Titlebook: Studying Mathematics; The Beauty, the Toil Marco Bramanti,Giancarlo Travaglini Book 2018 Springer International Publishing AG, part of Spri

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Negations and Indirect Proofs with an indirect proof, that is, a proof by contradiction, or by the law of the contrapositive. Some classical examples will help the reader to discuss this method of proof and its connections with the distinction between constructive and non-constructive proofs.

Consequence

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To Understand, i.e., to Know How to Explaineck that they are correct, both in general and, if possible, through ad hoc examples. We then ask the reader to identify the idea(s) behind the proof (solution) and write the idea(s) in a conversational style. The solution of each exercise suggests several further questions, which are partially addressed in the final section ..

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Majorizations.” This is also termed “majorizing . (with .),” and “majorization” is the kind of inequality we find this way. This chapter focuses on the basic ideas and techniques involved in majorizations, starting with the triangle inequality for absolute values and moving on through examples and exercises which reproduce the most typical situations.

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Book 2018he first year of college, for personal study or for introductory courses. It aims to set a meeting between two relatives who rarely speak to each other: the Mathematics of Beauty, which shows up in some popular books and films, and the Mathematics of Toil, which is widely known. Toil can be overcome

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Quantifying (Level A) “there exist(s)” and “for every” are called “quantifiers” and play this role. “At least”, “at most”, “exactly” are other frequently used terms in a mathematical discourse, while other common terms of everyday language like “some” or “one” can hide some ambiguities. This chapter presents some simple

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Negations and Indirect Proofs issues through many examples and remarks which cover the basic logical possibilities (negation of connectives, negation of quantifiers and nested quantifiers, etc.). Negation is a purely syntactic operation which is also related to the complement of a set. Negation is important when we have to deal