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?


1 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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .