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set1

OpenMath Content Dictionary: set1
Canonical URL:
http://www.openmath.org/cd/set1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
set1.ocd
CD as XML Encoded OpenMath:
set1.omcd
Defines:
cartesian_product, emptyset, in, intersect, map, notin, notprsubset, notsubset, prsubset, set, setdiff, size, subset, suchthat, 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 set theory. It is intended to be `compatible' with the corresponding elements in MathML.

cartesian_product
Role:
application
Description:

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

Example:
An example to show the representation of the Cartesian product of sets: AxBxC. 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" 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="set1">cartesian_product</csymbol><ci>A</ci><ci>B</ci><ci>C</ci></apply>
</math>

set1.cartesian_product($A, $B, $C)

Signatures:
sts
emptyset
Role:
constant
Description:

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

Commented Mathematical property (CMP):
The intersection of A with the emptyset is the emptyset
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="set1" name="intersect"/>
      <OMV name="A"/>
      <OMS cd="set1" name="emptyset"/>
    </OMA>
    <OMS cd="set1" name="emptyset"/>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="set1">intersect</csymbol><ci>A</ci><csymbol cd="set1">emptyset</csymbol></apply>
  <csymbol cd="set1">emptyset</csymbol>
 </apply>
</math>

set1.intersect($A, set1.emptyset) = set1.emptyset

Commented Mathematical property (CMP):
The union of A with the emptyset 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="set1" name="union"/>
      <OMV name="A"/>
      <OMS cd="set1" 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="set1">union</csymbol><ci>A</ci><csymbol cd="set1">emptyset</csymbol></apply>
  <ci>A</ci>
 </apply>
</math>

set1.union($A, set1.emptyset) = $A

Commented Mathematical property (CMP):
the size of the empty set is zero
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="set1" name="size"/>
      <OMS cd="set1" name="emptyset"/>
    </OMA>
    <OMS cd="alg1" name="zero"/>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="set1">size</csymbol><csymbol cd="set1">emptyset</csymbol></apply>
  <csymbol cd="alg1">zero</csymbol>
 </apply>
</math>

set1.size(set1.emptyset) = alg1.zero

Signatures:
sts
map
Role:
application
Description:

This symbol represents a mapping function which may be used to construct sets, it takes as arguments a function from X to Y and a set over X in that order. The value that is returned is a set of values in Y. The argument list may be a set or an integer_interval.

Example:
The set of even values between 0 and 20, that is the values 2x, where x ranges over the integral interval [0,10]. 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="map"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMS cd="arith1" name="times"/>
        <OMI> 2 </OMI>
	<OMV name="x"/>
      </OMA>
    </OMBIND>
    <OMA>
      <OMS cd="interval1" name="integer_interval"/>
      <OMI> 0 </OMI>
      <OMI> 10 </OMI>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">map</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="arith1">times</csymbol><cn type="integer">2</cn><ci>x</ci></apply>
  </bind>
  <apply><csymbol cd="interval1">integer_interval</csymbol>
   <cn type="integer">0</cn>
   <cn type="integer">10</cn>
  </apply>
 </apply>
</math>

set1.map(fns1.lambda[$x -> 2 * $x], interval1.integer_interval(0, 10))

{ 2 ⁢ x | x ∈ [ 0 , 10 ] }

Signatures:
sts
size
Role:
application
Description:

This symbol is used to denote the number of elements in a set. 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 set{3,6,9} = 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="set1" name="size"/>
      <OMA>
        <OMS cd="set1" name="set"/>
        <OMI> 3 </OMI>
        <OMI> 6 </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="set1">size</csymbol>
   <apply><csymbol cd="set1">set</csymbol>
    <cn type="integer">3</cn>
    <cn type="integer">6</cn>
    <cn type="integer">9</cn>
   </apply>
  </apply>
  <cn type="integer">3</cn>
 </apply>
</math>

set1.size({3 , 6 , 9}) = 3

size ⁡ ( { 3 , 6 , 9 } ) = 3

Example:
The size of the set of integers is infinite 
<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="set1" name="size"/>
      <OMS cd="setname1" name="Z"/>
    </OMA>
    <OMS cd="nums1" name="infinity"/>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="set1">size</csymbol><csymbol cd="setname1">Z</csymbol></apply>
  <csymbol cd="nums1">infinity</csymbol>
 </apply>
</math>

set1.size(setname1.Z) = nums1.infinity

Signatures:
sts
suchthat
Role:
application
Description:

