# 7 2 points write the formal negation of x y x y your

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visit the stars”.7. (2 points) Write the formal negation ofxy(x > y). Your negation mustnot contain any explicit negation symbols.8. (4 points) Demonstrate using logical equivalences that (pq)(p→ ¬q)isnota contradiction. Identify all logical equivalences by name. You willnot receive credit for a truth table solution.9. (4 points) Use logical equivalences to simplify (pq)(¬p→ ¬q) untilyou have at most one occurrence of each variablep, qremaining. Identifyall logical equivalences by name. You will not receive credit for a truthtable solution.10. (2 points) Is the statementxy(xy= 0) true or false? The domain ofdiscourse is the set of real numbers. Explain.11. (3 points) IfPandQare predicates over some domain, and if it is truethatx(P(x)Q(x)), mustxP(x)∨ ∀xQ(x) also be true? Explain.12. (2 points) SupposePis the predicate defined byP(x, y) =xis friendswithy, wherexandyare people. (No one is considered to be friends withthemselves.)Translatethe formal expressionxyz(y6=zP(x, y)P(x, z)) into English.13. (2 points) LetPbe defined as in the previous problem. Isxyz(y6=zP(x, y)P(x, z)) true or false? Explain.c 2020 R. Boerner, ASU School of Mathematical and Statistical Sciences
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