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# 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, in, intersect, notin, notprsubset, notsubset, prsubset, set, setdiff, size, subset, union, emptyset, map, suchthat
Date:
2004-03-30
Version:
3
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.
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.
society at http://www.openmath.org.
```

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.
$A×B×C$
Signatures:
sts

 [Next: emptyset] [Last: notprsubset] [Top]

## 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):
$A\cap \varnothing =\varnothing$
Commented Mathematical property (CMP):
The union of A with the emptyset is A
Formal Mathematical property (FMP):
$A\cup \varnothing =A$
Commented Mathematical property (CMP):
the size of the empty set is zero
Formal Mathematical property (FMP):
$\mathrm{size}\left(\varnothing \right)=0$
Signatures:
sts

 [Next: map] [Previous: cartesian_product] [Top]

## 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].
$\left\{2x|x\in \left[0,10\right]\right\}$
Signatures:
sts

 [Next: size] [Previous: emptyset] [Top]

## 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
$\mathrm{size}\left(\left\{3,6,9\right\}\right)=3$
Example:
The size of the set of integers is infinite
$\mathrm{size}\left(\mathbb{Z}\right)=\infty$
Signatures:
sts

 [Next: suchthat] [Previous: map] [Top]

## 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.
$\left\{x\in \mathbb{Z}|\frac{x}{2}\in \mathbb{Z}\right\}$
Signatures:
sts

 [Next: set] [Previous: size] [Top]

## 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}
$\left\{3,6,9\right\}$
Signatures:
sts

 [Next: intersect] [Previous: suchthat] [Top]

## 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):
$A\cap B\subset A\wedge A\cap B\subset B$
Signatures:
sts

 [Next: union] [Previous: set] [Top]

## 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):
$A\subset A\cup B\wedge B\subset A\cup B$
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):
$\forall A,B,C.A\cup B\cap C=A\cup B\cap A\cup C$
Signatures:
sts

 [Next: setdiff] [Previous: intersect] [Top]

## 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):
$A\setminus B\subset A$
Signatures:
sts

 [Next: subset] [Previous: union] [Top]

## 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):
$B\subset A\wedge C\subset B⇒C\subset A$
Signatures:
sts

 [Next: in] [Previous: setdiff] [Top]

## 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):
$a\in A\wedge a\in B⇒a\in A\cap B$
Signatures:
sts

 [Next: notin] [Previous: subset] [Top]

## 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):
$a\in A⇒¬\left(a\notin A\right)$
Example:
4 is not in {1,2,3}
$4\notin \left\{1,2,3\right\}$
Signatures:
sts

 [Next: prsubset] [Previous: in] [Top]

## 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):
$A\subset B⇒A\subset B\wedge A\ne B$
Example:
{2,3} is a proper subset of {1,2,3}
$\left\{2,3\right\}\subset \left\{1,2,3\right\}$
Signatures:
sts

 [Next: notsubset] [Previous: notin] [Top]

## 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):
$A\not\subset B⇒\exists x.x\in B\wedge x\notin A$
Example:
{2,3,4} is not a subset of {1,2,3}
$\left\{2,3,4\right\}\not\subset \left\{1,2,3\right\}$
Signatures:
sts

 [Next: notprsubset] [Previous: prsubset] [Top]

## 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):
$A\not\subset B⇒¬\left(A\subset B\right)$
Example:
{1,2,3} is not a proper subset of {1,2,3}
$\left\{1,2,3\right\}\not\subset \left\{1,2,3\right\}$
Signatures:
sts

 [First: cartesian_product] [Previous: notsubset] [Top]

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