Using the diagonal of Cantor I'm given a function $g:\mathbb{N} \to \mathbb{N}^{\mathbb{N}}$ and I need to find a set $A$ whose cardinality equals to the cardinality of the continuum and $ A  \cap g[N] = \emptyset$ 
I don't really know how to start with solving this problem. I'd really like some hints/clues rather than an answer. Thank you! :)
 A: I suppose you want $A$ to be a subset of $\mathbb N\to\mathbb N$, too; otherwise there is no fun at all here.
Hint as requested: Do you know how to use a diagonalization argument to find one function that is not in the range of $g$?
Can you think of a way to create $2^{\mathbb N}$ different version of that diagonalization argument, such that you're sure they all create different functions outside the range of $g$?

As an alternative, cheat: If you know enough basic cardinal arithmetic, just set $A = (\mathbb N\to\mathbb N)\setminus g(\mathbb N)$ and argue that this must be uncountable because $\mathbb N\to\mathbb N$ is, and $g(\mathbb N)$ is at most countable.
A: I guess, $A$ is supposed to be $\subseteq \Bbb N^{\Bbb N}$ (the set of functions $\Bbb N\to\Bbb N$), else we could answer an arbitrary continuum big set.
Hints:


*

*Given $B\subseteq \Bbb N^{\Bbb N}$, what is the biggest subset $A\subseteq \Bbb N^{\Bbb N}$ with $B\cap A=\emptyset$?

*What can be the cardinality of $B:=g[\Bbb N]$?

*What is the cardinality of $\Bbb N^{\Bbb N}$?

*Continuum minus countable?

