Cardinality with Applications to Computability

1. Cardinality with Applications to Computability Lecture 33 Section 7.5 Thu, Mar 17, 2005

2. 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.

3. 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 • A having the same cardinality as B, or • A having a lesser cardinality than B, or • A having a greater cardinality than B.

4. 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. • Theorem: If |A| = |B| and |B| = |C|, then |A| = |C|.

5. Example: Equal Cardinalities • 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|.

6. 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|.

7. Definition of Lesser Cardinality • Set A has a cardinalityless 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.

8. 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|.

9. Definition of Greater Cardinality • We may define |A|  |B| to mean |B|  |A| and define |A| > |B| to mean |B| < |A|.

10. Definition of Greater Cardinality • Theorem: |A|  |B| if and only if there exists an onto function from A to B. B A

11. Definition of Greater Cardinality • Theorem: |A|  |B| if and only if there exists an onto function from A to B. f B A

12. Definition of Greater Cardinality • Theorem: |A|  |B| if and only if there exists an onto function from A to B. g B A

13. Definition of Greater Cardinality • Theorem: |A|  |B| if and only if there exists an onto function from A to B. g B A

14. 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|.

15. 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|.

16. 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 the elements in such a way that each element appears exactly once in the list.

17. 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.

18. 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?

19. 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?

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

21. Proof of Countability of Q • Then list the numbers by diagonals

22. 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?

23. 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.

24. Uncountable Sets • A set is uncountable if it is not countable. • 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.

25. Proof, continued • 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.

26. Proof, concluded • 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.

27. 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, …

28. Proof, concluded • 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.

29. Number of Computer Programs • Theorem: The number of 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.

30. Proof, concluded • 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.

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

32. Cardinality of the Power Set • Theorem: 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).

33. 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).

34. 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.

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

36. Hierarchy of Cardinalities • Beginning with Z+, consider the sets Z+, (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, …)