Cardinality with Applications to Computability Lecture 33 Section 7.5 Wed, Apr 12, 2006

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.

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.

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.

Definition of Same Cardinality Theorem: If |A| = |B| and |B| = |C|, then |A| = |C|.

Same Cardinality Theorem: |2 Z | = | Z |, where 2 Z represents the even integers. Proof: Define f : Z  2 Z by f(n) = 2n. Clearly, f is a one-to-one correspondence. Therefore, |2 Z | = | 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 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| < |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 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|.

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

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

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

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

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

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.

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 |.

Countable Sets A set is countable if it either is finite or has the same cardinality as Z +. Examples: 2 Z 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 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.

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?

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 ?

Example: Countable Sets Theorem: Q is countable. Proof: Arrange the positive rationals in an infinite two-dimensional array. 1/11/21/31/4… 2/12/22/32/4… 3/13/23/33/4… 4/14/24/34/4… ::::

Proof of Countability of Q Then list the numbers by diagonals 1/11/21/31/4… 2/12/22/32/4… 3/13/23/33/4… 4/14/24/34/4… ::::

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?

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.

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

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 1 st, 2 nd, 3 rd, and so on.

R is Uncountable Label them x 1, x 2, x 3, and so on. Represent each x i by its decimal expansion. x 1 = 0.d 11 d 12 d 13 … x 2 = 0.d 21 d 22 d 23 … x 3 = 0.d 31 d 32 d 33 … and so on, where d ij is the j-th decimal digit of x i.

R is Uncountable Form a number x = 0.d 1 d 2 d 3 … as follows. Define d i = 0 if d ii  0. Define d i = 1 if d ii = 0. Then x  (0, 1), but x is not in the list x 1, x 2, x 3, … 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 f 1, f 2, f 3, …

Functions from Z + to Z + Define a function f : Z +  Z + as follows. f(i) = 0 if f i (i)  0. f(i) = 1 if f i (i) = 0. Then f(i)  f i (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 Programs Theorem: The set of all computer programs is countable. Proof: Once compiled, a computer program is a finite string of 0 s and 1 s. The set of all computer programs is a subset of the set of all finite binary strings.

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.

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

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

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).

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).

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.

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

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, …)