What does X^k mean in mathematical logic

Question

I am reading this book https://www2.karlin.mff.cuni.cz/~krajicek/mendelson.pdf and it writes

If X is a set and k is a positive integer, we denote by $X^k$ the set of all ordered k-tuples $(b_1, ... , b_k)$ of elements $b_1, ... , b_k$ of $X$. In particular, $X^1$ is $X$.

I've looked online and I can't find any other source explain this.

Can someone explains how this works? For example, is $X^2$, the set of ordered pairs?

Also, since I can't find anything else about this, I am wondering if this is standard or whether this source is just making up some formatting rules?

Answer 1

$X^n$ denotes the $n$-th Cartesian power of the set $X$. Your guess is correct, $X^2$ is the set of ordered pairs $(x_1,x_2)$ with $x_1, x_2 \in X$. Similarly, $X^3$ is the set of ordered triples $(x_1,x_2,x_3)$ with $x_1, x_2, x_3 \in X$. Unless you choose to define pairs, triples and $k$-tuples in a very uncommon way, ordered pairs are 2-tuples and ordered triples are 3-tuples. Some examples that you might already be familiar with are $\mathbb{R}^2$,$\mathbb{R}^3$ and $\mathbb{R}^n$.

This notation, Cartesian powers and a $k$-tuple as a mathematical object are very universal in mathematics. Here is the Wikipedia entry about them.