Index Set

set-theory

Definition

Index Set

An index set is a set $I$ used to label elements of a family or collection $(x_i{)}_{i\in I}$. Formally, for a function $x:I\to X$, the set $I$ serves as the domain of indices.

Common examples:

• Finite: $I=\{1,2,\dots ,n\}$ for sequences $(x_1,x_2,\dots ,x_n)$
• Countable: $I=\mathbb{N}$ or $\mathbb{Z}$ for infinite sequences
• Uncountable: $I=\mathbb{R}$ for continuous families

The cardinality $|I|$ determines the size of the indexed collection.