linguistic:voice_change, information:highlighting
Formal definition

Let I {\displaystyle I} and X {\displaystyle X} be sets and f {\displaystyle f} a function such that

where i {\displaystyle i} is an element of I {\displaystyle I} and the image f (i) {\displaystyle f(i)} of i {\displaystyle i} under the function f {\displaystyle f} is denoted by x i {\displaystyle x_{i}}. For example, f (3) {\displaystyle f(3)} is denoted by x 3. {\displaystyle x_{3}.} The symbol x i {\displaystyle x_{i}} is used to indicate that x i {\displaystyle x_{i}} is the element of X {\displaystyle X} indexed by i ∈ I. {\displaystyle i\in I.} The function f {\displaystyle f} thus establishes a family of elements in X {\displaystyle X} indexed by I, {\displaystyle I,} which is denoted by (x i) i ∈ I, {\displaystyle \left(x_{i}\right)_{i\in I},} or simply (x i) {\displaystyle \left(x_{i}\right)} 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 f {\displaystyle f} with a domain I {\displaystyle I} induces a family (f (i)) i ∈ I {\displaystyle (f(i))_{i\in I}} 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 X {\displaystyle X} gives rise to a family (x x) x ∈ X, {\displaystyle \left(x_{x}\right)_{x\in X},} where X {\displaystyle X} is indexed by itself (meaning that f {\displaystyle 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 (x i) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} defines a set X = { x i: i ∈ I }, {\displaystyle {\mathcal {X}}=\{x_{i}:i\in I\},} that is, the image of I {\displaystyle I} under f. {\displaystyle f.} Since the mapping f {\displaystyle f} is not required to be injective, there may exist i, j ∈ I {\displaystyle i,j\in I} with i ≠ j {\displaystyle i\neq j} such that x i = x j. {\displaystyle x_{i}=x_{j}.} Thus, | X | ≤ | I | {\displaystyle |{\mathcal {X}}|\leq |I|}, where | A | {\displaystyle |A|} denotes the cardinality of the set A. {\displaystyle A.} For example, the sequence ((− 1) i) i ∈ N {\displaystyle \left((-1)^{i}\right)_{i\in \mathbb {N} }} indexed by the natural numbers N = { 1, 2, 3, … } {\displaystyle \mathbb {N} =\{1,2,3,\ldots \}} has image set { (− 1) i: i ∈ N } = { − 1, 1 }. {\displaystyle \left\{(-1)^{i}:i\in \mathbb {N} \right\}=\{-1,1\}.} In addition, the set { x i: i ∈ I } {\displaystyle \{x_{i}:i\in I\}} does not carry information about any structures on I. {\displaystyle 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.