Indexed family

In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects for each integer one real number (possibly the same). In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers is a collection of real numbers, where a given function selects for each integer one real number (possibly the same). More formally, an indexed family is a mathematical function together with its domain I and image X. Often the elements of the set X are referred to as making up the family. In this view indexed families are interpreted as collections instead of as functions. The set I is called the index (set) of the family, and X is the indexed set. Definition. Let I and X be sets and x {displaystyle x} a surjective function, such that then this establishes a family of elements in X indexed by I , which is denoted by (xi)i∈I or simply (xi), when the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, the latter with the risk of mixing-up families with sets. An indexed family can be turned into a set by considering the set X = { x i : i ∈ I } {displaystyle {mathcal {X}}={x_{i}:iin I}} , that is, the image of I under x. Since the mapping x is not required to be injective, there may exist i , j ∈ I {displaystyle i,jin I} with i ≠ j {displaystyle i eq j} such that x i = x j {displaystyle x_{i}=x_{j}} . Thus, | X | ≤ | I | , {displaystyle |{mathcal {X}}|leq |I|,} where |A| denotes the cardinality of the set A. The index set is not restricted to be countable, and, of course, a subset of a powerset may be indexed, resulting in an indexed family of sets. For the important differences in sets and families see below. Whenever index notation is used the indexed objects form a family. For example, consider the following sentence. Here (vi)i ∈ {1, …, n} denotes a family of vectors. The i-th vector vi only makes sense with respect to this family, as sets are unordered and there is no i-th vector of a set. Furthermore, linear independence is only defined as the property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. If we consider n = 2 and v1 = v2 = (1, 0), the set of them consists of only one element and is linearly independent, but the family contains the same element twice and is linearly dependent.