Tuples in a Set: the Set of n-Tuples and Their Components

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let $X$ be a set, let $n\in\mathbb{N}_{0}$, and let $[n]$ be as in Intervals of Natural Numbers §segment.

¶ $X^{n}$ denotes the set $X^{[n]}$ of maps from $[n]$ to $X$; an element of $X^{n}$ is an $n$-tuple in $X$.

¶ For $x\in X^{n}$ and $k\in[n]$, the $k$-th component of $x$ is $x_{k}=x(k)$, and $x$ is written $(x_{1},\dots,x_{n})$; for $n=2$ this denotes the map $x$, not the ordered pair of $x_{1}$ and $x_{2}$.