Is everything ok with this notation? $A=\{x\mid 2x,x\in\Bbb{N}\}$ I couldn't understand this notation. Is everything ok with this notation?

I would write it like this:
$$A=\left\{n: n=2k, k\in\mathbb Z_{>0}\right\}$$
or
$$A=\left\{n\mid n=2k, k\in\mathbb Z_{>0}\right\}$$
or
$$A=\left\{n: \text{n is an even natural number}\right\}$$
or
$$A=\left\{n\mid \text{n is an even natural number}\right\}$$
 A: *

*Most likely, the author meant $A=\{2x\mid x\in\Bbb{N}\},$ but absent-mindedly reverted to their habit of using set-builder notation that
starts with $$A=\{x\mid \,.$$
Then, still on autopilot, they recalled that their statement
ought to feature a multiple of $2$ somewhere, so continued by
writing $$2x$$ (instead of, say, $\text“x=2k\text”).$
Finally, remembering that their statement ought to finish off with
$\:\in\mathbb N,$ they conveniently completed it as $$,x\in\mathbb
N\}$$ (instead of, say, $\text{“for some }k\in\mathbb N\text”).$


*Making sense of the notation (these two sets are equivalent):

*

*$$A=\{2k\mid k\in\Bbb{N}\}$$ the set of numbers of the form $2k$ such that $k$ is natural;

*$$A=\{x\mid \exists k{\in}\Bbb{N}\:\: x=2k\}$$ the set of numbers such that each, for some natural $k,$ equals $2k$.



*As written, the given string $$A=\{x\mid 2x,\;x\in\Bbb{N}\},$$ is not well-formed /meaningful, due to the middle portion $2x$ being just a term instead of a logical formula.
A: Arguably, it is fine, but it is somewhat informal and very confusing.
We will often write something like "Let $m,n,k\in\Bbb N$ ..." as a shorthand of $m\in\Bbb N, n\in\Bbb N, k\in\Bbb N$. We are being informal, since we expect other humans to read what we say, and we expect the context is clear enough to understand that.
So, if I wanted to tell you that not only $2x$ is a natural number, but also $x$ itself is a natural number, I could tell you that $x,2x\in\Bbb N$.
If I had written only that $2x\in\Bbb N$, this would have included rational numbers such as $\frac12$ and $\frac32$, but by requiring that $x\in\Bbb N$ as well, I've avoided that.
Consequently, $\{x\mid 2x,x\in\Bbb N\}$ is just $\Bbb N$. Whether or not that was the intention. So, that is kind of a silly way writing it anyway.
Of course, if we wanted the set of even natural numbers, then this no longer works (but we can, arguably, substitute $2x$ by $\frac x2$ in that set-builder definition). Or, we could have written, as you suggest $\{x\mid x=2k, k\in\Bbb N\}$, or $\{x\mid\exists k\in\Bbb N, x=2k\}$, or much more clearly, $\{2x\mid x\in\Bbb N\}$.
A: I believe the most natural interpretation is $\{ x \, | \, 2x \in \mathbb{N} \text{ and } x \in \mathbb{N} \}$ (which of course means that the set is just $\mathbb{N}$ itself). Indeed, usually when we write $s,t \in A$, we mean $s \in A$ and $t \in A$. This is demonstrated in, say, how we write the span of two vectors: $\{ su+tv \, | \, s,t \in \mathbb{R} \}$. So, $2x, x \in \mathbb{N}$ should be interpreted similarly.
However, most people working with math would never write something like this, because it is redundant. This leads me to think that this is some sort of exercise for practicing working with sets and mathematical language. If not, then there is probably a typo, and you should check with whoever wrote it.
A: The best anyone can guess from that fragment is that it means all of the numbers $\frac12, 1, \frac32, 2, \frac52,...$ where $2x\in\mathbb{N}$.
The notation is ambiguous because of the repeated $x$, but most mathematicians would interpret, for example, $\{(a,b)|2a,b\in\mathbb{N}\}$ as meaning "the set of ordered pairs $(a,b)$ such that $2a\in\mathbb{N}$ and $b\in\mathbb{N}$".
For example, this page defines
$$
\mathbb{Q}=\{a/b \mid a, b \in \mathbb{Z}, b\neq 0\}\text{.}
$$
The notation $\{x|\text{condition}\}$ means "the set of $x$ which satisfy the condition". Here $\{\}$ means "the set of", and $\mid$ means "such that", so in words
$\{x|x\in\mathbb{R},x>1\}$ means "the set of $x$ such that $x$ is an element of  the set of real numbers, and $x$ is greater than one".
Whatever the fragment means, it is not possible in the normal understanding of this notation that $\{x|2x\in\mathbb{N}\}$ means "the set of even numbers". That would be $\{x|x/2\in\mathbb{N}\}$. Some people seem to be confusing the condition part of the statement as being able to reach leftwards and change the value of the $x$ part.
