Injection from $\Bbb N$ to the set of functions from the naturals to $\{0,1\}$. Let $\Bbb N$ be the set of natural numbers and let $F$ be the set of total functions from $\Bbb N$ to $\{0,1\}$. 
Construct a total injective function $g_1\colon \Bbb N\to F$.
Sounds easy, but sorta lost. Thank you.
 A: For each $n\in\Bbb N$, $g_1(n)$ must be a function from $\Bbb N$ to $\{0,1\}$, and you want to design $g_1$ so that if $m\ne n$, then $g_1(m)\ne g_1(n)$.
Suppose that $A$ is any subset of $\Bbb N$; I can define a function 
$$\chi_A:\Bbb N\to\{0,1\}:n\mapsto\begin{cases}
1,&\text{if }n\in A\\
0,&\text{if }n\notin A\;.
\end{cases}$$
This function $\chi_A$ is called the indicator function or characteristic function of $A$. Notice that if $A,B\subseteq\Bbb N$, and $A\ne B$, then $\chi_A\ne\chi_B$: there is at least one $n\in\Bbb N$ such that $\chi_A(n)\ne\chi_B(n)$. Thus, one way to solve the problem is to find for each $n\in\Bbb N$ a subset $A_n$ of $\Bbb N$ such that $A_m\ne A_n$ whenever $m\ne n$, and let $g_1(n)=\chi_{A_n}$: since the sets $A_n$ are all distinct, their indicator functions $\chi_{A_n}$ are all distinct, and $g_1$ will be injective.
A: Just take the function $g_1$ such that $g_1(n)$ is the function defined by $g_1(n)(k)=1$ if $k=n$, $g_1(n)(k)=0$ otherwise.
A: what does a $g : \mathbb{N} \to F$ even look like? Well, $g(n)$ is a $\{0,1\}$-valued function on the naturals. Evaluating such a function might look like $g(n)(m)$.
So, our goal is to pick a value in $\{0, 1\}$ for $g(n)(m)$, for every pair of natural numbers $n$ and $m$.
What do we need for $g$ to be injective? Let's appeal to the definitions! $g$ is injective if and only if $g(m) = g(n)$ implies $m=n$. But what does $g(m) = g(n)$ mean? It means that for every $k$, $g(m)(k) = g(n)(k)$.
So, your goal is to pick the values in $\{ 0, 1 \}$ for $g(m)(n)$ with the property that:


*

*If $g(a)(c) = g(b)(c)$ for all $c$, then $a = b$


The contrapositive might be easier to work with: $g$ is injective iff $m \neq n$ implies $g(m) \neq g(n)$. $g(m) \neq g(n)$ means there is at least one value $k$ so that $g(m)(k) \neq g(n)(k)$.
So, your goal is to pick the values for $g(m)(n)$ with the property that


*

*If $a \neq b$, then we can find a $c$ so that $g(a)(c) \neq g(b)(c)$.



The following picture I'm about to describe is more or less exactly what is said above: but being phrased in terms of a picture might help.
You have a grid of squares, infinite towards the South and East. The rows and columns are labelled with natural numbers. In each square of this grid, you want to place a $0$ or a $1$. The number you place in the square in the $m$-th row and $n$-th column is the value you choose for $g(m)(n)$.
Your goal is to place $0$'s and $1$'s so that every row of this grid is different.
