Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals (Chapman and Hall Mathematics) 1st Edition by V. Bryant (PDF)

Description

Combinatorics may very loosely be described as that branch of mathematics which is concerned with the problems of arranging objects in accordance with various imposed constraints. It covers a wide range of ideas and because of its fundamental nature it has applications throughout mathematics. Among the well-established areas of combinatorics may now be included the studies of graphs and networks, block designs, games, transversals, and enumeration problem s concerning permutations and combinations, from which the subject earned its title, as weil as the theory of independence spaces (or matroids). Along this broad front,various central themes link together the very diverse ideas. The theme which we introduce in this book is that of the abstract concept of independence. Here the reason for the abstraction is to unify; and, as we sh all see, this unification pays off handsomely with applications and illuminating sidelights in a wide variety of combinatorial situations. The study of combinatorics in general, and independence theory in particular, accounts for a considerable amount of space in the mathematical journais. For the most part, however, the books on abstract independence so far written are at an advanced level, ·whereas the purpose of our short book is to provide an elementary in­ troduction to the subject.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This review applies to the 1980 edition, the only edition that I am aware of.The proof of Theorem 2.6 on page 17-18 is wrong. I have submitted my own proof as a video addendum to this review. For typographical reasons, a video was the only way in Amazon.The proof does show that A is a subset of B’ — in fact, a proper subset since |A| < |B'|. This does imply, although this goes unsaid, that there exists a y in B'A such that A union {y} belongs to script-E. However, this does not establishes I(2), although the authors assert, merely assert, that it does. Rather, we need to see that there exists a y in BA (where B is the original arbitrary B, not B') such that A union {y} belongs to script-E.Throughout this subject, most proofs are by reductio ad absurdum, or proof by contradiction. In some context, we want to prove a certain proposition S, and so we suppose that not-S holds and show that, in our context, not-S leads to a contradiction. Therefore, we conclude that S holds, since not-S is impossible. We also use proofs by contradiction within a proof by contradiction. So the logic gets complicated. A breezy style obscures the logic.I see this general problem also in "Matroid Theory" by Welsh. Personally, I need to see much more detail more carefully presented.By the way, in a mostly superficial change to the theor, you can allow the underlying space to be infinite if the independent sets themselves are required to be finite with a fixed bound on their size. Then their largest size is that of the bases, and is the dimension of the space. This change would have the advantage of making it obvious how independence space theory applies to linear algebra and projective geometry. Without the requirement of a size bound, then you would be dealing with infinite dimensions and would need the axiom of choice. This could be done, I suppose, but would seem pointless to some.I prematurely called the book "excellent", but I would still call it very good as at least an attempt to cover a neglected but logically fundamental subject in a small book that originally had a low price, I think. It could be excellent if revised with better proofs. ⭐

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Download Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals (Chapman and Hall Mathematics) 1st Edition 1980 PDF Free
Independence Theory in Combinatorics: An Introductory Account with Applications to Graphs and Transversals (Chapman and Hall Mathematics) 1st Edition 1980 PDF Free Download
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