Proofs and Fundamentals: A First Course in Abstract Mathematics

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    • Bloch's book is one of a growing genre of transition texts focused on rigor and basics. In the introduction, Bloch cites five texts similar to his own and I've seen quite a few more as well. I have pedagogical concerns about the entire genre. See what you think about the following two points in Bloch's text.

    ERRATA FOR “Proofs and Fundamentals: A First Course in Abstract ...

    The proof of this theorem involves a forward reference to Theorem 8. My feeling is that this sort of material is not best for a wide variety of students who have just finished calculus. The elevation of rigor above all other mathematical virtues gives a skewed view of what mathematics is. My experience with other texts from this genre tells me that the danger of alienating students is very real.

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    Undergraduate Texts in Mathematics Editorial Board S. Axler K.A. Ribet For other titles Published in this series, go to giuliettasprint.konfer.eu Ethan D. “Proofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of.

    I prefer a rigor-and-survey transition course with twin goals. The first goal is to introduce the students to rigor. The second goal is to give students a better general feel for modern mathematics.

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    Both goals, not just the first, address notable gaps left by their K math education. The introduction to rigor should be gentle.

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    Students should be given the feeling that they are building on their many years of formal mathematical training, not suddenly entering a foreign land. Rigor should be communicated to them as a corrected version of common sense, not a whole new thing.

    We should stay entirely away from the vision of mathematics as a giant edifice built on a small number of axioms. Rather we should stay in relative contexts, where students know something about some mathematical objects and try to establish more about these objects using what they already know. The survey of mathematics should give students a feel for the types of questions asked and answered by modern mathematicians.

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    It should communicate the breadth and excitement of mathematics. It should get students involved in a few landmark theorems and a few open questions. Students should leave the course with some appreciation for what math professors do with their research time!

    I have no criticisms of Bloch's book beyond those I have of its entire genre. Ethan D. In an effort to make advanced mathematics accessible to a wide variety of students, and to give even the most mathematically inclined students a solid basis upon which to build their continuing study of mathematics, there has been a tendency in recent years to introduce students to the for mulation and writing of rigorous mathematical proofs, and to teach topics such as sets, functions, relations and countability, in a "transition" course, rather than in traditional courses such as linear algebra.

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    If Senator Bullnose votes himself a raise, then he is a sleazebucket. An analog to this would be the ways we have of combining numbers to get new ones, such as addition and multiplication; if we did not have these operations, then numbers would not be very interesting. Logical symbols were used in Chapter 1 to help us become familiar with informal logic. This fallacy is known as the fallacy of the inverse and is also known as the fallacy of denying the antecedent. I would very much like to thank the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, and especially Professor Emanuel Farjoun, for their very kind hospitality during a sabbatical when this edition was drafted. Hence, to prove that something is true, it would suffice to prove that it is not false; this strategy is very useful in some proofs.

    A transition course functions as a bridge between computational courses such as Calculus, and more theoretical courses such as linear algebra and abstract algebra. This text contains core topics that I believe any transition course should cover, as well as some optional material intended to give the instructor some flexibility in designing a course. The presentation is straightforward and focuses on the essentials, without being too elementary, too exces sively pedagogical, and too full to distractions.

    A First Course in Abstract Mathematics