Complete Sets
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DOI: 10.1007/s13226-020-0433-5
COMPLETE SETS Theophilus Agama Department of Mathematics, African Institute for Mathematical Science, Ghana e-mails: [email protected]; [email protected] (Received 24 May 2018; accepted 22 April 2019) In this paper we introduce the concept of completeness of sets. We study this property on the set of integers. We examine how this property is preserved as we carry out various operations compatible with sets. We also introduce the problem of counting the number of complete subsets of any given set. That is, given any interval of integers H := [1, N ] and letting C(N ) denotes the complete set counting function, we establish the lower bound C(N ) À N log N . Key words : Sets; completeness. 2010 Mathematics Subject Classification : 05Axx, 11Bxx, 11Axx.
1. I NTRODUCTION The development of set theory dates back to the days of the German mathematician George Cantor. Infact he was one of the major pioneers of set theory and it’s development, and so, he is thought today as the major force behind it [1]. Today it is widely studied in many areas of mathematics, including number theory, combinatorics, computer science, algebra etc. Intuititively, a set can be thought of as a collection of well-defined objects. The objects in the set can be seen as it’s members or elements. These elements do characterize and tell us more about the nature of the set in question. The elements of a set can either be finite or infinite. For example the set A := {2, 5, 9, 1, −54} denotes a finite set of integers, since all the elements are integers. The set of R of real numbers and the set Z of integers are examples of infinite sets. In what follows we set A ± B := {ai ± bi : ai ∈ A and bi ∈ B}, A · B := {ai bj : ai ∈ A, bi ∈ B} and c · A := {ca : a ∈ A}, A \ B := A − B for finite sets of integers A and B. We recall an arithmetic progression of length n to be the set A of the form A = {a0 , a0 + q, a0 + 2q, . . . , a0 + (n − 1)q}. In a more special case we have the A := {q, 2q, . . . , nq}, a homogenous arithmetic progression. For
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THEOPHILUS AGAMA
the set A := {a0 , a1 , . . . , an }, we call A(N ) := {a00 , a01 , . . . , a0n }, where a0i = d(A) = (a0 − a0 , a1 − a0 , . . . , an − a0 ) with 0 =
a00
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