I put all of the elements of $D_2$ into matrix form, giving $[\begin{matrix} 1 & 0\\ 0 & 1\end{matrix}]$ , $[\begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix}] $ , $[\begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix}]$, and $[\begin{matrix} -1 & 0\\ 0 & 1\end{matrix}]$
Now I have to create an isomorphism between this and the set {(1,1), (1,-1), (-1,1), (-1,-1)}
I was thinking f(x,y) = (Det(x), Det(y)) where $x,y \epsilon D_2$ but I don't know if I am allowed to use x,y at the same time.
I still have a very loose understanding of isomorphisms so any help is appreciated, thank you.
An isomorphism shoud map one element to another one, not two elements to one, so in this case we'd need a function that maps a diagonal matrix $$\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \in D_2$$ to an element $(x,y)$ of the set $S = \{ (1,1), (1,-1), (-1,1), (-1,-1)\}$ or vice versa. The "cannonical" way to do that is by defining
$$f\left( \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \right) := (a,b).$$
Now you just need to show that this is an isomorphism.