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Transcript
```Technical Journal of Engineering and Applied Sciences
Available online at www.tjeas.com
A note on a property of the Cantor set and algebraic remark
Department of Statistics, Faculty of Mathematical sciences, University of Guilan, Rasht, Iran
Corresponding author Email: [email protected]
ABSTRACT: In this paper, we consider the known property of the Cantor set ; C+C=[0, 2], by a new
method. It is also shown that every x  [0, 2] can be written as the sum of two elements of C either
in a finite or in an uncountable number of ways.
Keywords: Cantor set, Partition, Subgroup
INTRODUCTION
The Cantor, C, on the line has been studied extensively, and in particular it has been observed that C
subtracted from itself renders the closed interval [-1, 1], (Falconer, 1985; Kraft, 1994; Palis and Takens, 1992).
By applying simple (analytical) geometric methods rather than (purely) analytic methods, we show that C added
to itself constitutes the closed interval [0, 2] and therefore, it generates as a group under addition. Further,
show that every element of [0, 2] can be expressed as the sum of two elements of C either in a finite or in an
uncountable number of ways. We conclude by a remark on the proper uncountable subgroups of 
,+.
PRELIMINARIES
n
Let C be the Cantor set on the line. Let C n , n  1 , be the union of 2 closed intervals remained after the
C 
C
n 1 n
nth deletion of the middle third open intervals in the formation of the Cantor set, so that
In a
similar and obvious way the Cantor set in the plane, denoted by D, can be formed by successive deletions of
middle third horizontal and vertical open strips of closed squares, starting from the closed unit square (a closed
n
square meaning the Cartesian product of two closed intervals); denote by Dn the union of 4 closed squares
remained after the nth deletion so that
square. It is clear that
{ x , y  
D
D n  C n C n
: x  0, y  0, x  y  }
n 1
Dn
and
. We call each of squares constituting the union Dn an nD=C×C. For [0,2], let l be the line segment
.
RESULTS
Proposition1: Every element of the closed interval [0,2] is the sum of two (not necessarily distinct) elements of
the Cantor set C, that is C+C=[0,2].
Proof :
-square and if it
cuts an n-square like S n then it cuts an n+1-square like Sn+1, such that . Thus a decreasing sequence S n, ,
can be formed such that for each n, S n is an n-
Proposition 2: Every element of [0,2] can be expressed as a sum of two elements of the Cantor set C, either in
a finite or in an uncountable number of ways.
Proof :
-square at a corner and let k
be the number of r-square
exactly k s-squares each of which at a corner. It is clear that in this case k is the number of sequences , Sn
being an n-
Tech J Engin & App Sci., 3 (11): 992-993, 2013
-square that it intersects and let be
the number of such n-squares. The nondecreasing sequence, either remains constant from some stage forth,
or increases infinitely often. We note that if, for some n, then . A little observation shows that in the first case
the number of sequences (S n being an nhe number of such sequence is uncountable.
Corollary :
Proof : In view of the fact that C+C=[0,2], it is sufficient to observe that an interval [a, b] of positive length
subgroup of under addition, generated by [a, b]. It suffices to see
positive integer such that . Then, from which, and hence. Letting, we have.
CONCLUSION
In search for generating sets for uncountable proper subgroups of  , +, the Cantor set C may seem
as a candidate. The above result shows that C cannot be taken as generator. Nevertheless it can be shown by
elementary methods that 
In fact 
, + has an uncountable number of uncountable proper subgroups.
, + can be considered as a vector space over
. Suppose B is a basis for this space. Clearly B
is uncountable, and every uncountable proper subset of it generates an uncountable subsets of  , +.
By partitioning B into two uncountable subsets B1 and B2, we see that the union of each subset of B1
with B2 generates a different proper subgroup of  ,+. Therefore (in view of the continuum hypothesis) the
number of such subgroups is greater that the number of real numbers. It is perhaps interesting to note that a
particular uncountable proper subgroup of
 , + is the set H of all linear combinations of finite numbers of elements of B with integer coefficients. It is
clear that for any bB, and any non-integer rational
r , rb  H .
REFERENCES
Falconer K.J. 1985. The geometry of fractal sets. In: Cambridge University Press.
Kraft R. 1994. What’s the difference between Cantor sets?. Amer. Math. Monthly 101(7): 640-650.
Palis J, Takens F. 1992. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. In: Cambridge University
Press.
993
```
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