multivalued function

multivalued function

Let us recall that a

function

f

from a set

A

to a set

B

is an assignment that takes each element of

A

to a

unique

element of

B

. One way to generalize this notion is to remove the

uniqueness

aspect of this assignment, and what results is a

multivalued function

. Although a multivalued function is in general not a function, one may formalize this notion mathematically as a function:

Definition

. A

multivalued function

f

from a set

A

to a set

B

is a function

f

:

A

→

P

⁢

(

B

)

, the

power set

of

B

, such that

f

⁢

(

a

)

is non-empty for every

a

∈

A

. Let us denote

f

:

A

⇒

B

the multivalued function

f

from

A

to

B

.

A multivalued function is said to be

single-valued

if

f

⁢

(

a

)

is a singleton for every

a

∈

A

.

From this definition, we see that every function

f

:

A

→

B

is naturally associated with a multivalued function

f

*

:

A

⇒

B

, given by

f

*

⁢

(

a

)

=

{

f

⁢

(

a

)

}

.

Thus a function is just a single-valued multivalued function, and vice versa.

As another example, suppose

f

:

A

→

B

is a surjective function. Then

f

-

1

:

B

⇒

A

defined by

f

-

1

⁢

(

b

)

=

{

a

∈

A

∣

f

⁢

(

a

)

=

b

}

is a multivalued function.

Another way of looking at a multivalued function is to interpret it as a special type of a

relation

, called a

total relation

. A relation

R

from

A

to

B

is said to be

total

if for every

a

∈

A

, there exists a

b

∈

B

such that

a

⁢

R

⁢

b

.

Given a total relation

R

from

A

to

B

, the function

f

R

:

A

⇒

B

given by

f

R

⁢

(

a

)

=

{

b

∈

B

∣

a

⁢

R

⁢

b

}

is multivalued.

Conversely

, given

f

:

A

⇒

B

, the relation

R

f

from

A

to

B

defined by

a

⁢

R

f

⁢

b

iff

b

∈

f

⁢

(

a

)

is total.

Basic notions such as

functional

composition

, injectivity and surjectivity on functions can be easily translated to multivalued functions:

Definition

. A multivalued function

f

:

A

⇒

B

is

injective

if

f

⁢

(

a

)

=

f

⁢

(

b

)

implies

a

=

b

,

absolutely injective

if

a

≠

b

implies

f

⁢

(

a

)

∩

f

⁢

(

b

)

=

∅

, and

surjective

if every

b

∈

B

belongs to some

f

⁢

(

a

)

for some

a

∈

A

. If

f

is both injective and surjective, it is said to be

bijective

.

Given

f

:

A

⇒

B

and

g

:

B

⇒

C

, then we define the

composition

of

f

and

g

, written

g

∘

f

:

A

⇒

C

, by setting

(

g

∘

f

)

⁢

(

a

)

:=

{

c

∈

C

∣

c

∈

g

⁢

(

b

)

⁢

for some

⁢

b

∈

f

⁢

(

a

)

}

.

It is easy to see that

R

g

∘

f

=

R

g

∘

R

f

, where the

∘

on the right hand side denotes

relational composition

.

For a subset

S

⊆

A

, if we define

f

⁢

(

S

)

=

{

b

∈

B

∣

b

∈

f

⁢

(

s

)

⁢

for some

⁢

s

∈

S

}

, then

f

:

A

⇒

B

is surjective iff

f

⁢

(

A

)

=

B

, and functional composition has a simplified and familiar form:

(

g

∘

f

)

⁢

(

a

)

=

g

⁢

(

f

⁢

(

a

)

)

.

A bijective multivalued function

i

:

A

⇒

A

is said to be an

identity

(on

A

) if

a

∈

i

⁢

(

a

)

for all

a

∈

A

(equivalently,

R

f

is a

reflexive relation

). Certainly, the function

i

⁢

d

A

on

A

, taking

a

into itself (or equivalently,

{

a

}

), is an identity. However, given

A

, there may be more than one identity on it:

f

:

ℤ

→

ℤ

given by

f

⁢

(

n

)

=

{

n

,

n

+

1

}

is an identity that is not

i

⁢

d

ℤ

. An absolute identity on

A

is necessarily

i

⁢

d

A

.

Suppose

i

:

A

⇒

A

, we have the following

equivalent

characterizations

of an identity:

1.

i

is an identity on

A

2.

f

⁢

(

x

)

⊆

(

f

∘

i

)

⁢

(

x

)

for every

f

:

A

⇒

B

and every

x

∈

A

3.

g

⁢

(

y

)

⊆

(

i

∘

g

)

⁢

(

y

)

for all

g

:

C

⇒

A

and

y

∈

C

To see this, first assume

i

is an identity on

A

. Then

x

∈

i

⁢

(

x

)

, so that

f

⁢

(

x

)

⊆

f

⁢

(

i

⁢

(

x

)

)

. Conversely,

i

⁢

d

A

⁢

(

x

)

⊆

(

i

⁢

d

A

∘

i

)

⁢

(

x

)

implies that

{

x

}

⊆

i

⁢

d

A

⁢

(

i

⁢

(

x

)

)

=

{

y

∣

y

∈

i

⁢

(

x

)

}

, so that

x

∈

i

⁢

(

x

)

. This proved the equivalence of (1) and (2). The equivalence of (1) and (3) are established similarly.

A multivalued function

g

:

B

⇒

A

is said to be an

inverse

of

f

:

A

⇒

B

if

f

∘

g

is an identity on

B

and

g

∘

f

is an identity on

A

. If

f

possesses an inverse, it must be surjective. Given that

f

:

A

⇒

B

is surjective, the multivalued function

f

-

1

:

B

⇒

A

defined by

f

-

1

⁢

(

b

)

=

{

a

∈

A

∣

b

∈

f

⁢

(

a

)

}

is an inverse of

f

. Like identities, inverses are not unique.

Remark

. More generally, one defines a

multivalued partial function

(or

partial multivalued function

)

f

from

A

to

B

, as a multivalued function from a subset of

A

to

B

. The same notation

f

:

A

⇒

B

is used to mean that

f

is a multivalued partial function from

A

to

B

. A multivalued partial function

f

:

A

⇒

B

can be equivalently characterized, either as a function

f

′

:

A

→

P

⁢

(

B

)

, where

f

′

⁢

(

a

)

is undefined iff

f

′

⁢

(

a

)

=

∅

, or simply as a relation

R

f

from

A

to

B

, where

a

⁢

R

f

⁢

b

iff

f

⁢

(

a

)

is defined and

b

∈

f

⁢

(

a

)

. Every

partial function

f

:

A

→

B

has an associated multivalued partial function

f

*

:

A

⇒

B

, so that

f

*

⁢

(

a

)

is defined and is equal to

{

b

}

iff

f

⁢

(

a

)

is and

f

⁢

(

a

)

=

b

.

Title

multivalued function

Canonical name

MultivaluedFunction

Date of creation

2013-03-22 18:36:26

Last modified on

2013-03-22 18:36:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E20

Synonym

multi-valued

Synonym

multiple-valued

Synonym

multiple valued

Synonym

single-valued

Synonym

single valued

Synonym

partial multivalued function

Related topic

Multifunction

Defines

multivalued

Defines

singlevalued

Defines

total relation

Defines

multivalued partial function

Defines

injective

Defines

surjective

Defines

bijective

Defines

identity

Defines

inverse

Defines

absolutely injective

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