Abstract
In this chapter we study a few more elaborate counting techniques. We start by discussing the inclusion-exclusion principle, which, roughly speaking, is a formula for counting the number of elements of a finite union of finite sets. The presentation continues with the notion of double counting for, counting a certain number of configurations in two distinct ways, to infer some hidden result. Then, a brief discussion of equivalence relations and their role in counting problems follows. Among other interesting results, we illustrate it by proving a famous theorem of B. Bollobás, on extremal set theory. The chapter ends with a glimpse on the use of the language of metric spaces in certain specific counting problems.
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Notes
- 1.
After Édouard Lucas , French mathematician of the nineteenth century.
- 2.
After Alexandre-Theóphile Vandermonde , French mathematician of the eighteenth century. For another proof of Vandermonde’s identity, see Problem 3, page 355.
- 3.
For another approach to this problem, see Problem 21, page 134.
- 4.
This terminology comes from the fact that the inclusion relation is a partial order in the family of all subsets of a given set. In this respect, see the discussion at Sect. 4.3.
- 5.
For a formal definition of an algorithm, see the footnote at page 116.
- 6.
After Pierre S. de Fermat , French mathematician of the seventeenth century. For a different proof of this result, see Sect. 10.2.
- 7.
After the Hungarian, Chinese and German twentieth century mathematicians Paul Erdös , Chao Ko and Richard Rado , respectively.
- 8.
The reader with previous acquaintance with the theory of Metric Spaces will notice that what we call a ball is generally known in the literature as a closed ball. Nevertheless, in order to ease the writing we will stick to this slightly different terminology. We believe that, as far as these notes are concerned, this will be a harmless practice.
- 9.
After the American mathematician of the twentieth century Richard Hamming , who used such a concept to study coding problems in Computer Science and Telecommunications Engineering. The famous Hamming codes were also named after him.
References
Y. Kohayakawa, C.G.T. de A. Moreira, Tópicos em Combinatória Contemporânea (in Portuguese) (IMPA, Rio de Janeiro, 2001)
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Caminha Muniz Neto, A. (2018). More Counting Techniques. In: An Excursion through Elementary Mathematics, Volume III. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-77977-5_2
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