Kenneth H. Rosen

Question

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 terms of $C(x),D(x),F(x)$, quantifiers, and logical connectives.

Let the domain consist of all students in your class.

1. A student in your class has a cat, a dog, and a ferret.
2. All students in your class have a cat, a dog, or a ferret.
3. Some student in your class has a cat and a ferret, but not a dog.
4. No student in your class has a cat, a dog, and a ferret.
5. For each of the three animals, cats,dogs, and ferrets, there is a student in your class who has this animal as a pet.

Comments

can you tell me the topic or chapter number?

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Kenneth H. Rosen
Predicates and Quantifiers
page no 53

http://www2.fiit.stuba.sk/~kvasnicka/Mathematics%20for%20Informatics/Rosen_Discrete_Mathematics_and_Its_Applications_7th_Edition.pdf

Answer 1

a) ∃x (C(x)∧D(x)∧F(x)).

b)∀x(C (x)∨D (x)∨F (x))

c)∃x(C(x)∧F(x)∧¬D(x))

d) This is the negation of part(a): ¬∃x(C(x)∧D(x)∧F(x)).

e) Here the owners of these pets can be different: (∃xC(x))∧(∃xD(x))∧(∃x F(x)). There is no harm in using the same dummy variable, but this could also be written, for example, as : (∃xC(x))∧(∃yD(y))∧(∃z F(z)

Comments

D) No student in your class has a cat, a dog, and a ferret. Does it mean?                                                          

 All students in your class do not have a cat, a dog, and a ferret.

∀x(¬C(x)∧¬D(x)∧¬F(x)).

Please comment.

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¬∃x(C(x)∧D(x)∧F(x))   ≡   ∀x  ¬ ( C(x) ∧ D(x) ∧ F(x) )   ≡    ∀x (   ¬C(x) ∨ ¬D(x) ∨ ¬F(x) )

Using De Morgan's laws for quantifiers.