Collection of objects, each associated with an element from some index set
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 image (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.
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 inindexed 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. A family contains any element exactly once if and only if the corresponding function is injective.
An indexed family defines a set that is, the image of under Since the mapping is not required to be injective, there may exist with such that Thus, , where denotes the cardinality of the set For example, the sequence indexed by the natural numbers has image set In addition, the set 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.
An indexed family is a subfamily of an indexed family if and only if is a subset of and holds for all
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).
Suppose a text states the following:
A square matrix is invertible, if and only if the rows of are linearly independent.
As in the previous example, it is important that the rows of 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.)
Let be the finite set where is a positive integer.
An ordered pair (2-tuple) is a family indexed by the set of two elements, each element of the ordered pair is indexed by each element of the set