Abstract
In this first chapter, we develop the usual elementary tools for counting the number of distinct configurations corresponding to a certain combinatorial situation, without needing to list them one by one. As the reader will see, the essential ideas are the construction of bijections and the use of recursive arguments.
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Notes
- 1.
In view of this definition, in principle we have two distinct definitions for the elements of A × B: on the one hand, they consist of the ordered pairs (a,b) such that a ∈ A and b ∈ B; on the other, they are sequences (a,b) for which a ∈ A and b ∈ B. Since the ordered pair (a,b) is defined by (a,b) = {{a}, {a,b}} (cf. Problem 5) and the sequence (a,b) (with a ∈ A and b ∈ B) is the function f : {1, 2}→ A ∪ B such that f(1) := a ∈ A and f(2) := b ∈ B, we come to the conclusion that, although we have been using the same notation, they are distinct mathematical objects. However, for our purposes the identification of the ordered pair (a,b) to the sequence (a,b) is totally harmless and will be done, from now on, without further comments.
- 2.
Such a definition is due to Kazimierz Kuratowski, Polish mathematician of the twentieth century.
- 3.
In Set Theory, a family is a set whose elements are also sets.
- 4.
After Leonardo di Pisa , also known as Fibonacci , Italian mathematician of the eleventh century. Apart from its own contributions to Mathematics—as the problem we are presently describing—one of the greatest merits of Fibonacci was to help revive, in Middle Age Europe, the Mathematics of Classical Antiquity; in particular, Fibonacci’s famous book Liber Abaci introduced, in Western Civilization, the Hindu-Arabic algarisms and numbering system.
- 5.
After James Stirling , Scottish mathematician of the eighteenth century. With respect to Stirling numbers of second kind, see also Problem 4, page 57.
- 6.
- 7.
For another proof to this problem, see Example 4.18.
- 8.
For a generalization of this problem, see Problem 8, page 58.
- 9.
For the other half of the original problem, see Example 2.29.
References
A. Caminha, An Excursion Through Elementary Mathematics I - Real Numbers and Functions (Springer, New York, 2017)
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Caminha Muniz Neto, A. (2018). Elementary 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_1
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