Read e-book online A Logical Introduction to Proof PDF

By Daniel W. Cunningham

ISBN-10: 1461436303

ISBN-13: 9781461436300

The publication is meant for college kids who are looking to easy methods to turn out theorems and be larger ready for the trials required in additional enhance arithmetic. one of many key elements during this textbook is the advance of a strategy 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 components of an explanation. an evidence diagram presents a device for exhibiting scholars the best way to write right mathematical proofs.

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Therefore, {x ∈ R+ : (x − 1)2 > 1} = (2, ∞). (3) We need to find the negative real numbers x that satisfy x > 1x . From x > 1x , we conclude that x2 < 1 because x < 0. So we have −1 < x < 0 and, as a result, we obtain {x ∈ R− : x > 1x } = (−1, 0). 1 1. Let qp and rs be rational numbers where p, q, r, s are integers and q, s are nonzero. ps Suppose qp = rs and p = 0. 3, show that qs = pr ps . 2. Let R(x) be the predicate x > 1x . Indicate whether the statements R(2), R(−2), R( 12 ), R(− 12 ) are true or false.

3 1. Using quantifier negation laws and propositional logic laws, translate each of the following assertions into positive statements. ) (a) (b) (c) (d) (e) (f) ¬(∀x > 2)(x < 4 → x2 < 16). ¬(∀x < 2)(x < 4 → x2 < 16). ¬(∃x < 2)(x < 4 ∧ x2 < 4). ¬(∃x > 2)(x < 4 ∧ x2 < 4). ¬(∀x ∈ N)(x > 2 → 3x < 2x ). ¬(∀x ∈ N)(x > 4 → 3x < 2x ). 2. Express the negation of the following statement as √ a positive statement: For all real numbers x, if x is rational and positive, then x is irrational. State your √ result in English.

There is a black square. Every circle is white. There are no black circles. a is north of e. a is not north of j. Every circle is north of d. There is a circle that has the same color as d. d is west of every circle. Solution. We first note that statements 2 and 7 are the only ones that are false in the Tarskian world depicted in Fig. 1. We shall now express, in order, each of the eight sentences in logical form. 1. ” Thus, the logical form of the sentence is ∃x(B(x) ∧ S(x)). 2 Quantifiers 39 2.

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A Logical Introduction to Proof by Daniel W. Cunningham

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