OpenMath Content Dictionary: multiset1

Canonical URL:

http://www.openmath.org/cd/multiset1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

multiset1.ocd

CD as XML Encoded OpenMath:

multiset1.omcd

Defines:

cartesian_product, emptyset, in, intersect, multiset, notin, notprsubset, notsubset, prsubset, setdiff, size, subset, union

Date:

2004-03-30

Version:

3

Review Date:

2006-03-30

Status:

official

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     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
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          Dictionary Group whose name is, for example, `math' containing
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This CD defines the set functions and constructors for basic multiset theory. It is intended to be `compatible' with the corresponding elements in MathML i.e. set operations acting on sets of type=multiset.

Based on set1.ocd

size

Role:

application

Description:

This symbol is used to denote the number of elements in a multiset. It is either a non-negative integer, or an infinite cardinal number. The symbol infinity may be used for an unspecified infinite cardinal.

Example:

The size of the multiset{3,3,9} = 3

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="multiset1" name="size"/>
      <OMA>
        <OMS cd="multiset1" name="multiset"/>
        <OMI> 3 </OMI>
        <OMI> 3 </OMI>
        <OMI> 9 </OMI>
      </OMA>
    </OMA>
    <OMI> 3 </OMI>
  </OMA>
</OMOBJ>

eq (size (multiset ( 3 , 3 , 9 ) ) , 3 )

$$\mathrm{size}\left(\mathrm{multiset}(3,3,9)\right)=3$$

Signatures:

sts

cartesian_product

Role:

application

Description:

This symbol represents an n-ary construction function for constructing the Cartesian product of multisets. It takes n multiset arguments in order to construct their Cartesian product.

Example:

An example to show the representation of the Cartesian product of multisets: AxBxC.

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="cartesian_product"/>
    <OMV name="A"/>
    <OMV name="B"/>
    <OMV name="C"/>
  </OMA>
</OMOBJ>

cartesian_product ( A, B, C)

$$A\times B\times C$$

Signatures:

sts

emptyset

Role:

constant

Description:

This symbol is used to represent the empty multiset, that is the multiset which contains no members. It takes no parameters.

Commented Mathematical property (CMP):

The intersection of A with the empty (multi) set is the empty (multi) set

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="multiset1" name="intersect"/>
      <OMV name="A"/>
      <OMS cd="multiset1" name="emptyset"/>
    </OMA>
    <OMS cd="multiset1" name="emptyset"/>
  </OMA>
</OMOBJ>

eq (intersect ( A, emptyset) , emptyset)

$$A\cap \varnothing =\varnothing$$

Commented Mathematical property (CMP):

The union of A with the empty (multi) set is A

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="eq"/>
    <OMA>
      <OMS cd="multiset1" name="union"/>
      <OMV name="A"/>
      <OMS cd="multiset1" name="emptyset"/>
    </OMA>
    <OMV name="A"/>
  </OMA>
</OMOBJ>

eq (union ( A, emptyset) , A)

$$A\cup \varnothing =A$$

Signatures:

sts

multiset

Role:

application

Description:

This symbol represents the multiset construct. It is either an n-ary function, in which case the multiset entries are given explicitly, or it works on an elements construct. There is no implied ordering to the elements of a multiset.

Example:

The multiset {4, 1, 0, 1 4}

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="multiset"/>
    <OMI> 4 </OMI>
    <OMI> 1 </OMI>
    <OMI> 0 </OMI>
    <OMI> 1 </OMI>
    <OMI> 4 </OMI>
  </OMA>
</OMOBJ>

multiset ( 4 , 1 , 0 , 1 , 4 )

$$\mathrm{multiset}(4,1,0,1,4)$$

Signatures:

sts

intersect

Role:

application

Description:

This symbol is used to denote the n-ary intersection of multisets. It takes multisets as arguments, and denotes the multiset that contains all the elements that occur in all of them, with multiplicity the minimum of their multiplicities in all multisets.

Commented Mathematical property (CMP):

(A intersect B) is a subset of A and (A intersect B) is a subset of B

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="multiset1" name="subset"/>
      <OMA>
        <OMS cd="multiset1" name="intersect"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="subset"/>
      <OMA>
        <OMS cd="multiset1" name="intersect"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
      <OMV name="B"/>
    </OMA>
  </OMA>
</OMOBJ>

and (subset (intersect ( A, B) , A) , subset (intersect ( A, B) , B) )

$$A\cap B\subset A\wedge A\cap B\subset B$$

Signatures:

sts

union

Role:

application

Description:

This symbol is used to denote the n-ary union of multisets. It takes multisets as arguments, and denotes the multiset that contains all the elements that occur in any of them, with multiplicity the sum of all the multiplicities in the multiset arguments.

