Can someone explain me how to read members of a set to prove uncountability?

Question

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all $x\in\mathbb{N}$} \}. $$

I always have trouble in reading these definitions in a normal way. Can someone explain me how to read definitions (or sets) like these and what the members of this set are?

I have to prove whether this set is countable or not, but for that I first have to understand what the set actually means.

Thank you.

Answer 1

I like to take it apart. First, $V$ is the set of all functions from $\Bbb N \to \Bbb N$ that satisfy the given condition. That gets us thinking about the right stuff. What is the condition? That the function $f$ has some $N_f$ (that depends on $f$) so that all the values of $f(x)$ are less than or equal to $N_f$. This means (across all $x \in \Bbb N$) there is some number greater than all the $f(x)$, which is an upper bound. This is the set of all bounded functions.

Answer 2

It's the set of bounded functions from $\mathbb{N}$ to $\mathbb{N}$.

Answer 3

Interpret from the inside out.

"$f(x)\le N_f$": The value of $f$ at $x$ is at most $N_f$.

"$f(x)\le N_f$ for all $x\in\mathbb{N}$": Every value of $f$ is at most $N_f$. The graph of $f$ would lie completely below the line $y=N_f$. In short, $f$ is bounded above by $N_f$.

"there is some $N_f\in\mathbb{N}$ so that $f(x)\le N_f$ for all $x\in\mathbb{N}$": There is some line $y=N_f$ below which the graph of $f$ lies. That just means the graph of $f$ doesn't go up forever. In short, $f$ is bounded above. (The subscript $f$ on $N_f$ indicates that the value of $N_f$ might depend on $f$, that is, if we consider more than one $f$ at some point, different $f$s could have different $N_f$s. Maybe that will be important later.)

"$\{f\colon\mathbb{N}\to\mathbb{N}\;|\;\text{there is...}\}$": The set of functions $f$ mapping $\mathbb{N}$ to $\mathbb{N}$ such that $f$ is bounded above.

So, $V$ is the set of bounded functions from $\mathbb{N}$ to $\mathbb{N}$.

Looking back at that subscript, it seems that the author was concerned that the reader would maybe think that all the functions in $V$ were bounded by one and the same $N$, that every function's graph lay below the same line $y=N$. By writing $N_f$, the author has emphasized that that's not what they meant. (I think this was clear from the logical structure of what they wrote, but some mathematical communities — especially those made up of old-fashioned analysts, in my experience — prefer to write these subscripts.)