Notation for length of arbitrarily long sequences in a set

Question

Suppose I have the set of arbitrarily long sequences of dice rolls:

\begin{equation} \Omega = \bigcup_{n = 1}^\infty [6]^n = \{ (1),(2),(3),(4),(5),(6),(1,1),(1,2),\ldots \} \end{equation}

Given some sequence $\omega \in \Omega$, what's the notation to get the length of the sequence? Is it $|\omega|$?

Answer 1

I have seen each of the following notations used to represent the length of a finite sequence $\sigma$:

• $\vert\sigma\vert$

• $length(\sigma)$

• $lh(\sigma)$

Personally, I think all three are perfectly fine, although it's worth spending a sentence saying what your notation means.

Note that "$\vert\sigma\vert$" is actually perfectly correct for finite sequences: a sequence is a set of ordered pairs, so it's length is indeed its cardinality. That said, once we look at infinite sequences this breaks down - a sequence of ordertype $\mathbb{N}$ and a sequence of ordertype $\mathbb{N}+1$ have the same cardinality. Since I often deal with infinite sequences, I tend to prefer $length$ or $lh$.