Tuples in a Set

Let $X$ be a set, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, and let $[n]$ be the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$, that is, the set of natural numbers $k$ with $1\le k\le n$.

The set $X^{n}$ is the set of all maps from $[n]$ to $X$. An element $x\in X^{n}$ is called an \textbf{$n$-tuple} in $X$; for $k\in[n]$ the value of $x$ at $k$ is written $x_{k}$ and called the \textbf{$k$-th component} of $x$, and $x$ is also written $(x_{1},\dots,x_{n})$.