4 Lec 1c Cardinality 1

1

Lecture 1c

Cardinality of Sets

and some results on

Limits on Computability

Cardinality

Countability

Diagonalization

Limits on Computability

Generally, theory developed by Georg Cantor

2

Summary

Cardinality

Cardinality is a measure of the size of a set.

Questions like How big a set is, Which of two sets is bigger.

3

Cardinality

For infinite sets some very strange results

A proper subset of an infinite set can have the same cardinality as the set

It is possible that AB but still |A|=|B|

There are infinite sets with different cardinalities

4

5

Cardinality See 2 slides further

Cardinality

For finite sets everything is simple

The cardinality of a finite set is the number of the elements of the set

The cardinality of A={1,4} is 2

|A| = 2

6

Cardinality

To describe the cardinalities of sets, incl. infinite sets, we need a system of numbers that includes something more than just the natural numbers

Cantor proposed an extended version of natural numbers called cardinal numbers or cardinals

Cardinal numbers include natural numbers plus a new family of numbers called transfinite cardinal numbers. The latter specify the cardinality of infinite sets.

7

Cardinal numbers

8

Cardinal numbers

The smallest cardinal number is 0 (pronounced aleph null) aleph is the first letter in the Hebrew

alphabet.

0 is the cardinality of N the set of natural numbers {0,1,2,3,}

9

Cardinal numbers

The next cardinal number is

is the cardinality of R the set of real numbers

10

There are infinite sets of different sizes well see later

Cardinality - definition

Two sets A and B have the same cardinality,

|A| = |B|

iff there is a bijection A B

We also say A and B are equinumerous or equipotent.

Sometimes we also say size instead of cardinality.

11

Cardinality of a Finite Set

If A is finite and there is a bijection

A {1,2,3,,n} where nN then we say A has cardinality n.

For finite sets the cardinality of a set is the number of the elements of the set.

For infinite sets it is more complicated.

12

Cardinality of an Infinite Set

Let Odd be the set of odd natural numbers {1,3,5,).

f: N Odd: n 2n+1 is a bijection

Therefore, |Odd| = |N|

Cantor denotes the cardinality of N 0

13


N Odd

0 1

1 3

2 5

3 7

14

f: N Odd: n 2n+1 is a bijection Therefore, |Odd| = |N|


Odd N but still

|Odd| = |N|

15


The classic example used is that of the infinite hotel paradox, also called Hilbert's paradox or the Grand Hotel.

Suppose you are an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It's possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:

1 2

2 3

3 4

...

n n + 1

... 16

Countability

A set A is countable iff:

It is finite, or

There is a bijection A N (its cardinality is 0 )

In the latter case A is called countably infinite.

17

Countability

According to its cardinality, a set can be:

Finite

Countably infinite (denumerable)

Uncountably infinite

18

Countable

Infinite

If there is an injection A B, we denote this by:

|A| |B|

If there is an injection A B but no bijection between them:

|A| < |B|

19

Countability

Definitions of and < for cardinals

Enumerations

Strongly related to the notion of countability is the notion of enumeration.

Many authors even make no distinction between the two.

E.g. Wikipedia, Boolos,

20

Enumerations

An enumerable set is one whose elements can be enumerated: arranged in a single list with a rst entry, second entry, third entry, and so on, so that every member of the set appears sooner or later on the list.

This list is called enumeration of the set. If the list is infinite, the set is called enumerably infinite or

denumerable.

Actually, listed are not the actual elements of the set but rather some descriptions of these elements.

21

Enumerations

An enumeration of a set A is simply a list of the elements of A.

x1,x2,x3,,xn,

Every element of the set must appear on the list at least once.

More formally, a set A is enumerable iff either A is empty or else there is a surjective function

f: Z+ A where Z+ is the set of positive integers

Z+={1,2,3,}. [in other words, every element corresponds to at least one index]

We say that this function enumerates A.

