1

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?

J.-E. Pin
  • 40,163
Gqqnbig
  • 445
  • What about "dimension" ? – Peter Oct 13 '20 at 07:25
  • 1
    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 $1\le i\le k$". – Mark S. Oct 13 '20 at 10:36

2 Answers2

2

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.

J.-E. Pin
  • 40,163
K.defaoite
  • 12,536
1

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 $|u|$.

J.-E. Pin
  • 40,163