set1 http://www.openmath.org/cd/set1.ocd 2003-04-01 2001-03-12 2 0 official arith1 fns1 quant1 logic1 alg1 interval1 nums1 setname1 relation1 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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