A Logical Introduction to Proof - download pdf or read online

By Daniel W. Cunningham

ISBN-10: 1461436303

ISBN-13: 9781461436300

The e-book is meant for college kids who are looking to methods to end up theorems and be larger ready for the trials required in additional enhance arithmetic. one of many key parts during this textbook is the improvement of a technique to put naked the constitution underpinning the development of an evidence, a lot as diagramming a sentence lays naked its grammatical constitution. Diagramming an explanation is a manner of offering the relationships among a few of the elements of an explanation. an evidence diagram offers a device for exhibiting scholars the way to write right mathematical proofs.

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Example 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.

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A Logical Introduction to Proof by Daniel W. Cunningham

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