# What does X^k mean in mathematical logic

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?

$$X^n$$ 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.