8Exploring Infinity 8 By Christopher Imm Johnson County Community College

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8Exploring Infinity 8By Christopher
ImmJohnson County Community College
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What are the sizes of the following sets?
1. The Z mod n, group, . This is the set
formed by the remainders of the integers divided
by a positive integer n. 2. The set of all
Natural numbers, , 1,2,3,4,. 3. The set
of all positive rational numbers, , the
ratio of two integers in simplified form. 4. The
set of real numbers,
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Example 1
The Z mod n, , group is formed by the
remainders of the integers divided by a positive
integer n. This is intuitive by asking how many
elements would be in the set.
What are the possible remainders upon division by
n?
Counting the number of elements in the set, the
size of this is set is n.
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Example 2
What is the size of the set, , 1,2,3,4,?
This may be a bit harder to visualize, our first
question may be, is infinity allowed to represent
the size of a set?
If so, how do we represent this?
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Example 3

• What is the size of the set of , the
  positive ratio of
• two integers in simplified form?
• Again the answer seems to be infinity, however,
  this set
• seems a bit larger than the one in example 2.
• Why? Because you can think of many positive
  rational
• numbers between the first two natural numbers
• For example the sequence
• This means there are an infinite number of
• numbers between the first two numbers of another
• infinite set, the natural numbers.

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Example 4
What is the size of the set of the real numbers
on the interval 0,1?
The answer is infinity, again, however, the
underlying question is, how can we compare these
infinite sets?
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Introducing Georg Cantor
A German mathematician born in St. Petersburg,
Russia in 1845.
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Cantor introduced different sizes of infinity

• Cantor devised a system from the Hebrew letter
  aleph, called aleph numbers.
• These numbers were also called Cardinal numbers
  or Cardinals, for short.
• The smallest infinite set was described as
  Cardinal aleph-naught or aleph-null.
• It was denoted as
• The set of the natural numbers, , as mentioned
  in Example 2, have Cardinal

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Discerning between sizes of infinity

• Cantor used the idea of a bijection between a set
  and the natural numbers, , to describe all
  sets Cardinality
• A bijection is a one to one, onto mapping between
  two sets.
• Sets of this type were sometimes called countably
  infinite or countable.
• The positive rational numbers, , in example
  3, are another example Cardinality

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There is a bijection between and
The idea of the 1-1, onto correspondence follows
in the diagram below.
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Cont.

• If any fractions not in reduced form are
  eliminated and we follow the arrow, the set of
  is in one to one and onto
  correspondence with . Therefore is
  countable.
• and two good examples of sets of
  Cardinality

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The Song
To the tune 99 bottles of beer on the wall.
? Aleph-null cups of coffee on the
wall, Aleph-null cups of coffee. Take one down,
pass it around, Aleph-null cups of coffee on the
wall. ? (And repeat)
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Larger sizes of infinity.

• Cantor realized that there were sets larger than
  aleph- naught. The next Cardinal number he
  defined was aleph-one.
• These sets were called the Cardinality of the
  continuum represented by the real numbers,
• It was denoted as or c, for the continuum.
• The set of real numbers, , or the subset on
  the interval 0,1 as mentioned in Example 4,
  have Cardinality
• Sets of this size are referred to as uncountable.

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Cantors diagonalization argument
Prove that the set S is
uncountable.
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Cont.

• Cantors original diagonal argument was done with
  a binary representation of the real numbers in
  decimal form. Thus, the new decimal
  representation was chosen to be the complement of
  each diagonal element, forming a new number not
  in the set.

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The Song
To the tune 99 bottles of beer on the wall.
?Aleph-one cups of coffee on the
wall, Aleph-one cups of coffee. Take infinity
down, pass infinity around, Aleph-one cups of
coffee on the wall. ? (And repeat)
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The Cantor Set

• To create the Cantor set, take the interval 0,1
  on the real number line, call this the initial
  stage or C0.
• In the first stage, C1, we remove the middle
  third of the segment.
• For each additional stage continue to remove the
  middle third of each segment, call the nth stage
  Cn.
• The Cantor set is C where,

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Graphical Representation of C

• The first few stages below, C0, C1, C2

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What is the length of C?

• To figure this out, consider these questions
• How many segments are taken away in each stage?
• What is the length of each segment taken away in
  each stage?
• How can we represent the sum of all the segments
  taken away in each stage?
• What is the limit of this sum as n-gt8.

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Cont.

• The number of segments taken away at each stage
  can be represented by
• The length of each segment taken away at each
  stage can be represented by
• The total length taken away at each stage can be
  represented by the series
• The total length taken away is given by

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Conclusion

• Since the Cantor set is constructed by a set of
  length one and the sum of the segments taken away
  is one, the Cantor set has a length of zero.
• Length of sets are referred to as measure.
• Thus, the Cantor set has measure 0.

