I have short questions about notation. Since the questions are too short, I hope it is not against the rules to ask them in one question. I try to write these notations:
We have the set of functions (not a sequence) number of countable infinity $f_1(n), f_2(n), f_3(n) \cdots $
Can I write this notation like this?
$$f_{\mathbb{N}}(n)$$
2) Are these notations are equivalent?
For the function $f(n)$,
$f: \mathbb{N}\longrightarrow \left\{0,1,2\right\}$ and $\left\{f:f\in \left\{0,1,2\right\}, n\in\mathbb{N}\right\}$
3) For the function $f(n)$ ,where $n\in\mathbb{N}$ is this notation correct?
$$f(n)_{n\in\mathbb{N}}$$
4) Are there difference between these notations?
$\left\{m,n\right\}\in\mathbb{N}$ and $(m,n)\in\mathbb{N}$
And for the sequences $a_n$
$\left\{a_n\right\}_{n\in\mathbb{N}}$ and $(a_n)_{n\in\mathbb{N}}$
Thank you very much.
Notation is always a matter of context and of convention. That being said ...
No. $f_{\Bbb N}(n)$ looks mor like a single function $f_{\Bbb N}$ (so just happening to have the set $\Bbb N$ as an index), evaluated at $n$. This differs from $\{f_1(n),f_2(n),f_3(n),\ldots\}$ which suggests (though I am no fan of "$\ldots$" occuring anywhere) an infinite set (namely for each function $f_i$, $i\in\Bbb N$, its value at $n$). I suppose what you really mean is the set of these functions (and in particular, not the set of all function values at some fixed $n$), which can be written $$\{\,f_n\mid n\in\Bbb N\,\}.$$
No. The fist expresses that $f$ is a function with domain $\Bbb N$ and range $\{0,1,2,\}$. The second is, after some simplications, just the set $\{0,1,2\}$.
No. This can be interpreted in a few ways, but certainly not what you intend. If you want to express that $f$ is a function with domain $\Bbb N$, you could write "$f\colon\Bbb N\to X$ for suitable $X$" or "$\operatorname{dom}(f)=\Bbb N$" or simply "$f$ is a function with domain $\Bbb N$".
Yes. $\{m,n\}$ denotes the set that has $m$ and $n$ as elements and nothing else, so it is a set two (or perhaps even just one!) elements. In particular $\{m,n\}=\{n,m\}$. Whereas $(m,n)$ denotes the ordered pair with first component $m$ and second component $n$, so $(m,n)\ne (n,m)$ unless $m=n$. In general (i.e., except for some special cases and ontological constructions of the set $\Bbb N$), neither $\{m,n\}$ nor $(m,n)$ will be an element of $\Bbb N$, hence probably neither $\{m,n\}\in\Bbb N$ nor $(m,n)\in\Bbb N$ mean what you perhaps intend to express.
For the sequences, I have seen both notations, though I find the second much more suggestive as the first gives conflicting signals to the reader: Does the order matter because of the indexing? Or does it not because of the set-like notation?