By Saharon Shelah

ISBN-10: 3540164480

ISBN-13: 9783540164487

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**Sample text**

We first identify the two predicates that appear in sentences 1–3. ” 1. ” In logical form, we have ∀x(C(x) → A(x)). 2. ” In logical form, we have ∃x(C(x) ∧ A(x)). 3. There are two equivalent ways to restate sentence 3. First, this sentence means that “it is false that some cat is an animal,” that is, ¬(some cat is an animal). In logical form, we obtain ¬∃x(C(x) ∧ A(x)). Secondly, the sentence also means that “every cat fails to be an animal;” that is, ∀x(C(x) → ¬A(x)). We now symbolize the predicates that appear in sentences 4–6.

Every cat is an animal. ”] 2. No cat is an animal. 3. Someone in this class does not do their homework. ”] Solution. Using our solutions in Example 4 on page 36, we shall first translate each of the English statements into logical form. We will take negations of these logical forms and “push through” the negation symbol using quantifier negation laws and propositional logic laws. The result will then be expressed in English. 1. ” LOGICAL FORM: ∀x(C(x) → A(x)). ” LOGICAL NEGATION: ¬∀x(C(x) → A(x)).

4. Evaluate the truth sets: (a) (b) (c) (d) 5. Let {x ∈ R : x2 < 9}. {x ∈ Z : x2 < 9}. {x ∈ R : 2x + 9 ≤ 5}. {x ∈ R : x > 0 and x3 < p q and Suppose p q r s 16 x }. be rational numbers where p, q, r, s are integers and q, s are nonzero. = rs . 3, show that 2p+q 2q = 2r+s 2s . 2 Quantifiers Given a statement P(x), which says something about the variable x, we want to express the fact that every element x in the universe makes P(x) true. In addition, we may want to express the fact that at least one element x in the universe makes P(x) true.

### Around Classification Theory of Models by Saharon Shelah

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