HW #4 — §1.4: Nested Quantifiers

All solutions must be handwritten.

PUT YOUR NAME, DATE, RECITATION SECTION & "HW #4" AT TOP RIGHT OF EACH PAGE; STAPLE MULTIPLE PAGES

1. (3 points each; total = 21 points)

   p. 59: 10 a–b, d, f–i

   • This problem asks you to translate 7 English sentences into the language of FOL
     (using one, 2-place predicate).

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2. (3 points each; total = 12 points)

   p. 59: 12 a, d, f–g

   • This problem asks you to translate 4 English sentences into the language of FOL
     (using one, 1-place predicate and one, 2-place predicate).

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3. (3 points each; total = 15 points)

   p. 61: 28 a–e

   • This problem gives you 5 FOL propositions whose domain is R (the set of all real numbers),
     and asks you to determine their truth values.

   • If a universally quantified proposition is false,
     give a counterexample!

   • You may have to do a bit of elementary algebra;
     if you need algebra hints, send me email.

   • It may help if you translate the FOL into English.

   • You will get partial credit only if you show all your work.
     (And by now you should know what "only if" means :-)

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4. (3 points each; total = 12 points)

   p. 61: 32 a–d

   • This problem gives you 4 FOL propositions,
     and asks you to compute their negations in such a way that all negation signs appear only in front of predicates.

     Here's an example: As you learned in lecture and read about in the text,

     ¬∀xP(x) ≡ ∃x[¬P(x)]

     So, the negation of ∀xP(x) can be written in either of those two ways.
     But only the one on the right-hand side of the logical-equivalence sign has its negation sign in front of the predicate.

   • Hint for part (c): Do you remember what the negation of a biconditional is?

Total points = 60

Tentative grading scheme:

A       57-60
A-      54-56
B+      51-53
B       47-50
B-      44-46
C+      41-43
C       34-40
C-      27-33
D+      21-26
D       11-20
F        0-10