This symbol represents the suchthat function which may be used to construct sets, it takes two arguments. The first argument should be the set which contains the elements of the set we wish to represent, the second argument should be a predicate, that is a function from the set to the booleans which describes if an element is to be in the set returned.

Example:
This example shows how to construct the set of even integers, using the suchthat constructor. 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="suchthat"/>
    <OMS cd="setname1" name="Z"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="x"/>
      </OMBVAR>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMA>
	  <OMS cd="arith1" name="divide"/>
	  <OMV name="x"/>
	  <OMI> 2 </OMI>
	</OMA>
	<OMS cd="setname1" name="Z"/>
      </OMA>
    </OMBIND>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">suchthat</csymbol>
  <csymbol cd="setname1">Z</csymbol>
  <bind><csymbol cd="fns1">lambda</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="set1">in</csymbol>
    <apply><csymbol cd="arith1">divide</csymbol><ci>x</ci><cn type="integer">2</cn></apply>
    <csymbol cd="setname1">Z</csymbol>
   </apply>
  </bind>
 </apply>
</math>

set1.suchthat(setname1.Z, fns1.lambda[$x -> set1.in($x / 2, setname1.Z)])

Signatures:
sts
set
Role:
application
Description:

This symbol represents the set construct. It is an n-ary function. The set entries are given explicitly. There is no implied ordering to the elements of a set.

Example:
The set {3, 6, 9} 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="set"/>
    <OMI> 3 </OMI>
    <OMI> 6 </OMI>
    <OMI> 9 </OMI>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">set</csymbol>
  <cn type="integer">3</cn>
  <cn type="integer">6</cn>
  <cn type="integer">9</cn>
 </apply>
</math>
Signatures:
sts
intersect
Role:
application
Description:

This symbol is used to denote the n-ary intersection of sets. It takes sets as arguments, and denotes the set that contains all the elements that occur in all of them.

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="set1" name="subset"/>
      <OMA>
        <OMS cd="set1" name="intersect"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
      <OMV name="A"/>
    </OMA>
    <OMA>
      <OMS cd="set1" name="subset"/>
      <OMA>
        <OMS cd="set1" 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="set1">subset</csymbol>
   <apply><csymbol cd="set1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
   <ci>A</ci>
  </apply>
  <apply><csymbol cd="set1">subset</csymbol>
   <apply><csymbol cd="set1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
   <ci>B</ci>
  </apply>
 </apply>
</math>

set1.subset(set1.intersect($A, $B), $A) and set1.subset(set1.intersect($A, $B), $B)

Signatures:
sts
union
Role:
application
Description:

This symbol is used to denote the n-ary union of sets. It takes sets as arguments, and denotes the set that contains all the elements that occur in any of them.

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="set1" name="subset"/>
      <OMV name="A"/>
      <OMA>
        <OMS cd="set1" name="union"/>
	<OMV name="A"/>
	<OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="set1" name="subset"/>
      <OMV name="B"/>
      <OMA>
        <OMS cd="set1" 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="set1">subset</csymbol>
   <ci>A</ci>
   <apply><csymbol cd="set1">union</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="set1">subset</csymbol>
   <ci>B</ci>
   <apply><csymbol cd="set1">union</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

