cardinality with applications to computability lecture 33 section 7.5 wed, apr 12, 2006

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  • Cardinality with Applications to ComputabilityLecture 33Section 7.5Wed, Apr 12, 2006

  • Cardinality of Finite SetsFor 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.

  • Cardinality of Infinite SetsWe 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, orA having a lesser cardinality than B, orA having a greater cardinality than B.

  • Definition of Same CardinalityTwo 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.

  • Definition of Same CardinalityTheorem: If |A| = |B| and |B| = |C|, then |A| = |C|.

  • Same CardinalityTheorem: |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|.

  • Cardinality of Z+Theorem: |Z+| = |Z|, where Z+ represents the positive integers.Proof:Define f : Z Z+ by f(n) = 2n if n > 0 f(n) = 1 2n if n 0.Verify that f is a one-to-one correspondence.Therefore, |Z+| = |Z|.

  • Definition of Lesser CardinalitySet 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| < |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.

  • Order Relations Among Infinite SetsCorollary: 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|.

  • Definition of Greater CardinalityWe may define |A| |B| to mean |B| |A| and define |A| > |B| to mean |B| < |A|.

  • Definition of Greater CardinalityTheorem: |A| |B| if and only if there exists an onto function from A to B.AB

  • Definition of Greater CardinalityTheorem: |A| |B| if and only if there exists an onto function from A to B.ABfone-to-onefunction

  • Definition of Greater CardinalityTheorem: |A| |B| if and only if there exists an onto function from A to B.ABgits inverse

  • Definition of Greater CardinalityTheorem: |A| |B| if and only if there exists an onto function from A to B.ABgonto function

  • Order Relations Among Infinite SetsCorollary: If |A| |B| and |B| |C|, then |A| |C|.Corollary: If |A| |B| and |B| |A|, then |A| = |B|.Etc.

  • Cardinality of the Interval (0, 1)Theorem: The interval (0, 1) has the same cardinality as R.Proof:The function f(x) = (x ) establishes that |(0, 1)| = |(/2, /2)|.The function g(x) = tan x establishes that |(/2, /2)| = |R|.Therefore, |(0, 1)| = |R|.

  • Countable SetsA 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.

  • Example: Countable SetsTheorem: 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.

  • Canonical OrderingThis 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?

  • Incorrect ProofIncorrect 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?

  • Example: Countable SetsTheorem: Q is countable.Proof:Arrange the positive rationals in an infinite two-dimensional array.

    1/11/21/31/42/12/22/32/43/13/23/33/44/14/24/34/4::::

  • Proof of Countability of QThen list the numbers by diagonals

    1/11/21/31/42/12/22/32/43/13/23/33/44/14/24/34/4::::

  • Proof of Countability of QWe get the list1/1, 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 ones1/1, 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?

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

  • Uncountable SetsA set is uncountable if it is not countable.

  • R is UncountableTheorem: 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.

  • R is UncountableLabel them x1, x2, x3, and so on.Represent each xi by its decimal expansion.x1 = 0.d11d12d13x2 = 0.d21d22d23x3 = 0.d31d32d33and so on, where dij is the j-th decimal digit of xi.

  • R is UncountableForm 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.

  • 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,

  • 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.

  • Number of Computer ProgramsTheorem: 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.

  • Number of Computer ProgramsThis 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.

  • Computability of FunctionsCorollary: There exists a function f : Z+ Z+ which cannot be computed by any computer program.

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

  • Cardinality of the Power SetTheorem: For any set A,|A| < |(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).

  • Proof, continuedThat 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).

  • Proof, continuedIs 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.

  • Proof, concludedEither way, we have a contradiction.Therefore, no such one-to-one function exists.Thus, |A| < |(A)|.

  • Hierarchy of CardinalitiesBeginning with Z+, consider the setsZ+, (Z+), ((Z+)), Each set has a cardinality strictly greater than its predecessor.|Z+| < |(Z+)| < |((Z+))| < These cardinalities are denoted 0,1,2, (aleph-naught, aleph-one, aleph-two, )