How to denote a set of sequences?

Question

How do you denote a set of sequences of a certain length, consisting of real numbers? For example, is this correct? $$S=\{A:A=\{a_k\}_{k=1}^{10}\wedge (\forall k\in\Bbb{N},k\le 10:a_k\in\Bbb{R})\}$$ Or: $$S=\{\{a_k\}_{k=1}^{10}:(\forall k\in\Bbb{N},k\le 10:a_k\in\Bbb{R})\}$$

Answer 1

There might be a number of different ways, which you choose is essentially a choice of style, and the appropriate choices will vary depending on context.

What is a sequence of real numbers of length $n $? It is essentially a tuple so it might be convenient to say

Let $a =(a_k)\in\mathbb{R}^n$ be a sequence of length $n $.

(Observe that $\mathbb{R}^n$ is precisely the set of all such sequences.)

If you would rather the indexing be more explicit/forced (as in your suggestions) I would opt for something like

$$S=\{ \{a_k\in\mathbb{R} :k\in\mathbb{N}\wedge k\le n\} \in \mathbb{R}^n\}.$$

But I do not think you gain much from this over the first option...

This all using the standard set building notation $$A=\{ x\in X:\varphi (x)\}, $$ where $X $ is some bounding set and $\varphi $ is a formula with free variable $x $ such that $x\in A\leftrightarrow (x\in X\wedge\varphi (x) )$.

Edit: another valid option would be this. Write $[n]:=\{0,1,\ldots, n-1\} $. Then the set of real sequences of length $n $ is the set $\mathbb{R}^{[n]} $. (The notation $A^B $ is interpreted as the set of functions $f\colon B\to A $.)