3 Use logical equivalences to demonstrate that the inverse and the converse of

# 3 use logical equivalences to demonstrate that the

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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? The domain of discourse is the set of real numbers. 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 ) ∨ ∀ xQ ( 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 6 = z P ( x, y ) P ( x, z )) into English. 8. Let P be defined as in the previous problem. Is x y z ( y 6 = z P ( x, y ) P ( x, z )) true or false? Explain. c 2018 R. Boerner, ASU School of Mathematical and Statistical Sciences