Properties of Cantor's Ternary Set

Properties of Cantor's Ternary Set

First I describe two different constructions for Cantor's Set, then investigate some of it's very special properties.

It is not hard to prove they are equivalent.

I am very limited here with typesetting in places it is a bit crude so bear with me and use your imagination. Please refer to the End Note.

A n = { a n }

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First, as intersecting closed  intervals. Consider the interval [0,1] i.e all real numbers between 0 and 1 and including the endpoints 0 and 1.

The first step is removing the middle open third interval ( 1/3 , 2/3 ) resulting in two closed intervals [0, 1/3] and [2/3, 1]

Then each of these in turn, leaving 2 x 2 = 4

________________________         S 0 = [0,1]

________                ________         S 1
 
__      __                   __       __         S 2  

…                                          ..
…  ..  .                          .  ..  …         S n
 

The procedure is repeated indefinitely (to infinitely) leaving a resulting set of real numbers not containing any intervals.

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Each real number b in the interval [0,1] has a ternary expansion, meaning it has an expansion in digits 0, 1 and 2 as the sum of powers of 1/3 with bn the n'th digit

Such as 0. b1  b2  b3  b4 … with bn = 1, 2, or 3 and the number b is

b = b1/ 3 + b2 /9 + b3/ 27 + … (with 9 = 3^2 27 = 3^3 etc.)

The Cantor Set is then all the ternary expansions in [0,1] without a digit 1

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A set of real numbers is called “countably infinite” if it is infinite but can not be written as a sequence. Cantor's diagonal argument proves that the Cantor Set is uncountable. I give a detailed proof. 

Suppose that the Cantor Set can be written as a a sequence b i, we will demonstrate a number that is not in the set 

b1  = b11   b12  b13  b14  b15 …
b2  = b21   b22  b23  b24  b25 …
b3  = b31   b32  b33  b34  b35 …
…
bi  = bi1  bi2  bi3 … 
… 
b5 = b51  b52   b53   b54  b55 … 

Each digit (entry) bij  is now either a zero or a one because it is a Cantor Set. When working down the diagonals (right–>left) of the “infinite matrix”

b11     b12 – b21     b13 – b22 – b31     b14 –  b23 – b32 – b41 
    b42 – b33 –  b42 –  …  … 

down the main diagonal, construct the number a as a = 0. a1  a2  a3 a4  … 
where aj = 1 if bjj=0 and aj=0 if bjj=1 

But now a cannot be in the sequence bi since it differs from each number in the sequence (in the place i = j ). and thus a not in the sequence.

This is a contradiction it is not, since Cantor's Set as assumed and cannot be written as a sequence and is uncountably infinite. 

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There are further very interesting (desirable) properties and examples.

The property of compactness is very important in tropological spaces and for the real line is characterised by a subset of real numbers being closed and bounded.

Clearly for the Cantor Set each element is between zero and one,  thus the set is bounded. Also a closed interval (containing endpoints) is topologically “closed” so that Cantor's Set is compact as being the intersection of closed sets. 

These are very basic ideas.

We have already brought in the concept as a countable set being a sequence. Further examples are the integers, indeed the union of countably many sets is also, it is easy to prove.

Surprisingly the rational numbers (fractions) can also be written (exhausted) in a sequence and it is easy to demonstrate. A countable union of countable sets is also countable.

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The concept of a Null Set (as a set of of measure zero) enables one to study the Lebesgue integral for real numbers without formal Measure Theory.

Given any positive epsilon ( eps > 0 ), if a set B of real numbers can be covered by a sequence of intervals {I n } with total length less than epsilon the set B is a null Set.

It means each element of B is contained in an interval I n ( { I n } covers B), as in the union of the intervals A n, and the total sum of lengths of the A n is less than any given eps.

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Suppose B is countable (eg. integers) thus is a sequence { b n }. Let b n be contained in an interval I n chosen such that length ( I n ) is less than eps/ 2^n.

The sum of lengths of the I n is  less than eps.( ½ + ¼ + 1/8 + … ) as a geometric series,
and is less than epsilon. So a countable set (eg. rationals) has measure zero

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Note that for each n the union { I n } = S n covers the Cantor Set. Note that the total length of S n is length (S n ) = (2/3)^n from the construction of  S n.

Given eps > 0 choose N such that eps n < eps N < eps for all n > N 

For any eps > 0 Cantor's Set may be covered with intervals the sum of which is less then epsilon and it is thus a Null Set from the definition.

So we have an example of an (over) countable set that is a Null Set.

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As another property the Cantor Set also is a fractal. One can see that it has scaled subsets, proper subsets which are identical as smaller and smaller sets and subsets ad infinitum (to infinitesimal).

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End Note

Sets will be in capitals, elements in small type and digits indexed and doubled index. We work with variables as integers, the set of real numbers, subsets, intervals and sets of intervals.

As slightly more familiar notation n is an integer ; a n = a_n is a real number, (sub) sets of real numbers is A.

A = { a n } = { a_n } a sequence, I n = I_n an interval ; b mn = b_mn each a digit, a 0, 1 or 2

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My Links  Strongly related also read my AbcTales stories “Naive Set Theory”,  “Basic Set Theory” in my Numbers and Infinities collection.

Literature as standard references, a comprehensive treatment of real analysis “Mathematical Analysis” by Tom Apostol one of the best formal textbooks I have come across, and then more of a curiosity, “Lebesgue Integration & Measure”, Alan J. Weir.

Comments and replies, as remarks or observations most welcome.