CS
What is the truth table for the negation (NOT) operator?
A
Which of the following is the truth table for the implication (->) connective in propositional logic?
A
Which of the following is a valid propositional equivalence? (Propositional Equivalences)
A) (p → q) ∧ (q → p) ↔ p ↔ q
Which of the following best defines the concept of soundness in propositional logic?
A) A proof system is sound if and only if every provable formula is a tautology.
What is not the result of a completed semantic tableau with no open branches?
A) A tautology
What is the final step in constructing a semantic tableau?
A) Check if there are any open branches
Which of the following is a theorem of conjunction in a Hilbert deduction system?
A) Conjunction Elimination: p ∧ q ⊢ p
Which of the following is a theorem of disjunction in a deduction system?
A) Disjunction Introduction: p ⊢ p ∨ q
What is the inverse of the proposition "If it rains, then the ground is wet"? (Inverse)
A) If it doesn't rain, then the ground is dry
Which of the following is a logically equivalent statement to "If P, then Q" in propositional logic?
A) If not Q, then not P
What is the contrapositive of the proposition "If it rains, then the ground is wet"? (Contrapositive)
A) If the ground is dry, then it must not have rained.
What is the converse of the proposition "If it rains, then the ground is wet"? (Converse)
A) If the ground is wet, then it must have rained.
Let φ be a formula and Γ be a set of formulas in propositional logic. Which of the following statements is true?
A) If Γ ⊢ φ, then Γ ⊨ φ.
Let P and Q be propositional variables. Which of the following is a sound inference rule?
A) Modus Tollens: From P → Q and ¬Q, infer ¬P
Which of the following is a soundproof system for propositional logic?
A) Natural Deduction
Which of the following is a contradiction in propositional logic?
A) P and ~P
Which of the following is a tautology?
A) P ∨ ¬P
Which of the following is a valid tableau for the formula ~(p -> q) -> (p /\ ~q)? | ~(p -> q) -> (p /\ ~q) | 1 | ~(p -> q) | 2 | p /\ ~q 3 | p 4 | ~q 5 | p -> q 6 | q 7 | contradiction (3,4) 8 | contradiction (2,7) 9 | contradiction (1,8)
A) Valid
What is the rule for terminating a semantic tableau construction?
A) When all branches are closed
What is the first step in constructing a semantic tableau?
A) Write the negation of the proposition to be analyzed
Which of the following is a theorem in the Hilbert Deduction System for implication?
A) p → q, p ⊢ q
Which of the following is a theorem in the Hilbert Deduction System for implication?
A) p → q, q → r ⊢ p → r
Which of the following is a valid propositional equivalence? (Propositional Equivalences)
A) p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r)
Which of the following is a valid application of resolution? (Resolution)
A) p ∧ q, ¬q ∴ p
Which of the following is a theorem in the Hilbert Deduction System for disjunction?
A) p ∨ q, ¬p ⊢ q
Which of the following is in CNF? (CNF)
B) (p ∧ q) ∨ r
Which of the following is in DNF? (DNF
B) (p ∧ q) ∨ r
Which of the following is in CNF? (CNF)
B) (p ∨ q) ∧ (p ∨ ¬q)
Which of the following is a disjunctive normal form (DNF) of the statement "If P and Q are true, then R is true" in propositional logic?
B) (~P v ~Q) v R
What is the result of a completed semantic tableau that has only closed branches?
B) A contradiction
Which of the following best defines the concept of completeness in propositional logic?
B) A proof system is complete if and only if every tautology is provable.
Which of the following conditions must be met for a semantic tableau to be terminated?
B) All branches are closed.
What is the next step in constructing a semantic tableau after writing the negation of the proposition to be analyzed?
B) Apply logical rules to the root of the tree
When constructing a semantic tableau, what should you do if a branch contains p and ¬p?
B) Close the branch
What is the truth value of the conjunction (AND) of two propositions if one is false?
B) False
Which of the following is a tautology in propositional logic?
B) P v ~P
Which of the following is a contradiction?
B) P ∧ ¬P
Which of the following statements is true regarding soundness and completeness in propositional logic?
B) Soundness implies completeness, but completeness does not imply soundness.
What is the purpose of semantic tableaux in propositional logic?
B) To find all contradictory propositions
Which of the following is a contradiction? (Contradictions)
B) p ∧ ¬p
Which of the following is a tautology? (Tautologies)
B) p ∨ ¬p
Which of the following is the correct construction of a semantic tableau for the formula ~(p -> q) v ~(q -> r) v (p -> r)?
B) ~(~p v q) v ~(~q v r) v (~p v r)
Which of the following is a theorem in the Hilbert Deduction System for negation?
B) ¬(p ∨ q), ¬p ⊢ ¬q
What is the result of a completed semantic tableau with at least one open branch?
C) A contingency
Which of the following is a complete proof system for propositional logic?
C) Both A and B
Let P and Q be propositional variables. Which of the following is a complete inference rule?
C) Hypothetical Syllogism: From P → Q and Q → R, infer P → R
Which of the following is the contrapositive of the statement "If it is raining, then the ground is wet" in propositional logic?
C) If the ground is not wet, then it is not raining.
Which of the following is a theorem of implication in a Hilbert deduction system?
C) Modus Tollens: (p → q) ∧ ¬q ⊢ ¬p
Which of the following is equivalent to ~(P -> Q) in propositional logic?
C) P ^ ~Q
What is the negation of the statement "P and Q are both true" in propositional logic?
C) P is false, or Q is false
Which of the following is a theorem in the Hilbert Deduction System for negation?
C) p → q, ¬q ⊢ ¬p
Which of the following is the negation of "P or Q" in propositional logic?
C) ~P and ~Q
Which of the following is a contingent formula?
D) All of the above
Which of the following is a propositional logic connective?
D) And
Which of the following is a tautology? (Tautologies)
D) None of the above
Which of the following is a theorem of negation in a deduction system?
D) Transposition: (p → q) ≡ (¬q → ¬p)