Small set
In mathematics, the term small set is sometimes used to refer to any set that is small, a subjective concept. However, it also has specialized meanings in two branches of mathematics.
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Combinatorics
In combinatorics, a small set of positive integers
- <math>S=\{s_1,s_2,s_3,\dots\}<math>
is one such that the infinite sum
- <math>\frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}+\cdots<math>
converges. A large set is any other set of positive integers.
For example, the set <math>\{1,2,3,4,5,\dots\}<math> of all positive integers is known to be a large set (see Harmonic series (mathematics)), and the set <math>\{1,2,4,8,\dots\}<math> of powers of 2 is known to be a small set. There are many sets about which it is not known whether they are large or small.
A union of small sets is small, as the sum of two convergent series is a convergent series. Also, a large set minus a small set is still large.
The set of prime numbers has been proven to be large.
The Müntz-Szasz theorem is that a set <math>S=\{s_1,s_2,s_3,\dots\}<math> is large iff
- <math>\{1,x^{s_1},x^{s_2},x^{s_3},\dots\}<math>
is dense in the uniform norm topology of continuous functions on a closed interval. This is a generalization of the Weierstrass approximation theorem.
Another known fact is that the set of numbers whose decimal representations exclude 7 (or any digit one prefers) is small. That is, for example, the set
- <math>\{\dots, 6, 8, \dots, 16, 18, \dots, 66, 68, 69, 80, \dots, \}<math>
is small. (This has been generalized to other bases as well.)
References
- A. D. Wadhwa (1975). An interesting subseries of the harmonic series. American Mathematical Monthly 82 (9) 931–933.
See also
Category theory
In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is the category of all sets one cares to consider. This is used when one does not wish to bother with set-theoretic concerns of what is and what is not considered a set, which concerns would arise if one tried to speak of the category of "all sets".
In this context, a large set is any set that is not small.
A small set is not to be confused with a small category, which is a category whose collection of objects forms a set. For more on small categories, see Category theory.
References
- S. Mac Lane, I. Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, ISBN 0–387–97710–4, ISBN 3–540–97710–4, the chapter on "Categorical preliminaries"
See also
Categories: Category theory | Combinatorics | Number sequences | Mathematical series