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3. Use logical equivalences to demonstrate that the inverse and the converseofp→qare logically equivalent. Identify all logical equivalences by name.You will not receive credit for a truth table solution.4. Rephrase verbally in equivalentonly if,sufficient,necessary,contrapositiveandunlessform: ”if we had an FTL drive, then we could visit the stars”.5. Is the statement∃x∀y(xy= 0) true or false? The domain of discourse isthe set of real numbers. Explain.6. IfPandQare predicates over some domain, and if it is true that∀x(P(x)∨Q(x)), must∀xP(x)∨ ∀xQ(x) also be true? Explain.7. SupposePis the predicate defined byP(x, y) =xis friends withy, wherexandyare people. (No one is considered to be friends with themselves.)Translatethe formal expression∀x∃y∃z(y6=z∧P(x, y)∧P(x, z)) intoEnglish.8. LetPbe defined as in the previous problem. Is∀x∃y∃z(y6=z→P(x, y)∧P(x, z)) true or false? Explain.c 2018 R. Boerner, ASU School of Mathematical and Statistical Sciences
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Logic,Homework Week 1,ASU School of Mathematical and Statistical Sciences,R. Boerner