Cardinality with Applications to Computability

1
Cardinality with Applications to Computability

• Lecture 33
• Section 7.5
• Wed, Apr 12, 2006

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Cardinality of Finite Sets

• For finite sets, the cardinality of a set is the
  number of elements in the set.
• For a finite set A, let A denote the
  cardinality of A.

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Cardinality of Infinite Sets

• We wish to extend the notion of cardinality to
  infinite sets.
• Rather than talk about the number of elements
  in an infinite set, for infinite sets A and B, we
  will speak of the cardinality of A.
• A having the same cardinality as B, or
• A having a lesser cardinality than B, or
• A having a greater cardinality than B.

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Definition of Same Cardinality

• Two sets A and B have the same cardinality if
  there exists a one-to-one correspondence from A
  to B.
• Write A B.
• Note that this definition works for finite sets,
  too.

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Definition of Same Cardinality

• Theorem If A B and B C, then A
  C.

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Same Cardinality

• Theorem 2Z Z, where 2Z represents the
  even integers.
• Proof
• Define f Z ? 2Z by f(n) 2n.
• Clearly, f is a one-to-one correspondence.
• Therefore, 2Z Z.

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Cardinality of Z

• Theorem Z Z, where Z represents the
  positive integers.
• Proof
• Define f Z ? Z by
• f(n) 2n if n gt 0
• f(n) 1 2n if n ? 0.
• Verify that f is a one-to-one correspondence.
• Therefore, Z Z.

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Definition of Lesser Cardinality

• Set A has a cardinality less than or equal to the
  cardinality of a set B if there exists a
  one-to-one function from A to B.
• Write A ? B.
• Then A lt B means that there is a one-to-one
  function from A to B, but there is not a
  one-to-one correspondence from A to B.

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Order Relations Among Infinite Sets

• Corollary If A ? B and B ? C, then
  A ? C.
• Corollary If A ? B, then A ? B.
• Proof
• Let A ? B.
• Define the function f A ?? B by f(a) a.
• Clearly, f is one-to-one.
• Therefore, A ? B.

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Definition of Greater Cardinality

• We may define A ? B to mean B ? A and
  define A gt B to mean B lt A.

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Definition of Greater Cardinality

• Theorem A ? B if and only if there exists an
  onto function from A to B.

B
A
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Definition of Greater Cardinality

• Theorem A ? B if and only if there exists an
  onto function from A to B.

f
one-to-one function
B
A
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Definition of Greater Cardinality

• Theorem A ? B if and only if there exists an
  onto function from A to B.

g
its inverse
B
A
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Definition of Greater Cardinality

• Theorem A ? B if and only if there exists an
  onto function from A to B.

g
onto function
B
A
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Order Relations Among Infinite Sets

• Corollary If A ? B and B ? C, then
  A ? C.
• Corollary If A ? B and B ? A, then
  A B.
• Etc.

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Cardinality of the Interval (0, 1)

• Theorem The interval (0, 1) has the same
  cardinality as R.
• Proof
• The function f(x) (x ½)? establishes that
• (0, 1) (?/2, ?/2).
• The function g(x) tan x establishes that
• (?/2, ?/2) R.
• Therefore, (0, 1) R.

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Countable Sets

• A set is countable if it either is finite or has
  the same cardinality as Z.
• Examples 2Z and Z are countable.
• To show that an infinite set is countable, it
  suffices to give an algorithm for listing, or
  enumerating, the elements in such a way that each
  element appears exactly once in the list.

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Example Countable Sets

• Theorem The number of strings of finite length
  consisting of the characters a, b, and c is
  countable.
• Correct proof
• Group the strings by length ?, a, b, c,
• aa, ab, , cc,
• Arrange the strings alphabetically within groups.

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Canonical Ordering

• This gives the canonical order
• ?, a, b, c, aa, ab, ac, ba, , cc,
• aaa, aab, , ccc, aaaa, aaab, ,
• where ? denotes the empty string.
• Consider the string bbabc.
• How do we know that it will appear in the list?
• In what position will it appear?

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Incorrect Proof

• Incorrect Proof
• Group the strings by their first letter a, aa,
  ab, , b, ba, bb, , c, ca, cb, .
• Within those groups, group those words by their
  second letter, and so on.
• List the a-group first, the b-group second, and
  the c-group last.
• In what position will we find the string bbabc?
  the string abc? the string aaaab?

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Example Countable Sets

• Theorem Q is countable.
• Proof
• Arrange the positive rationals in an infinite
  two-dimensional array.

