In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. Suppose X is an arbitrary set and for each i in I let A_i be an element of X. The subset Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

Y = { A_i ∈ X : i ∈ I }

of X is said to be an indexed set which is indexed by I. In mathematical literature, one often encounters statements like “let {A_i} be a family of objects indexed by I”. Here it is to be understood that the index i runs over the entire set I.

Formally, one may regard A as a function from I to X with A(i) = A_i. From this point of view an index set is nothing more than the domain of a function and an indexed set is nothing more than the range of that function. The change of notation and terminology here is used to shift the emphasis away from the function A itself to the elements A_i of the range. The function A is merely serving as a way a labeling the objects one is actually interested in. In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, the domain of a function is the set of all input values to the function. ... In mathematics, the range of a function is the set of all values produced by a function. ...

An indexed set (together with its indexing function) is sometimes called a family. Note that a family is more than just a set of elements; the index is important. Two different functions from I to X with the same range define different families even though the underlying set of elements is the same. Also in contrast to a set of elements, a family can contain an element more than once. That is, one may have A_i = A_j for i ≠ j. This happens precisely when the underlying function A is not injective. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...

Examples

• An ordered pair is a family indexed by the two element set {1, 2}.
• More generally, an n-tuple is a family indexed by the finite set {1, 2, …, n}
• A sequence is a family indexed by the natural numbers.
• Every set can be considered as a family indexed by itself and the identity function.

An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ... See also tuple (music) as in duple and triple. ... This is a page about mathematics. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

Operations on families

Index sets are often used in sums and other similiar operations. For example, if {a_i} is an family of numbers indexed by I, the addition of all those numbers is denoted

When the {A_i} is a family of sets one denotes the union of all the A_i by In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...

Likewise for intersections and cartesian products. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

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. In category theory, a functor is a special type of mapping between categories. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

See also

• coproduct
• disjoint union
• tagged union
• index notation
• array
• net