properties of bijections

properties of bijections

Let

A

,

B

,

C

,

D

be sets. We write

A

∼

B

when there is a

bijection

from

A

to

B

. Below are some

properties of bijections

.

1.

A

∼

A

. The

identity function

is the bijection from

A

to

A

.

2.

If

A

∼

B

, then

B

∼

A

. If

f

:

A

→

B

is a bijection, then its

inverse function

f

-

1

:

B

→

A

is also a bijection.

3.

If

A

∼

B

,

B

∼

C

, then

A

∼

C

. If

f

:

A

→

B

and

g

:

B

→

C

are bijections, so is the

composition

g

∘

f

:

A

→

C

.

4.

If

A

∼

B

,

C

∼

D

, and

A

∩

C

=

B

∩

D

=

∅

, then

A

∪

B

∼

C

∪

D

.

Proof.

If

f

:

A

→

B

and

g

:

C

→

D

are bijections, so is

h

:

A

∪

C

→

B

∪

D

, defined by

h

⁢

(

x

)

=

{

f

⁢

(

x

)

if

⁢

x

∈

A

,

g

⁢

(

x

)

if

⁢

x

∈

C

.

Since

A

∩

C

=

∅

,

h

is a

well-defined

function

.

h

is onto since both

f

and

g

are. Since

f

,

g

are

one-to-one

, and

B

∩

D

=

∅

,

h

is also one-to-one.
∎

5.

If

A

∼

B

,

C

∼

D

, then

A

×

C

∼

B

×

D

. If

f

:

A

→

B

and

g

:

C

→

D

are bijections, so is

h

:

A

×

C

→

B

×

D

, given by

h

⁢

(

x

,

y

)

=

(

f

⁢

(

x

)

,

g

⁢

(

y

)

)

.

6.

A

×

B

∼

B

×

A

. The function

f

:

A

×

B

→

B

×

A

given by

f

⁢

(

x

,

y

)

=

(

y

,

x

)

is a bijection.

7.

If

A

∼

B

and

C

∼

D

, then

A

C

∼

B

D

.

Proof.

Suppose

ϕ

:

A

→

B

and

σ

:

C

→

D

are bijections. Define

F

:

A

C

→

B

D

as follows: for any function

f

:

A

→

C

, let

F

⁢

(

f

)

=

σ

∘

f

∘

ϕ

-

1

:

B

→

D

.

F

is a well-defined function. It is one-to-one because

σ

and

ϕ

are bijections (hence are cancellable). For any

g

:

B

→

D

, it is easy to see that

F

⁢

(

σ

-

1

∘

g

∘

ϕ

)

=

g

, so that

F

is onto. Therefore

F

is a bijection from

A

C

to

B

D

.
∎

8.

Continuing from

property

8, using the bijection

F

, we have

Mono

⁡

(

A

,

B

)

∼

Mono

⁡

(

C

,

D

)

,

Epi

⁡

(

A

,

B

)

∼

Epi

⁡

(

C

,

D

)

, and

Iso

⁡

(

A

,

B

)

∼

Iso

⁡

(

C

,

D

)

, where

Mono

⁡

(

A

,

B

)

,

Epi

⁡

(

A

,

B

)

, and

Iso

⁡

(

A

,

B

)

are the sets of

injections

,

surjections

, and bijections from

A

to

B

.

9.

P

⁢

(

A

)

∼

2

A

, where

P

⁢

(

A

)

is the

powerset

of

A

, and

2

A

is the set of all functions from

A

to

2

=

{

0

,

1

}

.

Proof.

For every

B

⊆

A

, define

φ

B

:

A

→

2

by

φ

B

⁢

(

x

)

=

{

1

if

⁢

x

∈

B

,

0

otherwise

.

Then

φ

:

P

⁢

(

A

)

→

2

A

, defined by

φ

⁢

(

B

)

=

φ

B

is a well-defined function. It is one-to-one: if

φ

B

=

φ

C

for

B

,

C

⊆

A

, then

x

∈

B

iff

x

∈

C

, so

B

=

C

. It is onto: suppose

f

:

A

→

2

, then by setting

B

=

{

x

∈

A

∣

f

⁢

(

x

)

=

1

}

, we see that

φ

B

=

f

. As a result,

φ

is a bijection.
∎

Remark

. As a result of property 9, we sometimes denote

2

A

the powerset of

A

.

Title

properties of bijections

Canonical

name

PropertiesOfBijections

Date of creation

2013-03-22 18:50:41

Last modified on

2013-03-22 18:50:41

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 03-00

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