set1.subset($A, set1.union($A, $B)) and set1.subset($B, set1.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="set1" name="union"/>
        <OMV name="A"/>
        <OMA>
          <OMS cd="set1" name="intersect"/>
          <OMV name="B"/>
          <OMV name="C"/>
        </OMA>
      </OMA>
      <OMA>
        <OMS cd="set1" name="intersect"/>
        <OMA>
          <OMS cd="set1" name="union"/>
          <OMV name="A"/>
          <OMV name="B"/>
        </OMA>
        <OMA>
          <OMS cd="set1" 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="set1">union</csymbol>
    <ci>A</ci>
    <apply><csymbol cd="set1">intersect</csymbol><ci>B</ci><ci>C</ci></apply>
   </apply>
   <apply><csymbol cd="set1">intersect</csymbol>
    <apply><csymbol cd="set1">union</csymbol><ci>A</ci><ci>B</ci></apply>
    <apply><csymbol cd="set1">union</csymbol><ci>A</ci><ci>C</ci></apply>
   </apply>
  </apply>
 </bind>
</math>

quant1.forall[$A, $B, $C -> set1.union($A, set1.intersect($B, $C)) = set1.intersect(set1.union($A, $B), set1.union($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 set difference of two sets. It takes two sets as arguments, and denotes the set that contains all the elements that occur in the first set, but not in the second.

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="set1" name="subset"/>
    <OMA>
      <OMS cd="set1" 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="set1">subset</csymbol>
  <apply><csymbol cd="set1">setdiff</csymbol><ci>A</ci><ci>B</ci></apply>
  <ci>A</ci>
 </apply>
</math>

set1.subset(set1.setdiff($A, $B), $A)

Signatures:
sts
subset
Role:
application
Description:

This symbol has two (set) arguments. It is used to denote that the first set is a subset of 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="set1" name="subset"/>
        <OMV name="B"/>
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="subset"/>
        <OMV name="C"/>
        <OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="set1" 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="set1">subset</csymbol><ci>B</ci><ci>A</ci></apply>
   <apply><csymbol cd="set1">subset</csymbol><ci>C</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="set1">subset</csymbol><ci>C</ci><ci>A</ci></apply>
 </apply>
</math>

set1.subset($B, $A) and set1.subset($C, $B) ==> set1.subset($C, $A)

Signatures:
sts
in
Role:
application
Description:

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

Commented Mathematical property (CMP):
if a is in A and a is in B then a is in A intersect 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="set1" name="in"/>
        <OMV name="a"/>
        <OMV name="A"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="in"/>
        <OMV name="a"/>
        <OMV name="B"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="set1" name="in"/>
      <OMV name="a"/>
      <OMA>
        <OMS cd="set1" 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="set1">in</csymbol><ci>a</ci><ci>A</ci></apply>
   <apply><csymbol cd="set1">in</csymbol><ci>a</ci><ci>B</ci></apply>
  </apply>
  <apply><csymbol cd="set1">in</csymbol>
   <ci>a</ci>
   <apply><csymbol cd="set1">intersect</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

set1.in($a, $A) and set1.in($a, $B) ==> set1.in($a, set1.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 set. It is used to denote that the element is not in the given set.

Commented Mathematical property (CMP):
if a is a member of a then it is not true that a is not a member 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="set1" name="in"/>
    <OMV name="a"/>
    <OMV name="A"/>
  </OMA>
  <OMA>
    <OMS cd="logic1" name="not"/>
    <OMA>
      <OMS cd="set1" name="notin"/>
      <OMV name="a"/>
      <OMV name="A"/>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">in</csymbol><ci>a</ci><ci>A</ci></apply>
  <apply><csymbol cd="logic1">not</csymbol>
   <apply><csymbol cd="set1">notin</csymbol><ci>a</ci><ci>A</ci></apply>
  </apply>
 </apply>
</math>

set1.in($a, $A) ==> not(set1.notin($a, $A))

Example:
4 is not in {1,2,3} 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="notin"/>
    <OMI> 4 </OMI>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">notin</csymbol>
  <cn type="integer">4</cn>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

set1.notin(4, {1 , 2 , 3})

Signatures:
sts
prsubset
Role:
application
Description:

