1
$\begingroup$

Wikipedia says "An $n$-tuple is a sequence (or ordered list) of $n$ elements, where $n$ is a non-negative integer."

so I have a $n$-tuple $m$, what of $m$ is $n$?

If I write $f(m)=n$, which function should $f$ be? Is there already a mathematical notation available?

$\endgroup$
2
  • $\begingroup$ What about "dimension" ? $\endgroup$ – Peter Oct 13 '20 at 7:25
  • 1
    $\begingroup$ As kind of implied in K.defaoite's answer, you wouldn't normally have a tuple out of the blue without declaring a name for its length at the same time. Like, at worst maybe something like "let $x_i\in\mathbb R^{n_i}$ for $1\le i\le k$". $\endgroup$ – Mark S. Oct 13 '20 at 10:36
2
$\begingroup$

An $n$ tuple comes from the $n$ fold Cartesian product of a set with itself. That is to say given $x_1 \in S,...,x_n\in S$, this is equivalent to $$\underline{x}=(x_1,...,x_n)\in \underbrace{S \times {}\dotsm{} \times S}_{n \text{ times}}$$ Which is often denoted $$\underline{x}\in S^n$$ It is technically incorrect to talk about the dimension of $\underline{x}$, i.e the expression $\dim{\underline{x}}=n$ it is however correct to say $\dim(S^n)=n$ but this is redundant because the $n$ superscript already implies the dimension of the set.

$\endgroup$
1
$\begingroup$

Following the title of your question, I would call it the length of the tuple, in analogy with the length of a word of a free monoid, since a word is an ordered sequence of letters. As for notation, the length of a word $u$ is usually denoted by $|u|$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.