What would a function $F:\mathbb{N} \times \mathbb{N} \rightarrow \{0, 1\}$ look like?

A function $F:\mathbb{N}\times \mathbb{N} \to \{0,1\}$ (“from $\mathbb{N}\times \mathbb{N}$ to $\{0,1\}$”) assigns to each element of the set $\mathbb{N}\times \mathbb{N}$, ordered pairs of integers, an element in the set {0,1}, either 0 or 1. So if $m$ and $n$ are natural numbers, $F((m,n))=0$ or $F((m,n))=1$.

Some examples: let $F:\mathbb{N}\times \mathbb{N} \to \{0,1\}$ be given by $(m,n)\mapsto 0$. On the left of “$\to$”, we have the domain of our function. On the right, we have our codomain. On the left of “$\mapsto$”, we have some arbitrary element in the domain. On the right, we have the explicit element in the codomain that $(m,n)$ will map to. So, in particular, $F((m,n))=0$ for all natural numbers $m,n$.

Let $F: \mathbb{N}\times \mathbb{N} \to \{0,1\}$ be given by $(m,n)\mapsto 1$ if $m=n$ and $(m,n)\mapsto 0$ if $m\not = n$.

Visually for these examples, we can think of

0,0,0, ...

0,0,0, ...

...

...

0,0,0, ...

...

and

1,0,0, ...

0,1,0, ...

0,0,1, ...

...

...

0,0,0, ...

...

where the locations represent elements in $\mathbb{N}\times \mathbb{N}$, i.e.

(1,1), (1,2), (1,3), ...

(2,1), (2,2), (2,3), ...

...

...

($m$,1), ($m$,2), ... , ($m$,$n-1$), ($m$, $n$), ...

...

and the values that appear in those locations represent the element of $\{0,1\}$ which $F$ maps its location to.