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multiset1

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 (Revision 1)
Review Date:
2006-03-30
Status:
official 
     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     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
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

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 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="multiset1">size</csymbol>
   <apply><csymbol cd="multiset1">multiset</csymbol>
    <cn type="integer">3</cn>
    <cn type="integer">3</cn>
    <cn type="integer">9</cn>
   </apply>
  </apply>
  <cn type="integer">3</cn>
 </apply>
</math>

multiset1.size(multiset1.multiset(3, 3, 9)) = 3

size ⁡ ( multiset ⁡ ( 3 , 3 , 9 ) ) = 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. 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">cartesian_product</csymbol><ci>A</ci><ci>B</ci><ci>C</ci></apply>
</math>

multiset1.cartesian_product($A, $B, $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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="multiset1">intersect</csymbol><ci>A</ci><csymbol cd="multiset1">emptyset</csymbol></apply>
  <csymbol cd="multiset1">emptyset</csymbol>
 </apply>
</math>

multiset1.intersect($A, multiset1.emptyset) = multiset1.emptyset

Commented Mathematical property (CMP):
The union of A with the empty (multi) set is A
Formal Mathematical property (FMP): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="multiset1">union</csymbol><ci>A</ci><csymbol cd="multiset1">emptyset</csymbol></apply>
  <ci>A</ci>
 </apply>
</math>

multiset1.union($A, multiset1.emptyset) = $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} 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">multiset</csymbol>
  <cn type="integer">4</cn>
  <cn type="integer">1</cn>
  <cn type="integer">0</cn>
  <cn type="integer">1</cn>
  <cn type="integer">4</cn>
 </apply>
</math>

multiset1.multiset(4, 1, 0, 1, 4)

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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="multiset1">subset</csymbol>
   <apply><csymbol cd="multiset1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
   <ci>A</ci>
  </apply>
  <apply><csymbol cd="multiset1">subset</csymbol>
   <apply><csymbol cd="multiset1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
   <ci>B</ci>
  </apply>
 </apply>
</math>

multiset1.subset(multiset1.intersect($A, $B), $A) and multiset1.subset(multiset1.intersect($A, $B), $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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="multiset1">subset</csymbol>
   <ci>A</ci>
   <apply><csymbol cd="multiset1">union</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="multiset1">subset</csymbol>
   <ci>B</ci>
   <apply><csymbol cd="multiset1">union</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

multiset1.subset($A, multiset1.union($A, $B)) and multiset1.subset($B, multiset1.union($A, $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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <bind><csymbol cd="quant1">forall</csymbol>
  <bvar><ci>A</ci></bvar>
  <bvar><ci>B</ci></bvar>
  <bvar><ci>C</ci></bvar>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="multiset1">union</csymbol>
    <ci>A</ci>
    <apply><csymbol cd="multiset1">intersect</csymbol><ci>B</ci><ci>C</ci></apply>
   </apply>
   <apply><csymbol cd="multiset1">intersect</csymbol>
    <apply><csymbol cd="multiset1">union</csymbol><ci>A</ci><ci>B</ci></apply>
    <apply><csymbol cd="multiset1">union</csymbol><ci>A</ci><ci>C</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$A, $B, $C -> multiset1.union($A, multiset1.intersect($B, $C)) = multiset1.intersect(multiset1.union($A, $B), multiset1.union($A, $C))]

∀   A , B , C . A ∪ B ∩ C = A ∪ B ∩ A ∪ 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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">subset</csymbol>
  <apply><csymbol cd="multiset1">setdiff</csymbol><ci>A</ci><ci>B</ci></apply>
  <ci>A</ci>
 </apply>
</math>

multiset1.subset(multiset1.setdiff($A, $B), $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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="multiset1">subset</csymbol><ci>B</ci><ci>A</ci></apply>
   <apply><csymbol cd="multiset1">subset</csymbol><ci>C</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="multiset1">subset</csymbol><ci>C</ci><ci>A</ci></apply>
 </apply>
</math>

multiset1.subset($B, $A) and multiset1.subset($C, $B) ==> multiset1.subset($C, $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): 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="multiset1">in</csymbol><ci>a</ci><ci>A</ci></apply>
   <apply><csymbol cd="multiset1">in</csymbol><ci>a</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="multiset1">in</csymbol>
   <ci>a</ci>
   <apply><csymbol cd="multiset1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

multiset1.in($a, $A) and multiset1.in($a, $B) ==> multiset1.in($a, multiset1.intersect($A, $B))

a ∈ A ∧ a ∈ B ⇒ a ∈ A ∩ 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} 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">notin</csymbol>
  <cn type="integer">4</cn>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

multiset1.notin(4, multiset1.multiset(1, 1, 2, 3))

4 ∉ 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} 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">prsubset</csymbol>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">2</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

multiset1.prsubset(multiset1.multiset(2, 3), multiset1.multiset(2, 2, 3))

multiset ⁡ ( 2 , 3 ) ⊂ 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} 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">notsubset</csymbol>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
   <cn type="integer">3</cn>
  </apply>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

multiset1.notsubset(multiset1.multiset(2, 3, 3), multiset1.multiset(1, 2, 3))

multiset ⁡ ( 2 , 3 , 3 ) ⊄ 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} 
<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>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="multiset1">notprsubset</csymbol>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">1</cn>
  </apply>
  <apply><csymbol cd="multiset1">multiset</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">1</cn>
  </apply>
 </apply>
</math>

multiset1.notprsubset(multiset1.multiset(1, 2, 1), multiset1.multiset(1, 2, 1))

multiset ⁡ ( 1 , 2 , 1 ) ⊄ multiset ⁡ ( 1 , 2 , 1 )

Signatures:
sts

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