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 one real number for each integer (possibly the same). More formally, an indexed family is a mathematical function together with its domain and image . 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 family indexed by the 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. Mathematical statement. Definition. Let and be sets and a function such that where formula_2 is an element of and the image formula_3 of formula_2 under the function is denoted by formula_5. For example, formula_6 is denoted by formula_7. The symbol formula_5 is used  to indicate that formula_5 is the element of indexed by formula_10. The function thus establishes a family of elements in indexed by , which is denoted by formula_11, 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 formula_12 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. A family contains any element exactly once if and only if the corresponding function is injective. An indexed family formula_11 defines a set formula_14, that is, the image of under . Since the mapping is not required to be injective, there may exist formula_15 with  such that . Thus, formula_16, where denotes the cardinality of the set . For example, the sequence formula_17 indexed by the natural numbers formula_18 has image set formula_19. In addition, the set formula_20 does not carry information about any structures on . Hence, by using a set instead of the family, some information might be lost. For example, an ordering on the index set of a family induces an ordering on the family, but no ordering on the corresponding image set. Examples. Indexed vectors. For example, consider the following sentence: Here denotes a family of vectors. The -th vector only makes sense with  respect to this family, as sets are unordered so there is no -th vector of a set. Furthermore, linear independence is defined as a property of a collection; it therefore is important if those vectors are linearly independent as a set or as a family. For example, if we consider and as the same vector, then the "set" of them consists of only one element (as a set is a collection of unordered distinct elements) and is linearly independent, but the family contains the same element twice (since indexed differently) and is linearly dependent (same vectors are linearly dependent). Matrices . Suppose a text states the following: As in the previous example, it is important that the rows of "A" are linearly independent as a family, not as a set. For example, consider the matrix The "set" of the rows consists of a single element as a set is made of unique elements so it is linearly independent, but the matrix is not invertible as the matrix determinant is 0. On the other hands, the "family" of the rows contains two elements indexed differently such as the 1st row and the 2nd row so it is linearly dependent. The statement is  therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows. (The statement is also correct when "the rows" is interpreted as referring to a multiset, in which the elements are also kept distinct but which lacks some of the structure of an indexed family.) Other examples. Let be the finite set , where is a positive integer. Operations on indexed families. Index sets are often used in sums and other similar operations. For example, if is an indexed family of numbers, the sum of all those numbers is denoted by When  is a family of sets, the union of all those sets is denoted by Likewise for intersections and Cartesian products. Indexed subfamily. An indexed family is a subfamily of an indexed family , if and only if is a subset of and holds for all in . Usage in category theory. The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category , indexed by another category , and related by morphisms depending on two indices.