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M325K_Quiz1_Soln - p →∼ q q → r r ∠q ∴ ∼ p p q...

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M325K Quiz 1-Solutions 2/3/2010 Name: 1. (4pts) Give the converse, inverse, contrapositive, and negation of the following statement. “If it is night and the moon is full then Jane turns into a wolf.” [ p q r ] Converse: “If Jane turns into a wolf, then it is night and the moon is full.” [ r p q ] Inverse: “If it is not night or the moon is not full, then Jane doesn’t turn into a wolf.” [ p ∨ ∼ q →∼ r ] Contrapositive: “If Jane does not turn into a wolf, then it is not night or the moon is not full.” [ r →∼ p ∨ ∼ q ] Negation: “It is night and the moon is full and Jane does not turn into a wolf.” [ p q ∧ ∼ r ] Note: Negation of an if-then statement does not have the word ‘if’, see p.21. 2. (3pts) Use the truth table to determine whether the following argument is valid. Include a few words of explanation to support your answer.
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Unformatted text preview: p →∼ q q → r r ∧ q ∴ ∼ p p q r ∼ q ∼ r p →∼ q ∼ r → q r ∧ ∼ q ∼ p T T T F F F T F F T T F F T F T F F T F T T F T T T F T F F T T T F F F F T T F F T T F T F T F F T T T F T F F T T F T T T T F F F T T T F F T This argument is invalid since there exists a critical row with the false conclusion. (Critical row is a row where all the premises are true.) 3. (3pts) Simplify the following (using Theorem 1.1.1). Supply a reason for each step. ∼ ( p ∨ ∼ q ) ∨ ( ∼ p ∧ ∼ q ) ∼ ( p ∨ ∼ q ) ∨ ( ∼ p ∧ ∼ q ) ≡ ( ∼ p ∧ ∼ ( ∼ q )) ∨ ( ∼ p ∧ ∼ q ) De Morgan’s Law ≡ ( ∼ p ∧ q ) ∨ ( ∼ p ∧ ∼ q ) Double Negative ≡∼ p ∧ ( q ∨ ∼ q ) Distributive ≡∼ p ∧ t Negation ≡∼ p Identity...
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