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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The cardinality of some symmetric differences


Authors: Po-Yi Huang, Wen-Fong Ke and Günter F. Pilz
Journal: Proc. Amer. Math. Soc. 138 (2010), 787-797
MSC (2010): Primary 05A05; Secondary 11N05, 94B05
Published electronically: October 23, 2009
MathSciNet review: 2566544
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Abstract: In this paper, we prove that for positive integers $ k$ and $ n$, the cardinality of the symmetric differences of $ \{1,2,\dots,k\}$, $ \{2,4,\dots,2k\}$, $ \{3,6,\dots,3k\}$, ..., $ \{n,2n,\dots,kn\}$ is at least $ k$ or $ n,$ whichever is larger. This solved a problem raised by Pilz in which binary composition codes were studied.


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Additional Information

Po-Yi Huang
Affiliation: Department of Mathematics and National Center for Theoretical Sciences (South), National Cheng Kung University, 1 University Road, Tainan 701, Taiwan
Email: pyhuang@mail.ncku.edu.tw

Wen-Fong Ke
Affiliation: Department of Mathematics and National Center for Theoretical Sciences (South), National Cheng Kung University, 1 University Road, Tainan 701, Taiwan
Email: wfke@mail.ncku.edu.tw

Günter F. Pilz
Affiliation: Department of Algebra, Johannes Kepler Universität Linz, Altenberger Strasse 69, 4040 Linz, Austria
Email: guenter.pilz@jku.at

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10189-2
PII: S 0002-9939(09)10189-2
Received by editor(s): June 1, 2009
Published electronically: October 23, 2009
Additional Notes: The first author was supported by the National Science Council, Taiwan, grant #96-2115-M-006-003-MY3
The second author was partially supported by the National Science Council, Taiwan, grant #97-2923-M-006-001-MY2
The third author was supported by grant P19463 of the Austrian National Science Fund (FWF)
Communicated by: Jim Haglund
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.