Commented Mathematical property (CMP):

A is a subset of (A union B) and B is a subset of (A union B)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="multiset1" name="subset"/>
      <OMV name="A"/>
      <OMA>
        <OMS cd="multiset1" name="union"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="subset"/>
      <OMV name="B"/>
      <OMA>
        <OMS cd="multiset1" name="union"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

and (subset ( A, union ( A, B) ) , subset ( B, union ( A, B) ) )

$$A\subset A\cup B\wedge B\subset A\cup B$$

Commented Mathematical property (CMP):

for all sets A,B and C union(A,intersect(B,C)) = intersect(union(A,B),union(A,C))

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMBIND>
    <OMS cd="quant1" name="forall"/>
    <OMBVAR>
      <OMV name="A"/>
      <OMV name="B"/>
      <OMV name="C"/>
    </OMBVAR>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="multiset1" name="union"/>
        <OMV name="A"/>
        <OMA>
          <OMS cd="multiset1" name="intersect"/>
          <OMV name="B"/>
          <OMV name="C"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="multiset1" name="intersect"/>
        <OMA>
          <OMS cd="multiset1" name="union"/>
          <OMV name="A"/>
          <OMV name="B"/>
        </OMA>
        <OMA>
          <OMS cd="multiset1" name="union"/>
          <OMV name="A"/>
          <OMV name="C"/>
        </OMA>
      </OMA>
    </OMA>    
  </OMBIND>
</OMOBJ>

forall [ A B C ] . (eq (union ( A, intersect ( B, C) ) , intersect (union ( A, B) , union ( A, C) ) ) )

$$\forall \hspace{0.17em}A,B,C.A\cup B\cap C=A\cup B\cap A\cup C$$

Signatures:

sts

setdiff

Role:

application

Description:

This symbol is used to denote the multiset difference of two multisets. It takes two multisets as arguments, and denotes the multiset that contains all the elements that occur in the first multiset with strictly greater multiplicity than in the second. The multiplicity in the result is the difference of the two.

Commented Mathematical property (CMP):

the difference of A and B is a subset of A

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="subset"/>
    <OMA>
      <OMS cd="multiset1" name="setdiff"/>
      <OMV name="A"/>
      <OMV name="B"/>
    </OMA>
    <OMV name="A"/>
  </OMA>
</OMOBJ>

subset (setdiff ( A, B) , A)

$$A\setminus B\subset A$$

Signatures:

sts

subset

Role:

application

Description:

This symbol has two (multiset) arguments. It is used to denote that the first set is a subset of the second, i.e. every element of the first occurs with multiplicity at least as much in the second.

Commented Mathematical property (CMP):

if B is a subset of A and C is a subset of B then C is a subset of A

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="multiset1" name="subset"/>
        <OMV name="B"/>
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="multiset1" name="subset"/>
        <OMV name="C"/>
        <OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="subset"/>
      <OMV name="C"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMOBJ>

implies (and (subset ( B, A) , subset ( C, B) ) , subset ( C, A) )

$$B\subset A\wedge C\subset B\Rightarrow C\subset A$$

Signatures:

sts

in

Role:

application

Description:

This symbol has two arguments, an element and a multiset. It is used to denote that the element is in the given multiset.

Commented Mathematical property (CMP):

if a is in A and a is in B then a is in A intersection B

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="logic1" name="implies"/>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="multiset1" name="in"/>
        <OMV name="a"/>
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="multiset1" name="in"/>
        <OMV name="a"/>
        <OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="in"/>
      <OMV name="a"/>
      <OMA>
        <OMS cd="multiset1" name="intersect"/>
        <OMV name="A"/>
        <OMV name="B"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

implies (and (in ( a, A) , in ( a, B) ) , in ( a, intersect ( A, B) ) )

$$a\in A\wedge a\in B\Rightarrow a\in A\cap B$$

Signatures:

sts

notin

Role:

application

Description:

This symbol has two arguments, an element and a multiset. It is used to denote that the element is not in the given multiset.

Example:

4 is not in {1,1,2,3}

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="notin"/>
    <OMI> 4 </OMI>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 1 </OMI>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

notin ( 4 , multiset ( 1 , 1 , 2 , 3 ) )

$$4\notin \mathrm{multiset}(1,1,2,3)$$

Signatures:

sts

prsubset

Role:

application

Description:

This symbol has two (multiset) arguments. It is used to denote that the first multiset is a proper subset of the second, that is a subset of the second multiset but not actually equal to it.

Example:

{2,3} is a proper subset of {2,2,3}

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="prsubset"/>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 2 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

prsubset (multiset ( 2 , 3 ) , multiset ( 2 , 2 , 3 ) )

$$\mathrm{multiset}(2,3)\subset \mathrm{multiset}(2,2,3)$$

Signatures:

sts

notsubset

Role:

application

Description:

This symbol has two (multiset) arguments. It is used to denote that the first multiset is not a subset of the second.

Example:

{2,3,3} is not a subset of {1,2,3}

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="notsubset"/>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
      <OMI> 3 </OMI>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

notsubset (multiset ( 2 , 3 , 3 ) , multiset ( 1 , 2 , 3 ) )

$$\mathrm{multiset}(2,3,3)\not\subset \mathrm{multiset}(1,2,3)$$

Signatures:

sts

notprsubset

Role:

application

Description:

This symbol has two (multiset) arguments. It is used to denote that the first multiset is not a proper subset of the second. A proper subset of a multiset is a subset of the multiset but not actually equal to it.

Example:

{1,2,1} is not a proper subset of {1,2,1}

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="multiset1" name="notprsubset"/>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 1 </OMI>
    </OMA>
    <OMA>
      <OMS cd="multiset1" name="multiset"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 1 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

notprsubset (multiset ( 1 , 2 , 1 ) , multiset ( 1 , 2 , 1 ) )

$$\mathrm{multiset}(1,2,1)\not\subset \mathrm{multiset}(1,2,1)$$

Signatures:

sts