In mathematics, an index set is another name for a function domain. A collection indexed by I, often written Ai for i in I (can be said 'for irunning overI ') is in effect a functionA(i) into some codomain.
Index sets are often used in sums (sigma notation) and other such operations; and are common when the Ai are themselves sets rather than numbers, in indexed intersections and unions.
Families
A family is another description of an indexed collection, often used of a family of sets. In contrast to a set of elements, a family can contain an element more than once (that is, the underlying function need not be injective).
Examples
An n-tuple can be considered as a family over the finite index set {1, 2, ..., n}
More generally, a functor can be considered as giving rise to an indexed family of objects in a categoryD, indexed by another category C, and related by morphisms depending on two indices.
Family background characteristics have a considerable influence on minority participation and achievement in science and mathematics education.
Mathematics classes with a high proportion of minorities are less likely than those with a low proportion of minorities to have mathematics teachers with majors in the field.
The teachers in science and mathematics classes that have a high percentage of minority students are more likely to emphasize preparing students for standardized tests and are less likely than those having fewer minority students to emphasize preparing students for further study in science or mathematics.
Formally, a family is a triple (X, I, ι) of sets X and I and a surjective function ι: I → X.
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.
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.
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