# 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, )

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