examples of countable sets

examples of countable sets

This entry lists some common

examples of countable sets

.

Derived Examples

1.

any

finite set

, including the

empty set

∅

(proof (

http://

planetmath

.org/AlternativeDefinitionsOfCountable

)).

2.

any subset of a

countable set

(proof (

http://planetmath.org/SubsetsOfCountableSets

)).

3.

any finite

product of countable sets

(proof (

http://planetmath.org/UnionOfCountableSets

)).

4.

any

countable

union of countable sets

(proof (

http://planetmath.org/ProductOfCountableSets

)).

5.

the set of all finite subsets of a countable set.

Proof.

Let

A

be a countable set, and

F

⁢

(

A

)

the set of all finite subsets of

A

. Let

A

n

be the set of all subsets of

A

of

cardinality

at most

n

. Then

A

1

is countable, since

A

is. Suppose now that

A

n

is countable. The

function

f

:

A

n

×

A

1

→

A

n

+

1

where

f

⁢

(

X

,

Y

)

=

X

∪

Y

is easily seen to be onto. Since

A

n

×

A

1

is countable, so is

A

n

+

1

. Now,

F

⁢

(

A

)

is just the union of all the countable sets

A

i

, and this union is a countable union, we see that

F

⁢

(

A

)

is countable too.
∎

6.

the set of all cofinite subsets of a countable set. This is true, because there is a

one-to-one correspondence

between the set

F

⁢

(

A

)

of finite sets and the set

co

-

⁡

F

⁢

(

A

)

of cofinite sets:

X

↦

A

-

X

.

7.

the set of all

finite sequences

over a countable set.

Proof.

Let

A

be a countable set, and

A

F

the set of all finite sequences over

A

. An element of

A

F

can be identified with an element of

A

n

, and vice versa (the bijection is clear). Therefore,

A

F

can be identified with the union of

A

i

, for

i

=

0

,

1

,

2

,

…

. Since each

A

i

is countable (because

A

is), and we are taking a countable union,

A

F

is countable as a result.
∎

8.

fix countable sets

A

,

B

. The set

X

of all functions from finite subsets of

B

into

A

is countable.

Proof.

For each finite subset

C

of

B

, the set of all functions from

C

to

A

is just

A

C

, which has cardinality

|

A

|

|

C

|

, and thus is countable since

A

is. Since

X

is just the union of all

A

C

, where

C

ranges over the finite subsets of

B

, and there are countably many of them (as

B

is countable),

X

is also countable.
∎

9.

fix countable sets

A

,

B

and an

element

a

∈

A

. The set

Y

of all functions from

B

to

A

such that

f

⁢

(

b

)

=

a

for all but a finite number of

b

∈

B

is countable.

Proof.

For any

f

:

B

→

A

, call the

support

of

f

the set

{

b

∈

B

∣

f

⁢

(

b

)

≠

a

}

, and denote it by

supp

⁡

(

f

)

. Then every

f

∈

Y

has

finite support

. The map

G

:

Y

→

X

(where

X

is defined in the last example) given by

G

(

f

)

=

f

|

supp

(

f

)

is an

injection

: if

G

⁢

(

f

)

=

G

⁢

(

h

)

, then

f

⁢

(

b

)

=

h

⁢

(

b

)

for any

b

∈

supp

⁡

(

f

)

=

supp

⁡

(

h

)

, and

f

⁢

(

b

)

=

a

=

h

⁢

(

b

)

otherwise, whence

f

=

g

. But since

X

is countable, so is

Y

.
∎

Concrete Examples

1.

the sets

ℕ

(

natural numbers

),

ℤ

(

integers

), and

ℚ

(

rational numbers

)

2.

the set of all

algebraic numbers

Proof.

Let

𝔸

be the set of all algebraic numbers over

ℚ

. For each

polynomial

p

(in one

variable

X

) over

ℚ

, let

R

p

be the set of roots of

p

over

ℚ

. By

definition

,

𝔸

is the union of all

R

p

, where

p

ranges over the set

P

of all polynomials over

ℚ

. For any

p

∈

P

of degree

n

, we may

associate

a vector

v

p

∈

ℚ

n

+

1

:

p

=

a

0

+

a

1

⁢

X

+

⋯

+

a

n

⁢

X

n

⟺

v

p

=

(

a

0

,

a

1

,

…

,

a

n

)

.

The association can be reversed. So the set

P

n

⊂

P

of all polynomials of degree

n

is equinumerous to

ℚ

n

+

1

, and therefore countable. As

P

is just the countable union of all

P

n

,

P

is countable, which means

𝔸

=

⋃

{

R

p

∣

p

∈

P

}

is countable also.
∎

3.

the set of all

algebraic integers

, because every algebraic integer is an algebraic number.

4.

the set of all words over an

alphabet

, because ever word can be thought of as a finite sequence over the alphabet, which is finite.

Title

examples of countable sets

Canonical

name

ExamplesOfCountableSets

Date of creation

2013-03-22 19:02:59

Last modified on

2013-03-22 19:02:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Example

Classification

msc 03E10

Related topic

AlgebraicNumbersAreCountable

Generated on Thu Feb 8 20:09:09 2018 by

LaTeXML