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What is the Cardinality of the Cantor set?

• To discover this, ask some other questions.
• What are some elements remaining in the Cantor
  set?
• Is there a convenient representation for the
  entire Cantor set?
• Is there a bijection between the Cantor set and
  the natural numbers, ?

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What are some elements remaining in the Cantor
set?

• All the endpoints of the remaining intervals.
• For example 0, 1/3, 2/3, 1, 1/9, 2/9, 7/9, 8/9,
  and so on

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Is there a convenient representation for the
entire Cantor set?

• Notice that all elements in the Cantor set are
  powers of 1/3. A unique way to represent all
  elements is to use base 3 or the ternary
  representation.
• Exs 003, 1/30.13, 2/30.23, 10.2223,
• 1/90.013, 2/90.023,7/90.213,8/90.223,
• and so on

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Is there a bijection between the Cantor set and
the natural numbers, ?

• The answer to this question is no, by viewing the
  ternary representation of the Cantor set, C.
• What about the real numbers,
• Consider the real numbers on the interval 0,1.
  
• Try to find a surjective (onto) mapping, f , from
  C to the real numbers on 0,1.
• To this end, represent the real numbers on 0,1
  in base 2 or binary.

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Find the function f C-gt0,1

• We can express every number in C in its ternary
• representation only consisting of 0s and 2s
• (repeating).
• For example for the first few stages
• 003, 10.2223
• 1/30.130.02223, 2/30.23
• 1/90.0130.002223, 2/90.023,
• 7/90.2130.202223, 8/90.223,
• and so on

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Cont.

• Similarly, we can express all reals on the
  interval 0,1 in their binary representation,
  from 002, to 10.1112.
• Thus, replacing all 2s in the numbers in C by
  1s, creates a surjective (onto) mapping from C
  to the real numbers in 0,1.
• Define f as follows

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Conclusion

• Since, C is surjective to the reals on 0,1,
• C must have at least the cardinality of c or
• Aleph-1. However since C is a subset of the
  reals on 0,1, it must be at most that
  Cardinality as well. Thus C has Cardinality of c
  or Aleph-1.
• It is worth noting, f is not bijective (1-1).
  For example,
• Hence, f (7/9)f (8/9), however 7/9?8/9.

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Summary of the Cantor Set

• The Cantor set has measure 0.
• The Cantor set is uncountably infinite, with
  Cardinality of c or Aleph-1.
• The Cantor set is an example of a set which you
  can take an uncountably infinite number of
  elements away from an uncountable set and still
  have an uncountably infinite set.
• The Cantor set has the added property of being
  closed and bounded.

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Enter Waclaw Sierpinski
A Polish mathematician born in Warsaw, Poland in
1882.
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Sierpinskis Carpet

• The process
• To build Sierpinski's Carpet, S, start with a
  square with side length 1 unit, completely
  shaded. (Iteration 0, or the initiator)
• Divide each square into nine equal squares and
  cut out the middle one. (the generator)
• Repeat this process on all shaded squares.

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Graphical representation of S

• Interactive Sierpinski's Carpet

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The size of S.

• Start with an area of 1 square unit.
• The number of squares taken away at iteration n
  is
• The size of each square taken away at iteration n
  is
• The total area taken away at iteration n is
• The total area taken away from S is

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Conclusion

• The area of Sierpinskis Carpet is 0.

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The Menger Sponge

• The process
• To build the Menger Sponge, M, start with a cube
  edge 1 unit. (the initiator)
• Divide the cube into twenty-seven equal cubes and
  cut out the middle one. (the generator)
• Repeat this process on all remaining cubes.

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Graphical representation of M

• Visualizing the Menger Sponge

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The size of M

• Start with an cube of volume 1 cubic unit.
• The number of cubes at iteration n is
• The volume of each cube at iteration n is
• The total volume at iteration n is
• The total volume of M is

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Extensions to higher dimensions

• The procedure then for creating an N dimensional
  pyramid can be summarized by the following rules.
  
• Start with an N-1 dimensional cube centered at
  the origin.
• Pull the midpoint of the cube (origin) into the
  Nth dimension.
• Make edges from the midpoint to each vertex of
  the N-1 cube.
• Make faces using the midpoint and each edge of
  the N-1 cube.
• Using these rules the 4D pyramid (hyper-pyramid)
  is constructed by taking a 3D cube and pulling
  its midpoint into the 4th dimension.

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A 4-D Hyper-Gasket
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The process can be continued by forming another
pyramid with the hyper-cube, and so on