Converting to english I am working on homework right now, and I am not sure of how to solve this problem. I am not sure of how to come up with the translation. Any help is greatly appreciated. This is the problem:
Write this in English:   ∀k   ∈ 3Z  ,∃S   ⊆ N  ,|S  | = k  . (Is it true?) What is the   negation of this statement? (Is the negation true?)
I have made an attempt and think that it says, "For every integer, there exists 3 integers such that S is a subset of N, and |S| is equal to k."
I am not sure if this is correct or not.
 A: Let's take each part individually:


*

*$\forall k \in 3 \mathbb{Z}$: By definition, $3\mathbb{Z} = \{3n : n \in \mathbb{Z}\} = \{..., -6, -3, 0, 3, 6, ...\}$ is the set of all multiples of $3$.

*$\exists S \subseteq \mathbb{N}$: There exists a subset $S$ of $\mathbb{N}$, with properties to be specified in

*$|S| = k$: The cardinality (number of elements of $S$) is equal to $k$.
Putting it all together, the translation is:

For every integer $k$ which is a multiple of $3$, there is a subset of $\mathbb{N}$ whose cardinality is equal to $k$.

A: @T.Bongers already translated the profoundly boring statement, so I'm not going to.
Regarding the negation:
You can see the statement is false (Can the number of elements in a set be a negative number, like $-3$ for instance?), so It's negation being false would lead to a contradiction, so It's true and it reads:

"There exists an integer multiple of 3 ($k$), for which all subsets of $\mathbb{N}$ have cardinality different from k"

A: For statements of the form "$\forall \text{(foo)},\exists \text{(bar)}, \text{(blah)}$" you can say "For every foo there is a bar such that blah". So in your case, maybe something like
"For every integral multiple $k$ of $3$ there is a set of natural numbers whose size is $k$."
You have misinterpreted the pieces of the statement, I believe. "$k\in 3\mathbb{Z}$" means $k$ is in the set $3\mathbb{Z}=\{\ldots -12, -9, -6, -3, 0, 3, 6, 9, 12, \ldots \}$. So the first part of the statement is "For every $k$ in $3\mathbb{Z}$", which can be said more plainly as "For every integral multiple of $3$".
The second part is "There is a subset $S$ of $\mathbb{N}$ such that $|S|$ is $k$." Said more plainly, this is "There is a set of natural numbers of size $k$."
