Family (mathematics)

In mathematics, a family is a collection. It is a formal version of a lookup table. It consists of a set, called the index set, containing the keys, and a mapping from those keys onto the elements of the family. Each key points to exactly one element of the family and each element belongs to at least one key. As different keys may point to the same element, a family can, unlike a set, contain the same element several times. Furthermore any additional structure of the index set extends to the family. Hence, an ordered family is a family with an ordered index set.
Formally, a family is a triple (X, I, ι) of sets X and I and a surjective function ι: I → X.
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Notation
A family is usually denoted by (A_{i})_{i∈I}. In this case I is the index set, ι(i)=A_{i} the mapping and A_{i} the element belonging to the key i, which is sometimes also called the ith element of the family.
It is also common to use {A_{i}}_{i∈I}, with curly brackets instead of parentheses, for a family. But this can be misleading, as it is easily confused with {A_{i}  i∈I}, which is an unstructured' set.
Implicit usage
Often a family is not mentioned explicitly, but used implicitly. Sometimes the lack to mention the family can lead to misunderstandings or even subtle errors.
Examples
Index Notation
Whenever index notation is used the indexed objects form a family.
 The vectors v_{1}, …, v_{n} are linearly independent.
(v_{i})_{i ∈ {1, …, n}} is a family of vectors. The ith vector v_{i} only makes sense with respect to this family, as sets are unordered and there is no ith 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 v_{1} = v_{2} = (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.
It is not clear if the author claims the vectors are linear independent as a family or as set.
Matrices
 A matrix A is invertible, if and only if the rows of A are linearly independent.
As in the above example it is important whether the rows of A are linearly independent as a family or as a set.
If we consider the matrix
 <math> A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, <math>
the set of rows only consists of a single element (1, 1) and is linearly independent, but the matrix is not invertible. The family of rows contains two elements and is lineraly dependent.
The statement is therefore correct if it refers to the family of rows, but wrong if it refers to the set of rows.
Functions, sets and families
There is a onetoone correspondence between surjective functions and families, as any function f with domain I induces a family (f(i))_{i∈I}. But, unlike a function, a family is viewed as a collection and being an element of a family is equivalent with being in the range of the corresponding function. A family does contain any element exactly once, if and only if the corresponding function is injective.
Like a set, a family is a container and any set X gives rise to a family (x)_{x∈X}. Thus any set naturally becomes a family. For any family (A_{i})_{i∈I} there is the set of all elements {A_{i}  i∈I}, but this does not carry any information on multiple containment or the structure of I. Hence, by using a set instead of the family, some information might be lost.
Examples
 An ordered pair is a family indexed by the two element set {1, 2}.
 More generally, an ntuple is a family indexed by the finite set {1, 2, …, n}
 A sequence is a family indexed by the natural numbers.
 A nbym matrix is a family indexed by the cartesian product {1, 2, …, n} × {1, 2, …, m}.
 A net is a family indexed by a directed set.
Operations on families
Index sets are often used in sums and other similar operations. For example, if (a_{i})_{i∈I} is an family of numbers, the sum of all those numbers is denoted by
 <math>\sum_{i\in I}a_i<math>
When (A_{i})_{i∈I} is a family of sets, the union of all those sets is denoted by
 <math>\bigcup_{i\in I}A_i<math>
Likewise for intersections and cartesian products.
Usage in category theory
More generally, a functor can be considered as giving rise to an indexed family of objects in a category D, indexed by another category C, and related by morphisms depending on two indices.