Chapter 5: Logic 2022 moup Discussion 5.1 CONJUNCTION AND DISJUNCTION OURSE: \( \qquad \) jolve the following problems. Show your Solution. SCORE: \( \qquad \) CHEDULE: \( \qquad \) Table 1. \begin{tabular}{|c|c|c|c|c|} \hline\( p \) & \( q \) & \( \sim p \) & \( \sim p \wedge q \) & \( \sim(\sim p \wedge q) \) \\ \hline T & T & & & \\ \hline T & F & & & \\ \hline F & T & & & \\ \hline F & F & & & \\ \hline \end{tabular} Table 2. \begin{tabular}{|c|c|c|c|} \hline\( p \) & \( q \) & \( \sim q \) & \( \sim p \vee \sim q \) \\ \hline\( T \) & \( T \) & \( F \) & \\ \hline\( T \) & \( F \) & \( T \) & \\ \hline\( F \) & \( T \) & \( F \) & \\ \hline\( F \) & \( F \) & \( T \) & \\ \hline \end{tabular}
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1. Calculate \( \sim p \) (negation of \( p \)): - If \( p = T \), then \( \sim p = F \). - If \( p = F \), then \( \sim p = T \). 2. Calculate \( \sim p \wedge q \) (conjunction of \( \sim p \) and \( q \)): - \( \sim p \wedge q \) is true if both \( Show more…
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3. Use logical equivalences to demonstrate that the inverse and the converse of p -> q are logically equivalent. Identify all logical equivalences by name. You will not receive credit for a truth table solution. 4. Rephrase verbally in equivalent only if, sufficient, necessary, contrapositive and unless form: "if we had an FTL drive, then we could visit the stars". 5. Is the statement ∃x∀y(xy = 0) true or false? Explain. 6. If P and Q are predicates over some domain, and if it is true that ∀x(P(x) ∨ Q(x)), must ∀xP(x) ∨ ∀Q(x) also be true? Explain. 7. Suppose P is the predicate defined by P(x, y) = x is friends with y, where x and y are people. (No one is considered to be friends with themselves.) Translate the formal expression ∀x∃y∃z(y ≠ z ∧ P(x, y) ∧ P(x, z)) into English. 8. Let P be defined as in the previous problem. Is ∀x∃y∃z(y ≠ z → P(x, y) ∧ P(x, z)) true or false? Explain.
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