set3 http://www.openmath.org/cd http://www.openmath.org/cd/set3.ocd 2003-04-01 1999-05-10 1 1 experimental This CD defines more set functions. big_intersect 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. big_union 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. cartesian_power 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. powerset 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. The intersection over all subsets of a given set X is the empty set. k_subsets 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. map_with_target 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). One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 } 2 1 10 The definition of a product of subsets X and Y of a group G: map_with_condition 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. One may form a set in the following way. This gives the set {2^2, 4^2, ... , 10^2 } 2 1 10 2 map_with_target_and_condition 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. One may form a set in the following way. This gives the set {1^2, 2^2, ... , 10^2 } 2 1 10 2