alternative definitions of countable

alternative definitions of countable

The following are alternative ways of characterizing a

countable set

.

Proposition 1

.

Let

A

be a set and

N

the set of natural numbers. The following are

equivalent

:

1.

there is a

surjection

from

ℕ

to

A

.

2.

there is an

injection

from

A

to

ℕ

.

3.

either

A

is finite or there is a

bijection

between

A

and

ℕ

.

Proof.

First notice that if

A

were the

empty set

, then any map to or from

A

is empty, so

(

1

)

⇔

(

2

)

⇔

(

3

)

vacuously

. Now, suppose that

A

≠

∅

.

(

1

)

⇒

(

2

)

. Suppose

f

:

ℕ

→

A

is a surjection. For each

a

∈

A

, let

f

-

1

⁢

(

a

)

be the set

{

n

∈

ℕ

∣

f

⁢

(

n

)

=

a

}

. Since

f

-

1

⁢

(

a

)

is a subset of

ℕ

, which is

well-ordered

,

f

-

1

⁢

(

a

)

itself is well-ordered, and thus has a least

element

(keep in mind

A

≠

∅

, the

existence

of

a

∈

A

is guaranteed, so that

f

-

1

⁢

(

a

)

≠

∅

as well). Let

g

⁢

(

a

)

be this least element. Then

a

↦

g

⁢

(

a

)

is a

well-defined

mapping

from

A

to

ℕ

. It is one-to-one, for if

g

⁢

(

a

)

=

g

⁢

(

b

)

=

n

, then

a

=

f

⁢

(

n

)

=

b

.

(

2

)

⇒

(

1

)

. Suppose

g

:

A

→

ℕ

is one-to-one. So

g

-

1

⁢

(

n

)

is at most a

singleton

for every

n

∈

ℕ

. If it is a singleton, identify

g

-

1

⁢

(

n

)

with that element. Otherwise, identify

g

-

1

⁢

(

n

)

with a designated element

a

0

∈

A

(remember

A

is non-empty). Define a

function

f

:

ℕ

→

A

by

f

⁢

(

n

)

:=

g

-

1

⁢

(

n

)

. By the discussion above,

g

-

1

⁢

(

n

)

is a well-defined element of

A

, and therefore

f

is well-defined.

f

is onto because for every

a

∈

A

,

f

⁢

(

g

⁢

(

a

)

)

=

a

.

(

3

)

⇒

(

2

)

is clear.

(

2

)

⇒

(

3

)

. Let

g

:

A

→

ℕ

be an injection. Then

g

⁢

(

A

)

is either finite or

infinite

. If

g

⁢

(

A

)

is finite, so is

A

, since they are equinumerous. Suppose

g

⁢

(

A

)

is infinite. Since

g

⁢

(

A

)

⊆

ℕ

, it is well-ordered. The (

induced

) well-ordering on

g

⁢

(

A

)

implies that

g

⁢

(

A

)

=

{

n

1

,

n

2

,

…

}

, where

n

1

<

n

2

<

⋯

.

Now, define

h

:

ℕ

→

A

as follows, for each

i

∈

ℕ

,

h

⁢

(

i

)

is the element in

A

such that

g

⁢

(

h

⁢

(

i

)

)

=

n

i

. So

h

is well-defined. Next,

h

is injective. For if

h

⁢

(

i

)

=

h

⁢

(

j

)

, then

n

i

=

g

⁢

(

h

⁢

(

i

)

)

=

g

⁢

(

h

⁢

(

j

)

)

=

n

j

, implying

i

=

j

. Finally,

h

is a surjection, for if we pick any

a

∈

A

, then

g

⁢

(

a

)

∈

g

⁢

(

A

)

, meaning that

g

⁢

(

a

)

=

n

i

for some

i

, so

h

⁢

(

i

)

=

g

⁢

(

a

)

.
∎

Therefore, countability can be defined in terms of either of the above three statements.

Note that the

axiom of choice

is not needed in the proof of

(

1

)

⇒

(

2

)

, since the selection of an element in

f

-

1

⁢

(

a

)

is definite, not arbitrary.

For example, we show that

ℕ

2

is

countable

. By the

proposition

above, we either need to find a surjection

f

:

ℕ

→

ℕ

2

, or an injection

g

:

ℕ

2

→

ℕ

. Actually, in this case, we can find both:

1.

the function

f

:

ℕ

→

ℕ

2

given by

f

⁢

(

a

)

=

(

m

,

n

)

where

a

=

2

m

⁢

(

2

⁢

n

+

1

)

is

surjective

. First, the function is well-defined, for every

positive integer

has a unique representation as the

product

of a power of

2

and an

odd number

. It is surjective because for every

(

m

,

n

)

, we see that

f

⁢

(

2

m

⁢

(

2

⁢

n

+

1

)

)

=

(

m

,

n

)

.

2.

the function

g

:

ℕ

2

→

ℕ

given by

f

⁢

(

m

,

n

)

=

2

m

⁢

3

n

is clearly injective.

Note that the injectivity of

g

, as well as

f

being well-defined, rely on the unique factorization of

integers

by

prime numbers

. In this entry (

http://planetmath.org/ProductOfCountableSets

), we actually find a bijection between

ℕ

and

ℕ

2

.

As a corollary, we record the following:

Corollary 1

.

Let

A

,

B

be sets,

f

:

A

→

B

a function.

•

If

f

is an injection, and

B

is countable, so is

A

.

•

If

f

is a surjection, and

A

countable, so is

B

.

The proof is left to the reader.

Title

alternative definitions of countable

Canonical

name

AlternativeDefinitionsOfCountable

Date of creation

2013-03-22 19:02:49

Last modified on

2013-03-22 19:02:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E10

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