CardinalityIntroduction to Analysis

December 1, 2008Samantha Wong

Cardinality

Cardinality is the number of elements in a set.

For Example:S = {1, 5, 8, 10}.Then this set contains four elements.

Some Definitions

Two sets S and T are called equinumerous if there exists a bijective function from S onto T. We write S~T.

The cardinal number of a set In is n, and if S ~ In, we say that S has n elements.

nI

Notation

We denote the cardinal number of a set S, as |S|.

As in the previous example:S = {1, 5, 8, 10}.Then |S| = 4

Ordinal NumbersAn ordinal number tells us the position of an element in a set.

Going back to our example:S = {1, 5, 8, 10}. Then,1 is the first ordinal5 is the second ordinal8 is the third ordinal10 is the fourth ordinal.

Ordinal Numbers

Cardinal Ordinal

One First

TwoThree...

SecondThird...

Ordinal Numbers

Example:A = {a, b, c}. a is the first element, b the

second, c the third. So, we have three elements, and |A| = 3.

Some Definitions

Finite: A set S is finite if S is equal to the empty set, or if there exists n an element of the natural numbers, and a bijection f:{1,2,…n} S.

Infinite: A set is infinite if it is not finite.

Some Definitions (cont’d)

Denumerable:A set S is denumerable if there exists a bijection f:NS.

Countable: A set is countable if it is finite or denumerable.

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

A Bit of Cardinal Arithmetic

Let s=|S|, and w=|W|. Then:

s + w=|S|U|W|=|SUW|s x w = |S| x |W| = |S x W|

sw= |S||W| = |SW|

Cardinal Numbers

Back to our example:

S = {1, 5, 8, 10}. |S|=4.

S is finite, because it has finitely many elements.

The Cardinality of Natural Numbers

The set of natural numbers is not finite, but it is countable.

|N| = 0

Example One

The cardinality of the natural numbers and even natural numbers is the same.

Let E = even natural numbers.Let N = natural numbers.

Bijection f:NE, where f(n)=2n.

Then E has the same cardinality as N.

|E| = 0 = |N|

Example Two The cardinality of the odd natural numbers and

the even natural numbers are the same.

Let O = odd natural numbers.

Bijection f: O E, where f(n) = n+1.

Then O has the same cardinality as E (and N).

|O| = |E| = 0 = |N|

Example Three

E + O = N

Since we know: |E|=0 , |O|=0 , |N|=0

Then, |E|+|O|=|N| gives us 0 + 0 = 0.

Definition

Power set: Given any set S, let P(S) denote the collection of subsets of S. Then P(S) is called the power set of S.

For example:Let S = {1,2}. Then, P(S) = {, {1},{2},{1,2}}.

*Note that |S| < |P(S)|

Theorem

For any set S, |S| < |P(S)|.

Theorem

Any subset of a countable set is countable.

The Cardinality of Real NumbersTheorem:

The set of real numbers is uncountable.

We denote the cardinality of the real numbers as:

|R| = C

The Real Numbers are Uncountable (Proof)

Proving the real numbers are uncountable.

1. Assume that R is countable. 2. Construct a number that is not in

the set. 3. By constructing a number not in our

original set, we conclude that R is uncountable.

The Real Numbers are Uncountable (Proof)

1. Assume that the set of real numbers is countable.

2. Then any subset of the real numbers is countable (by the previous theorem).

So let us look at the set S = (0,1)

The Real Numbers are Uncountable (Proof)

Since we have defined S to be countable, we can list all elements of S.

So S = {s1, s2, … , sn}

The Real Numbers are Uncountable (Proof)

so we can write any element of S in its decimal expansion. Meaning,

s1 = 0.a11a12a13a14…s2 = 0.a21a22a23a24…and so on.

And each aij is an element of {0,1, 2, 3, 4, 5, 6, 7, 8, 9}.

The Real Numbers are Uncountable (Proof)

Let y = 0.b1b2b3b4…

Where: bi = {1, if ann ≠ 1; 8 if ann = 1}.

The Real Numbers are Uncountable (Proof)

For example, if

x1 = 0.32045….x2 = 0.44246…x3 = 0.57124…

Then y = 0.118…

The Real Numbers are Uncountable (Proof)

y is made up of 1’s and 8’s, so y is in S = (1,0)

But, y ≠ sn because it differs from sn at the nth decimal place.

S must be uncountable.Then the real numbers are uncountable.

Recall…

Since the real numbers are uncountable, and the natural numbers are countable:

|N| < |R|0 < C

There are more real numbers than natural numbers!

Hmm…

0 < ? < C

The Continuum Hypothesis

Cantor believed his sequence,0, 1, 2, …, 0 , 1, 2, …,

contained every cardinal number. But, which one is C?

0 is the number of finite ordinal numbers.

1 is the number of ordinal numbers that are either finite or in the 0 class.

And so on…

The Continuum Hypothesis (cont’d)

There are exactly C = 20 real

numbers and C > 0 .

But, does C = 1?Cantor believed so.

The Generalized Continuum Hypothesis

α+1 = 2α

for all α?

The Continuum Hypothesis

0 < ? < Cl

Georg Cantor suggested that no such set exists.

Kurt Godel showed that this couldn’t be disproved.

Paul Cohen showed that this couldn’t be proved either.

1900

1940

1963

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