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

Commented Mathematical property (CMP):
A is a proper subset of B implies that A is a subset of B and A not= 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="set1" name="prsubset"/>
    <OMV name="A"/>
    <OMV name="B"/>
  </OMA>
  <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA>
      <OMS cd="set1" name="subset"/>
      <OMV name="A"/>
      <OMV name="B"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="neq"/>
      <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="set1">prsubset</csymbol><ci>A</ci><ci>B</ci></apply>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="set1">subset</csymbol><ci>A</ci><ci>B</ci></apply>
   <apply><csymbol cd="relation1">neq</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

set1.prsubset($A, $B) ==> set1.subset($A, $B) and $A != $B

Example:
{2,3} is a proper subset of {1,2,3} 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="prsubset"/>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">prsubset</csymbol>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

set1.prsubset({2 , 3}, {1 , 2 , 3})

{ 2 , 3 } ⊂ { 1 , 2 , 3 }

Signatures:
sts
notsubset
Role:
application
Description:

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

Commented Mathematical property (CMP):
if A is not a subset of B then there exists an x in B s.t. x is not a member 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="implies"/>
  <OMA>
    <OMS cd="set1" name="notsubset"/>
    <OMV name="A"/>
    <OMV name="B"/>
  </OMA>
  <OMBIND>
    <OMS cd="quant1" name="exists"/>
    <OMBVAR>
      <OMV name="x"/>
    </OMBVAR>
    <OMA>
      <OMS cd="logic1" name="and"/>
      <OMA>
        <OMS cd="set1" name="in"/>
	<OMV name="x"/>
	<OMV name="B"/>
      </OMA>
      <OMA>
        <OMS cd="set1" name="notin"/>
	<OMV name="x"/>
	<OMV name="A"/>
      </OMA>
    </OMA>
  </OMBIND>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="set1">notsubset</csymbol><ci>A</ci><ci>B</ci></apply>
  <bind><csymbol cd="quant1">exists</csymbol>
   <bvar><ci>x</ci></bvar>
   <apply><csymbol cd="logic1">and</csymbol>
    <apply><csymbol cd="set1">in</csymbol><ci>x</ci><ci>B</ci></apply>
    <apply><csymbol cd="set1">notin</csymbol><ci>x</ci><ci>A</ci></apply>
   </apply>
  </bind>
 </apply>
</math>

set1.notsubset($A, $B) ==> quant1.exists[$x -> set1.in($x, $B) and set1.notin($x, $A)]

A ⊄ B ⇒ ∃   x . x ∈ B ∧ x ∉ A

Example:
{2,3,4} 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="set1" name="notsubset"/>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
      <OMI> 4 </OMI>
    </OMA>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">notsubset</csymbol>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
   <cn type="integer">4</cn>
  </apply>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

set1.notsubset({2 , 3 , 4}, {1 , 2 , 3})

{ 2 , 3 , 4 } ⊄ { 1 , 2 , 3 }

Signatures:
sts
notprsubset
Role:
application
Description:

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

Commented Mathematical property (CMP):
A is not a proper subset of B implies that it is not true that A is a proper 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="implies"/>
  <OMA>
    <OMS cd="set1" name="notprsubset"/>
    <OMV name="A"/>
    <OMV name="B"/>
  </OMA>
  <OMA>
    <OMS cd="logic1" name="not"/>
    <OMA>
      <OMS cd="set1" name="prsubset"/>
      <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="set1">notprsubset</csymbol><ci>A</ci><ci>B</ci></apply>
  <apply><csymbol cd="logic1">not</csymbol>
   <apply><csymbol cd="set1">prsubset</csymbol><ci>A</ci><ci>B</ci></apply>
  </apply>
 </apply>
</math>

set1.notprsubset($A, $B) ==> not(set1.prsubset($A, $B))

Example:
{1,2,3} is not a proper subset of {1,2,3} 
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="set1" name="notprsubset"/>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
    <OMA>
      <OMS cd="set1" name="set"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">notprsubset</csymbol>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
  <apply><csymbol cd="set1">set</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
   <cn type="integer">3</cn>
  </apply>
 </apply>
</math>

set1.notprsubset({1 , 2 , 3}, {1 , 2 , 3})

{ 1 , 2 , 3 } ⊄ { 1 , 2 , 3 }

Signatures:
sts

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