operations on multisets

operations on multisets

In this entry, we view

multisets

as

functions

whose ranges are the class

K

of

cardinal numbers

. We define

operations on multisets

that mirror the

operations

on sets.

Definition

. Let

f

:

A

→

K

and

g

:

B

→

K

be multisets.

•

The union of

f

and

g

, denoted by

f

∪

g

, is the multiset whose domain is

A

∪

B

, such that

(

f

∪

g

)

⁢

(

x

)

:=

max

⁡

(

f

⁢

(

x

)

,

g

⁢

(

x

)

)

,

keeping in mind that

f

⁢

(

x

)

:=

0

if

x

is not in the domain of

f

.

•

The

intersection

of

f

and

g

, denoted by

f

∩

g

, is the multiset, whose domain is

A

∩

B

, such that

(

f

∩

g

)

⁢

(

x

)

:=

min

⁡

(

f

⁢

(

x

)

,

g

⁢

(

x

)

)

.

•

The sum (or

disjoint union

) of

f

and

g

, denoted by

f

+

g

, is the multiset whose domain is

A

∪

B

(not the disjoint union of

A

and

B

), such that

(

f

+

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

+

g

⁢

(

x

)

,

again keeping in mind that

f

⁢

(

x

)

:=

0

if

x

is not in the domain of

f

.

Clearly, all of the operations described so far are

commutative

. Furthermore, if

+

is cancellable on both sides:

f

+

g

=

f

+

h

implies

g

=

h

, and

g

+

f

=

h

+

f

implies

g

=

h

.

Subtraction

on multisets can also be defined. Suppose

f

:

A

→

K

and

g

:

B

→

K

are multisets. Let

C

be the set

{

x

∈

A

∩

B

⁢

∣

f

⁢

(

x

)

>

⁢

g

⁢

(

x

)

}

. Then

•

the

complement

of

g

in

f

, denoted by

f

-

g

, is the multiset whose domain is

D

:=

(

A

-

B

)

∪

C

, such that

(

f

-

g

)

⁢

(

x

)

:=

f

⁢

(

x

)

-

g

⁢

(

x

)

for all

x

∈

D

.

For example, writing finite multisets (those with finite domains and finite

multiplicities

for all

elements

) in their usual notations, if

f

=

{

a

,

a

,

b

,

b

,

b

,

c

,

d

,

d

}

and

g

=

{

b

,

b

,

c

,

c

,

c

,

d

,

d

,

e

}

, then

•

f

∪

g

=

{

a

,

a

,

b

,

b

,

b

,

c

,

c

,

c

,

d

,

d

,

e

}

•

f

∩

g

=

{

b

,

b

,

c

,

d

,

d

}

•

f

+

g

=

{

a

,

a

,

b

,

b

,

b

,

b

,

b

,

c

,

c

,

c

,

c

,

d

,

d

,

d

,

d

,

e

}

•

f

-

g

=

{

a

,

a

,

b

}

We may characterize the union and intersection operations in terms of multisubsets.

Definition

. A multiset

f

:

A

→

K

is a

multisubset

of a multiset

g

:

B

→

K

if

1.

A

is a subset of

B

, and

2.

f

⁢

(

a

)

≤

g

⁢

(

a

)

for all

a

∈

A

.

We write

f

⊆

g

to mean that

f

is a multisubset of

g

.

Proposition 1

.

Given multisets

f

and

g

.

•

f

∪

g

is the smallest multiset such that

f

and

g

are multisubsets of it. In other words, if

f

⊆

h

and

g

⊆

h

, then

f

∪

g

⊆

h

.

•

f

∩

g

is the largest multiset that is a multisubset of

f

and

g

. In other words, if

h

⊆

f

and

h

⊆

g

, then

h

⊆

f

∩

g

.

Remark

. One may also define the

powerset

of a multiset

f

: the multiset such that each of its elements is a multisubset of

f

. However, the resulting multiset is just a set (the multiplicity of each element is

1

).

Title

operations on multisets

Canonical

name

OperationsOnMultisets

Date of creation

2013-03-22 19:13:23

Last modified on

2013-03-22 19:13:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03E99

Defines

multisubset

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