1/1 1/2 1/3 1/4
2/1 2/2 2/3 2/4
3/1 3/2 3/3 3/4
4/1 4/2 4/3 4/4

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Proof of Countability of Q

• Then list the numbers by diagonals

1/1 1/2 1/3 1/4
2/1 2/2 2/3 2/4
3/1 3/2 3/3 3/4
4/1 4/2 4/3 4/4

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Proof of Countability of Q

• We get the list
• 1/1, 2/1, 1/2, 3/1, 2/2, 1/3, 4/1, 2/3, 3/2,
• 1/4, 5/1, 4/2, 3/3, 2/4, 1/5,
• Then remove the repeated fractions, i.e., the
  unreduced ones
• 1/1, 2/1, 1/2, 3/1, 1/3, 4/1, 2/3, 3/2, 1/4,
• 5/1, 1/5,
• In what position will we find 3/5?

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False Proof of the Countability of Q

• Incorrect listing 1
• List the rationals from in order according to
  size.
• Incorrect listing 2
• List all fractions with denominator 1 first.
• Follow that list with all fractions with
  denominator 2.
• And so on.

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Uncountable Sets

• A set is uncountable if it is not countable.

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R is Uncountable

• Theorem R is uncountable.
• Proof
• It suffices to show that the interval (0, 1) is
  uncountable.
• Suppose (0, 1) is countable.
• Then we may list its members 1st, 2nd, 3rd, and
  so on.

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R is Uncountable

• Label them x1, x2, x3, and so on.
• Represent each xi by its decimal expansion.
• x1 0.d11d12d13
• x2 0.d21d22d23
• x3 0.d31d32d33
• and so on, where dij is the j-th decimal digit
  of xi.

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R is Uncountable

• Form a number x 0.d1d2d3 as follows.
• Define di 0 if dii ? 0.
• Define di 1 if dii 0.
• Then x ? (0, 1), but x is not in the list x1, x2,
  x3,
• This is a contradiction.
• Therefore, R is not countable.

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Functions from Z to Z

• Theorem The number of functions
• f Z ? Z is uncountable.
• Proof
• Suppose there are only countably many.
• List them f1, f2, f3,

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Functions from Z to Z

• Define a function f Z ? Z as follows.
• f(i) 0 if fi(i) ? 0.
• f(i) 1 if fi(i) 0.
• Then f(i) ? fi(i) for all i in Z.
• Therefore, f is not in the list.
• This is a contradiction.
• Therefore, the set is uncountable.

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Number of Computer Programs

• Theorem The set of all computer programs is
  countable.
• Proof
• Once compiled, a computer program is a finite
  string of 0s and 1s.
• The set of all computer programs is a subset of
  the set of all finite binary strings.

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Number of Computer Programs

• This set may be listed
• ?,
• 0, 1,
• 00, 01, 10, 11,
• 000, 001, 010, , 111,
• 0000, 0001, 0010, 0011, , 1111,
• Therefore, it is countable.
• As a subset of this set, the set of computer
  programs is countable.

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Computability of Functions

• Corollary There exists a function f Z ? Z
  which cannot be computed by any computer program.

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Subsets of N

• There are uncountably many subsets of N.
• However, there are countably many finite subsets
  of N.
• Can you prove it?

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Cardinality of the Power Set

• Theorem For any set A,
• A lt ?(A).
• Proof
• There is a one-to-one function f A ? ?(A)
  defined by f(x) x.
• Therefore, A ? ?(A).
• We must prove that there does not exist a
  one-to-one correspondence from A to ?(A).

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Proof, continued

• That is, we must prove that there does not exist
  an onto function from A to ?(A).
• Suppose g A ? ?(A) is onto.
• For every x ? A, either x ? g(x) or x ? g(x).
• Define a set B x ? A x ? g(x).
• Then B ? ?(A), since B ? A.
• So B g(a) for some a ? A (since g is onto, by
  assumption).

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Proof, continued

• Is a ? g(a)?
• Case 1 Suppose a ? g(a).
• Then a ? B, by the definition of B.
• But B g(a), so a ? g(a), a contradiction.
• Case 2 Suppose a ? g(a).
• Then a ? B, by the definition of B.
• But B g(a), so a ? g(a), a contradiction.

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Proof, concluded

• Either way, we have a contradiction.
• Therefore, no such one-to-one function exists.
• Thus, A lt ?(A).

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Hierarchy of Cardinalities

• Beginning with Z, consider the sets
• Z, ?(Z), ?(?(Z)),
• Each set has a cardinality strictly greater than
  its predecessor.
• Z lt ?(Z) lt ?(?(Z)) lt
• These cardinalities are denoted ?0,?1,?2,
  (aleph-naught, aleph-one, aleph-two, )