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Quantifiers

16 六月, 2015 - 16:59
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Exercise 4.4.8

Let P (x) be the statement "has been to Prague", where the domain consists of your classmates.

  1. Express each of these quantifcations in English.
    • x :(P (x))
    • x :(P (x))
    • ¬∃x :(P (x))
    • ¬∀x :(P (x))
    • x :(¬P (x))
    • x :(¬P (x))
    • ¬∃x :(¬P (x))
    • ¬∀x :(¬P (x))
  2. Which of these mean the same thing?

Exercise 4.4.9

Let C (x) be the statement "x has a cat", let D (x) be the statement "x has a dog", and let F (x) be the statement "x has a ferret". Express each of these statements in first-order logic using these relations. Let the domain be your classmates.

  1. A classmate has a cat, a dog, and a ferret.
  2. All your classmates have a cat, a dog, or a ferret.
  3. At least one of your classmates has a cat and a ferret, but not a dog.
  4. None of your classmates has a cat, a dog, and a ferret.
  5. For each of the three animals, there is a classmate of yours that has one.

Exercise 4.4.10

Determine the truth value of each of these statements if the domain is all real numbers. Where appropriate, give a witness.

  1. \exists x:(x^{2}=2)
  2. \exists x:(x^{2}=-1)
  3. \forall x:(x^{2}+2\geq 1)
  4. \forall x:(x^{2}\neq x)