Virtual Labs

Propositional Logic: Syntax and Inference

Which of the following is not an atomic proposition?

a: The sun is hot Explanation

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b: I take a taxi if I have enough money Explanation

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c: It was almost noon Explanation

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d: There was no water in the jug on the table in the kitchen Explanation

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If the weather is warm and the sky is clear, then we go boating. Pick the valid conclusion. [Hint: try to think of each proposition as a symbol]

a: We did not go boating, so the weather must not have been warm Explanation

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b: The sky was cloudy and the weather was warm, so we did not go boating. Explanation

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c: We went boating, so the weather must have been warm and the sky clear Explanation

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d: The sky was clear and the weather was warm. We did not go boating Explanation

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What is the logical form of the statement: 'Either it's raining or I'll go for a walk'?

a: p ∧ q Explanation

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b: p ∨ q Explanation

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c: p → q Explanation

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d: ¬p → q Explanation

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Which inference rule states that if we know 'p → q' and 'p' is true, then 'q' must be true?

a: Modus Ponens Explanation

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b: Modus Tollens Explanation

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c: Hypothetical Syllogism Explanation

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d: Disjunctive Syllogism Explanation

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What is the contrapositive of 'If it's sunny, then I'm happy'?

a: If I'm happy, then it's sunny Explanation

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b: If I'm not happy, then it's not sunny Explanation

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c: If it's not sunny, then I'm not happy Explanation

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d: If I'm not happy, then it's sunny Explanation

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In a truth table for p → (q → r), how many rows will have a final value of true?

a: 3 Explanation

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b: 4 Explanation

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c: 6 Explanation

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d: 7 Explanation

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Which of the following is a valid application of the Resolution rule?

a: (p ∨ q), (¬p ∨ r) ⊢ (q ∨ r) Explanation

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b: (p ∨ q), (¬p ∨ ¬q) ⊢ p Explanation

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c: (p ∨ q), (p ∨ ¬q) ⊢ p Explanation

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d: (p ∨ q), (¬q ∨ r) ⊢ p Explanation

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What is the conjunctive normal form (CNF) of p → (q ∨ r)?

a: (¬p ∨ q ∨ r) Explanation

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b: (p ∧ q) ∨ (p ∧ r) Explanation

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c: (¬p ∧ q) ∨ (¬p ∧ r) Explanation

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d: (¬p ∨ q) ∧ (¬p ∨ r) Explanation

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Which of the following formulas is satisfiable but not valid?

a: p ∨ ¬p Explanation

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b: p ∧ ¬p Explanation

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c: p → p Explanation

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d: p ∧ q Explanation

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