Family

set-theory

Definition

Family

A family of elements of a set $X$ indexed by an index set $I$ is a function

$$f:I\to X.$$

Writing $x_i=f(i)$, the family is denoted

$$(x_i{)}_{i\in I}.$$

The index set $I$ labels the elements; the codomain $X$ says where they live. The function viewpoint lets the same element appear for several indices, so a family need not be a set: it carries indexing and repetition that a set does not.