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set1
http://www.openmath.org/cd
http://www.openmath.org/cd/set1.ocd
2006-03-30
2004-03-30
3
1
Author: OpenMath Consortium
SourceURL: https://github.com/OpenMath/CDs
official
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
application
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.
An example to show the representation of the Cartesian product of sets: AxBxC.
emptyset
constant
This symbol is used to represent the empty set, that is the set which
contains no members. It takes no parameters.
The intersection of A with the emptyset is the emptyset
The union of A with the emptyset is A
the size of the empty set is zero
map
application
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.
The set of even values between 0 and 20, that is the values 2x, where
x ranges over the integral interval [0,10].
2
0
10
size
application
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.
The size of the set{3,6,9} = 3
3
6
9
3
The size of the set of integers is infinite
suchthat
application
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.
This example shows how to construct the set of even integers, using
the suchthat constructor.
2
set
application
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.
The set {3, 6, 9}
3
6
9
intersect
application
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.
(A intersect B) is a subset of A and (A intersect B) is a subset of B
union
application
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.
A is a subset of (A union B) and B is a subset of (A union B)
for all sets A,B and C
union(A,intersect(B,C)) = intersect(union(A,B),union(A,C))
setdiff
application
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.
the difference of A and B is a subset of A
subset
application
This symbol has two (set) arguments. It is used to denote that the
first set is a subset of the second.
if B is a subset of A and C is a subset of B then C is a subset of A
in
application
This symbol has two arguments, an element and a set. It is used to
denote that the element is in the given set.
if a is in A and a is in B then a is in A intersect B
notin
application
This symbol has two arguments, an element and a set. It is used to
denote that the element is not in the given set.
if a is a member of a then it is not true that a is not a member of A
4 is not in {1,2,3}
4
1
2
3
prsubset
application
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.
A is a proper subset of B implies that A is a subset of B and A not= B
{2,3} is a proper subset of {1,2,3}
2
3
1
2
3
notsubset
application
This symbol has two (set) arguments. It is used to denote that the
first set is not a subset of the second.
if A is not a subset of B then
there exists an x in B s.t. x is not a member of B
{2,3,4} is not a subset of {1,2,3}
2
3
4
1
2
3
notprsubset
application
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.
A is not a proper subset of B implies
that it is not true that A is a proper subset of B
{1,2,3} is not a proper subset of {1,2,3}
1
2
3
1
2
3