# OpenMath Content Dictionary: set3

Canonical URL:
http://www.openmath.org/cd/set3.ocd
CD Base:
http://www.openmath.org/cd
CD File:
set3.ocd
CD as XML Encoded OpenMath:
set3.omcd
Defines:
big_intersect, big_union, cartesian_power, k_subsets, map_with_condition, map_with_target, map_with_target_and_condition, powerset
Date:
1999-05-10
Version:
1 (Revision 1)
Review Date:
2003-04-01
Status:
experimental

This CD defines more set functions.

## big_intersect

Description:

This symbol is a unary function whose argument should be a collection C of subsets of a given set. When applied to C, it represents the intersection over all members of C.

Signatures:
sts

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## big_union

Description:

This symbol is a unary function whose argument should be a collection C of subsets of a given set. When applied to C, it represents the union over all members of C.

Signatures:
sts

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## cartesian_power

Description:

This symbol is a binary function whose first argument should be a set A and whose second argument should be a natural number k. When applied to A and k, it represents the Cartesian product of k copies of A.

Signatures:
sts

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## powerset

Description:

This symbol represents unary function whose argument should be a set. When applied to a set X, it represents the collection of all subsets of X.

Commented Mathematical property (CMP):
The intersection over all subsets of a given set X is the empty set.
Formal Mathematical property (FMP):
${\bigcap }_{\mathcal{P}\left(X\right)}=\varnothing$
Signatures:
sts

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## k_subsets

Description:

This symbol represents a binary function whose first argument should be a set and whose second argument should be a natural number. When applied to a set X and a number k, it represents the collection of all subsets of X of size k.

Signatures:
sts

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## map_with_target

Description:

This symbol represents a function with three arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X on which the first argument f is defined. The third argument specifies the range Y of the function. The symbol is used to denote the image {f(x) in Y | x in X} of application of the function f on the elements of X (so as to form a subset of Y).

Example:
One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 }
$" >\left\{{x}^{2}\in \mathbb{Z}|x\in \left[1,10\right]\right\}$
Example:
The definition of a product of subsets X and Y of a group G:
$XY=" >\left\{xy\in G|xy\in X×Y\right\}$
Signatures:
sts

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## map_with_condition

Description:

This symbol represents a function with three arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X. The third argument specifies a Boolean function P on X defining the subset Z of X (so Z = {x in X| P(x)}) on which the first argument f is defined, The symbol is used to denote the image {f(x) | x in X and P(x)} of application of the function f on the elements of Z.

Example:
One may form a set in the following way. This gives the set {2^2, 4^2, ... , 10^2 }
$\left\{{x}^{2}|x\in \left[1,10\right]\wedge \left(2|x\right)\right\}$
Signatures:
sts

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## map_with_target_and_condition

Description:

This symbol represents a function with four arguments. The first argument is a function assignment f (in the form of a lambda binding), the second argument is a set X on which the first argument f is defined. The third argument specifies the range Y of the function. The fourth argument specifies a Boolean function P on X defining the subset Z of X (so Z = {x in X| P(x)}) on which the first argument f is defined, The symbol is used to denote the image {f(x) in Y | x in X and P(x)} of application of the function f on the elements of Z.

Example:
One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 }
$\left\{{x}^{2}\in \mathbb{Z}|x\in \left[1,10\right]\wedge \left(2|x\right)\right\}$
Signatures:
sts

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