Formally, an indexed family is the same thing as a image of the family's underlying function is an element of the family.
2.
Sometimes it is useful to consider all functions with certain parameters as " parametric family ", i . e . as an indexed family of functions.
3.
For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation, the indexed family can be treated simply as a collection of sets.
4.
Given an indexed family of objects A _ i, indexed with i \ in I, the direct sum may be written \ textstyle A = \ bigoplus _ { i \ in I } A _ i.
5.
An object is the product of a family of objects iff there exist morphisms, such that for every object and a-indexed family of morphisms there exists a unique morphism such that the following diagrams commute for all:
6.
An indexed family,, where is in some index-set, may define a multiset, sometimes written } }, in which the multiplicity of any element is the number of indices such that a _ i = x.
7.
Hence, an indexed family of sets is conceptually different from a family of sets ( which is just a synonym for " set of sets " ), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family.
8.
Hence, an indexed family of sets is conceptually different from a family of sets ( which is just a synonym for " set of sets " ), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family.
9.
It states that for every indexed family ( S _ i ) _ { i \ in I } of nonempty sets there exists an indexed family ( x _ i ) _ { i \ in I } of elements such that x _ i \ in S _ i for every i \ in I.
10.
It states that for every indexed family ( S _ i ) _ { i \ in I } of nonempty sets there exists an indexed family ( x _ i ) _ { i \ in I } of elements such that x _ i \ in S _ i for every i \ in I.
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