Indexed family of sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets. This is technically true even when you write $\prod\mathscr{A}$ for some family $\mathscr{A}$ of sets: the implied index set is $\mathscr{A}$ itself, and a point in the product is a function $\endgroup$ – Brian M. Scott Sep 7 '13 at 22:04 Indexed Families of Sets The index we use for the subsets does not have to be an integer. It can be an element of any set. In this case, instead of calling it a sequence of sets, we call it an indexed family of sets. We give a precise de–nition. De–nition Let A and X be non-empty sets. An indexed family of subsets of X with This is technically true even when you write $\prod\mathscr{A}$ for some family $\mathscr{A}$ of sets: the implied index set is $\mathscr{A}$ itself, and a point in the product is a function $\endgroup$ – Brian M. Scott Sep 7 '13 at 22:04 The arguments in all answers below work for arbitrary index sets. $\endgroup$ – Hagen von Eitzen Oct 18 '12 at 6:25 $\begingroup$ @Hagen von Eitzen: Except when the family is empty in which case a separate argument is needed. $\endgroup$ – Shahab Aug 9 '17 at 11:40 3.7 Indexed families of sets 1. Introduction to set theory and to methodology and philosophy of mathematics and computer programming Indexed families of sets An overview by Jan Plaza c 2017 Jan Plaza Use under the Creative Commons Attribution 4.0 International License Version of March 10, 2017 2. I have to prove that the union for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-infinity,1). And that the intersection for n an element of the natural numbers of the indexed set D=(-n,1/n) is equal to (-1,0]. I've asked the professor twice now for help and he hasn't been able to explain it at all.
5 Jun 2019 Let Λ be a nonempty set and suppose that for each α∈∧, there is a corresponding set Aα. The family of sets {Aα | α
4 Mar 2017 21. A doubly indexed family of sets over S and T is any function (association), s.t. with any ordered pair s, t ∈ S × T we I don't like either notation: I would write {Ai:i∈I} or ⟨Ai:i∈I⟩. Technically there is no difference: each implies the existence of a function i↦Ai whose domain is I. Indexing. Given a family of sets F, it is often convenient to associate to each set in the family a "label" called an index, which need not be related in any way to the Much like the definitions of "countable", "sequence" and "natural numbers" so does the definition of "family" can be changed from one context to another.
F is sometimes called a family of elements of A indexed by I. (or with I as index class) and we write (F(i))iEI instead of F. In particular, if E is a set then a family of
F is sometimes called a family of elements of A indexed by I. (or with I as index class) and we write (F(i))iEI instead of F. In particular, if E is a set then a family of is called the term of index i of the family. A family with index set N is called a sequence. The union and the intersection of a family of sets {A_i}_(i in I) are denoted It is a more complex object, defined by a set of indexes , and a set for each index . Technically, one can define it as the graph of a function of domain . (I say a 10 Jan 2010 Indexed Families of Sets. ▻ What about unions and intersections of infinite collections of sets? We would also like to define the infinite union Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set. Families should be It is quite common for a family of sets to be indexed by simply being num- bered. Whenever our index set is the natural numbers (or a subset thereof) there is a
also generalize the definition to indexed families of sets where the index set is We use a notation similar to sequences that is we denote the indexed family.
PROOF INVOLVING SETS and INDEXED FAMILIES OF SETS By: Ahmad Wachidul Kohar, Sri Rejeki (IMPoME 2012) 2.1 Proof Involving Sets 2.1.1 Terms involving Sets 1. Definition 1 Suppose A and B are sets. means “if , then ”, that is 2. Definition 2 If are sets, we say that and , that is A = B ⇔ ( A ⊆ B ) ∧ ( B ⊆ A) In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets. This is technically true even when you write $\prod\mathscr{A}$ for some family $\mathscr{A}$ of sets: the implied index set is $\mathscr{A}$ itself, and a point in the product is a function $\endgroup$ – Brian M. Scott Sep 7 '13 at 22:04 Indexed Families of Sets The index we use for the subsets does not have to be an integer. It can be an element of any set. In this case, instead of calling it a sequence of sets, we call it an indexed family of sets. We give a precise de–nition. De–nition Let A and X be non-empty sets. An indexed family of subsets of X with This is technically true even when you write $\prod\mathscr{A}$ for some family $\mathscr{A}$ of sets: the implied index set is $\mathscr{A}$ itself, and a point in the product is a function $\endgroup$ – Brian M. Scott Sep 7 '13 at 22:04
First the matter of indexing and cardinality: the Morgan's law is sometimes formulated for a finite number of sets, but this is in fact inessential, as we shall see
is called the term of index i of the family. A family with index set N is called a sequence. The union and the intersection of a family of sets {A_i}_(i in I) are denoted It is a more complex object, defined by a set of indexes , and a set for each index . Technically, one can define it as the graph of a function of domain . (I say a 10 Jan 2010 Indexed Families of Sets. ▻ What about unions and intersections of infinite collections of sets? We would also like to define the infinite union Whenever available, set and combinatorial class operations (counting, iteration, listing) on the family are induced from those of the index set. Families should be It is quite common for a family of sets to be indexed by simply being num- bered. Whenever our index set is the natural numbers (or a subset thereof) there is a This paper is an adjunct to [2]. In [2, § 2], we remarked that the index set G(F) of a recursively enumerable family 57 of classes of r.e. sets can be En 0 complete What I wish to show now is that every indexed-set corresponds to a family, in such a way that it is natural simply to define indexed-sets as special sorts of families.
First the matter of indexing and cardinality: the Morgan's law is sometimes formulated for a finite number of sets, but this is in fact inessential, as we shall see set-indexed category set has, for any set I, ordinary I-indexed families of sets We want to define a relational database to be a J-indexed family of relations. Sal shows examples of intersection and union of sets and introduces some set notation.