Family

set-theory

Definition

Family

Let $I$ (index set) and $X$ be sets. A family of elements in $X$ indexed by $I$ is strictly defined as a function $f$:

$$f:I\to X$$

However, instead of using the standard function notation $f(i)$, we typically use subscript notation. We denote the image of an index $i\in I$ as $x_i$, where $x_i=f(i)$..

The family itself is denoted by:

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