"Sequence-builder" notation

I am somewhat familiar with set-builder notation, but I'm wonder what the equivalent notational conventions for sequences are. I know that one can denote a sequence as the following:

$$\{a_n\}_x^y \ \ \text{and} \ \ (a_n)_x^y$$

But what if you want to state something about how that sequence works as well? Consider the sequence of all even numbers from $0$ to $24$. If it was a set, I believe one would write:

$$\{x \in \Bbb N : x = 2n, \ n \in \Bbb N_0^{12}\}$$

Or equivalently:

$$\{x \in \Bbb N : n \in \Bbb N, \ 0 \le x = 2n \le 24 \}$$

How would one write this as a sequence? I'll be dealing with a lot more complex sequences (and sets) than this, so I want to be as fluent in the notation as possible. Are there any free and up-to-date resources that discuss the matter in detail?

Solution 1:

$ \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} $ A sequence can be seen as a function whose domain is a subset of natural numbers (or integers). This is also how an $n$-tuple is treated.

$$ (x_1, \cdots, x_n) = (x_i)_{i=1}^{n} $$

Consider the function $f:\{0, 1,\cdots, 12\} \to \Z$ defined by $\forall n \in \{0,1,\dots,12\}: f(n) = 2n$. Then, the sequence can be denoted by $(f_i)_{i=0}^{12}$, or simply $(2i)_{i=0}^{12}$. Therefore, it is natural to define a function and use it as a sequence by denoting $f(i) = f_i$.

Comment: I believe that your set-builder notation is not standard because the elements are not clear; $x$ or $n$? I would write $$ \{x \in \Z ~|~ \exists n \in \{0,1,\cdots,12\}: x = 2n\} $$ or simply $$ \{2(n-1) ~|~n\in \N \land n \le 13\} $$

Some authors distinguish $(x_i)$ from $\{x_i\}$; the latter denotes the image (set) of the sequence $(x_i)$, which is exactly what I wrote above.