˅ q ˅ s
(p ˅ q ˅ t) ˄ (r ˅ s ˅ u)
(p ˅ q ˅ t) ˄ (p ˅ q ˅ ¬t) ˄ (r ˅ s ˅ u) ˄ (r ˅ s ˅ ¬u)
8. Let p, q, r, and s be propositional variables. What is the resolvent of clauses C1 and C2?
C1 = pq̄r
C2 = q̄r̄s
pqs
q̄
pq̄s
F
9. Which sets of clauses are satisfiable? Select all that apply.
{pq̄, p, p̄q}
{pqr}
{p̄q̄, p, p̄q}
{p, p̄}
10. The proof that the set of clauses
is unsatisfiable is given below. Clauses in Steps 7 through 11 were obtained using the resolution rule. For each clause in Steps 7 through 11, please select the number of the clause that was used to obtain it.
(7): (1) and
(8): (3) and
(9): (4) and
(10): (5) and
(11): (9) and
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