A Geometric Approach to Linear Functions Graver, Jack E.
The College mathematics journal,
11/1/1995, 19951101, 1995-11-00, Letnik:
26, Številka:
5
Journal Article
The rational numbers are countable. The traditional proof demonstrates that there exists a one-to one function from the natural numbers onto the positive rational numbers or simply that there exists ...a list of all positive rationals (without repeats). But the list is not explicitly given. That is, there is no reasonable way to say which rational number is 150th in the list or where 21/13 appears in the list. There are several different listings of the rationals, some much easier to compute, some involving interesting mathematical constructions. We start this paper with a discussion of the traditional method, then introduce the bijection that is easiest to define and finally concentrate on the intriguing Calkin-Wilf algorithm.
In this paper we consider a variety of questions in the context of Boolean designs. For example, Erdös asked: How many subsets of an
n-set can be found so that pairwise their intersections are all ...even (odd)? E. Berlekamp 2 and the author both answered this question; the answer is approximately 2
1
2
n
. Another question which can be formulated in terms of Boolean designs was asked by J. A. Bondy and D. J. A. Welsh 1. For what values of
d can one find a connected binary matroid of rank
d which is identically self-dual? We prove that such matroids exist for all
d except 2, 3, and 5. The paper ends with a discussion of more general modular designs and with constructions of some identically self-dual matroids representable over the field of three elements.
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