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Concept
Indexed family
Summary
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 one real number for each integer (possibly the same) as indexing.
More formally, an indexed family is a mathematical function together with its domain and (that is, indexed families and mathematical functions are technically identical, just point of views are different). Often the elements of the set are referred to as making up the family. In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set is called the index set of the family, and is the indexed set.
Sequences are one type of families indexed by natural numbers. In general, the index set is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
Let and be sets and a function such that
where is an element of and the image of under the function is denoted by . For example, is denoted by The symbol is used to indicate that is the element of indexed by The function thus establishes a family of elements in indexed by which is denoted by or simply if the index set is assumed to be known. Sometimes angle brackets or braces are used instead of parentheses, although the use of braces risks confusing indexed families with sets.
Functions and indexed families are formally equivalent, since any function with a domain induces a family and conversely. Being an element of a family is equivalent to being in the range of the corresponding function. In practice, however, a family is viewed as a collection, rather than a function.
Any set gives rise to a family where is indexed by itself (meaning that is the identity function). However, families differ from sets in that the same object can appear multiple times with different indices in a family, whereas a set is a collection of distinct objects.
Official source
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