Indexed family

 ( Basic Concepts In Set Theory ) 

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 , indexed by the set of , 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 $I$ and image $X$ (that is, indexed families and mathematical functions are technically identical, just points of view are different). Often the elements of the set $X$ are referred to as making up the family. In this view, an indexed family is interpreted as a collection of indexed elements, instead of a function. The set $I$ is called the index set of the family, and $X$ is the indexed set.

Sequence are one type of families indexed by Natural number. In general, the index set $I$ is not restricted to be Countable set. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.

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Formal definition Let $I$ and $X$ be sets and $f$ a function such that $$\backslash begin\{align\}$$

f ~:~ &I \to X \\
      &i \mapsto x_i = f(i),
     

\end{align} where $i$ is an element of $I$ and the image $f(i)$ of $i$ under the function $f$ is denoted by $x\_i$. For example, $f(3)$ is denoted by $x\_3.$ The symbol $x\_i$ is used to indicate that $x\_i$ is the element of $X$ indexed by $i\; \backslash in\; I.$ The function $f$ thus establishes a family of elements in $X$ indexed by $I,$ which is denoted by $\backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\},$ or simply $\backslash left(x\_i\backslash right)$ if the index set is assumed to be known. Sometimes angle brackets or Bracket 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 $f$ with a domain $I$ induces a family $(f(i))\_\{i\; \backslash in\; I\}$ and conversely. (The terms "mapping" for functions and "indexing" for indexed families are equivalent.) 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 $X$ gives rise to a family $\backslash left(x\_t\backslash right)\_\{t\; \backslash in\; X\},$ where $X$ is indexed by itself (meaning that $f$ 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 $\backslash left(x\_i\backslash right)\_\{i\; \backslash in\; I\}$ defines a set $\backslash mathcal\{X\}\; =\; \backslash \{x\_i\; :\; i\; \backslash in\; I\backslash \},$ that is, the image of $I$ under $f.$ Since the mapping $f$ is not required to be injective, there may exist $i,\; j\; \backslash in\; I$ with $i\; \backslash neq\; j$ such that $x\_i\; =\; x\_j.$ Thus, $|\; \backslash mathcal\{X\}|\; \backslash leq\; |I|$, where $|A|$ denotes the cardinality of the set $A.$ For example, the sequence $\backslash left(\; (-1)^i\; \backslash right)\_\{i\backslash in\; \backslash N\}$ indexed by the natural numbers $\backslash N\; =\; \backslash \{1,\; 2,\; 3,\; \backslash ldots\backslash \}$ has image set $\backslash left\backslash \{(-1)^i\; :\; i\; \backslash in\; \backslash N\backslash right\backslash \}\; =\; \backslash \{-1,1\backslash \}.$ In addition, the set $\backslash \{\; x\_i\; :\; i\; \backslash in\; I\; \backslash \}$ does not carry information about any structures on $I.$ 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.

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Indexed subfamily An indexed family $\backslash left(B\_i\backslash right)\_\{i\; \backslash in\; J\}$ is a subfamily of an indexed family $\backslash left(A\_i\backslash right)\_\{i\; \backslash in\; I\},$ if and only if $J$ is a subset of $I$ and $B\_i\; =\; A\_i$ holds for all $i\; \backslash in\; J.$

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Examples

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Indexed vectors For example, consider the following sentence:

Here $\backslash left(v\_i\backslash right)\_\{i\; \backslash in\; \backslash \{1,\; \backslash ldots,\; n\backslash \}\}$ denotes a family of vectors. The $i$-th vector $v\_i$ only makes sense with respect to this family, as sets are unordered so there is no $i$-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 $n\; =\; 2$ and $v\_1\; =\; v\_2\; =\; (1,\; 0)$ 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).

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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 $(1,\; 1)$ 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 hand, the family of the rows contains two elements indexed differently such as the 1st row $(1,\; 1)$ and the 2nd row $(1,\; 1)$ 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.)

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Other examples Let $\backslash mathbf\{n\}$ be the finite set $\backslash \{1,\; 2,\; \backslash ldots\; n\backslash \},$ where $n$ is a positive integer.

• An ordered pair (2-tuple) is a family indexed by the set of two elements, $\backslash mathbf\{2\}\; =\; \backslash \{1,\; 2\backslash \};$ each element of the ordered pair is indexed by an element of the set $\backslash mathbf\{2\}.$
• An tuple is a family indexed by the set $\backslash mathbf\{n\}.$
• An infinite sequence is a family indexed by the natural numbers.
• A Tuple is an $n$-tuple for an unspecified $n,$ or an infinite sequence.
• An $n\; \backslash times\; m$ matrix is a family indexed by the Cartesian product $\backslash mathbf\{n\}\; \backslash times\; \backslash mathbf\{m\}$ which elements are ordered pairs; for example, $(2,\; 5)$ indexing the matrix element at the 2nd row and the 5th column.
• A net is a family indexed by a directed set.

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Operations on indexed families Index sets are often used in sums and other similar operations. For example, if $\backslash left(a\_i\backslash right)\_\{i\; \backslash in\; I\}$ is an indexed family of numbers, the sum of all those numbers is denoted by $$\backslash sum\_\{i\; \backslash in\; I\}\; a\_i.$$

When $\backslash left(A\_i\backslash right)\_\{i\; \backslash in\; I\}$ is a family of sets, the union of all those sets is denoted by $$\backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i.$$

Likewise for intersections and Cartesian products.

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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 theory , indexed by another category , and related by depending on two indices.

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See also

• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 vols., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993. Cited as EDM (volume).

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