cardinality

cardinality

Cardinality

Cardinality

is a notion of the size of a set which does not rely on numbers. It is a relative notion. For

instance

, two sets may each have an

infinite

number of

elements

, but one may have a greater cardinality. That is, in a sense, one may have a “more infinite” number of elements. See

Cantor diagonalization

for an example of how the reals have a greater cardinality than the

natural numbers

.

The formal definition of cardinality rests upon the notion

mappings

between sets:

Cardinality

.

The cardinality of a set

A

is greater than or equal to
the cardinality of a set

B

if there is a one-to-one function (an

injection

) from

B

to

A

.
Symbolically, we write

|

A

|

≥

|

B

|

.

and

Cardinality

.

Sets

A

and

B

have the same cardinality
if there is a one-to-one and onto

function

(a

bijection

) from

A

to

B

.
Symbolically, we write

|

A

|

=

|

B

|

.

It can be shown that if

|

A

|

≥

|

B

|

and

|

B

|

≥

|

A

|

then

|

A

|

=

|

B

|

.
This is the SchrÃÂ¶der-Bernstein

Theorem

.

Equality of cardinality is variously called

equipotence

,

equipollence

,

equinumerosity

, or

equicardinality

.
For

|

A

|

=

|

B

|

, we would say that “

A

is

equipotent

to

B

”,
“

A

is

equipollent

to

B

”, or “

A

is

equinumerous

to

B

”.

An

equivalent

definition of cardinality is

Cardinality (alt. def.)

.

The cardinality of a set

A

is the unique

cardinal number

κ

such that

A

is equinumerous with

κ

. The cardinality of

A

is written

|

A

|

.

This definition of cardinality makes use of a special class of numbers, called the cardinal numbers. This highlights the fact that, while cardinality can be understood and defined without appealing to numbers, it is often convenient and useful to treat cardinality in a “numeric” manner.

Results

Some results on cardinality:

1.

A

is equipotent to

A

.

2.

If

A

is equipotent to

B

, then

B

is equipotent to

A

.

3.

If

A

is equipotent to

B

and

B

is equipotent to

C

, then

A

is equipotent to

C

.

Proof.

Respectively:

1.

The

identity function

on

A

is a bijection from

A

to

A

.

2.

If

f

is a bijection from

A

to

B

, then

f

-

1

exists and is a bijection from

B

to

A

.

3.

If

f

is a bijection from

A

to

B

and

g

is a bijection from

B

to

C

, then

f

∘

g

is a bijection from

A

to

C

.

∎

Example

The set of

even integers

2

⁢

ℤ

has the same cardinality as the set of integers

ℤ

: if we define

f

:

2

⁢

ℤ

→

ℤ

such that

f

⁢

(

x

)

=

x

2

,
then

f

is a bijection, and therefore

|

2

⁢

ℤ

|

=

|

ℤ

|

.

Title

cardinality

Canonical name

Cardinality

Date of creation

2013-03-22 12:00:42

Last modified on

2013-03-22 12:00:42

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

25

Author

yark (2760)

Entry type

Definition

Classification

msc 03E10

Synonym

size

Related topic

OrderGroup

Related topic

GeneralizedContinuumHypothesis

Related topic

CardinalNumber

Related topic

DedekindInfinite

Defines

equipotence

Defines

equipotent

Defines

equicardinality

Defines

equipollence

Defines

equipollent

Defines

equinumerosity

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