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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.

- 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*))

- ∃
- 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 "

**has a ferret". Express each of these statements in first-order logic using these relations. Let the domain be your classmates.**

*x*- A classmate has a cat, a dog, and a ferret.
- All your classmates have a cat, a dog, or a ferret.
- At least one of your classmates has a cat and a ferret, but not a dog.
- None of your classmates has a cat, a dog, and a ferret.
- 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.

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