Formally, a list is such a function.

22

Enumerations vs. Counting

If the two notions are to be distinguished, in enumeration the emphasis is on listing the elements, in counting on counting the number of elements.

But, anyway, the two notions are actually equivalent because enumerable sets are countable, and vice versa.

23

Optional: Ordering in Enumerations

Instead of Z+ we could use any other index set with cardinality o. We call index set a set I with a surjection I A; A is indexed by means of I.

When an enumeration is used in an ordered list context, we impose some sort of ordering structure requirement on the index set. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite is that the index set be well-ordered. According to this characterization, an ordered enumeration is defined to be a surjection with a well-ordered domain. This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.

24

Enumerations

It is actually more convenient to allow f to be a partial function.

More general definition:

A set is enumerable iff it is empty or the range of some partial function of positive integers.

25

Enumerations

Some authors use:

N instead of Z+

Bijection instead of surjection

Total function f instead of partial function

All these definitions are equivalent.

E.g. if f is a surjection (an element may show more than once in the list) we can remove the repeating elements and get a bijection. If f is a partial function (there might be gaps in the list) we can remove the gaps). [e.g., Boolos 3-]

26

Enumerations - examples

Set of positive integers is enumerated as

1,2,3,4, f(n)=n

Set of natural numbers:

0,1,2,3, f(n)=n-1

Set of negative integers:

-1,-2,-3,-4, f(n)=-n

Set of integers:

0,1,-1,2,-2,3,-3,

f(n)=n/2, if n is even, -(n-1)/2 if odd 27


Set of integers:

0,1,-1,2,-2,3,-3,

28

index

Optional: Enumerations

There may be many different enumerations of a set.

E.g. 1,2,3,4,5,6, and

2,1,4,3,6,5, and

1,1,2,2,3,3,4,4, etc.

29

Optional: Enumerations - examples

30

is also enumerable


Set of ordered pairs of positive integers

31

f(m,n)=(m2+2mn+n2-m-3n+2)/2

Enumerating words (strings) of a

language

Lexicographic ordering = dictionary ordering

Standard ordering

32

Enumerating words of a language

,a,aa,aaa,,aab,aaba,aabaa,,ac,aca,acaa,,b,ba,baa,,zzzz,

33

There are infinite decreasing chains


Standard ordering

34

If A = {a,b}


Because the words of a language can be enumerated, the set of words over a finite or countable alphabet is countable.

A* is countable

35


Therefore, every word of a language can be coded as a positive integer (f-1)

Every (description of a) Turing machine can be coded as an integer we can enumerate all TMs or programs

36

Optional: Effective enumerability

In computability theory one often considers enumerations with the added requirement that the mapping from Z+ to the enumerated set must be computable.

37

Smith, http://en.wikipedia.org/wiki/Enumeration


A set A is called effectively enumerable (or computably enumerable in more contemporary language, or recursively enumerable referring to the use of recursion theory in formalizations of what it means for a function to be computable) iff either A= or else there is an effectively computable function that enumerates it.

[effectively computable function means there is an algorithm that computes it]. [See Chomsky hierarchy: the set can be generated/

recognized by an algorithm; see TMs 2]

38


A set may be effectively enumerable, or enumerable but not effectively so, or neither (not enumerable).

39

Countability = Enumerability

We will use this fact to both show that:

A set is countably infinite by finding an infinite enumeration of the set

The set is uncountably infinite by proving that no infinite enumeration of it exists

40

Countability = Enumerability

Generally, well use countable and enumerable as synonyms

41

Resources

Hein 121

Epp 4th ed. 428

Gopalakrishnan 37

Good Math 161, 131

Rich 788

Boolos 3,16

Lewis 20

Greenlaw 39

Smith (An itro to Godels theorems) 13

Wikipedia enumeration

Wikipedia pairing functions

42

Documents

4 Lec 1c Cardinality 1