Set builder notation I'm not sure of the correct notation if someone could please help. 
Say you wish to have a set comprised of the union of two sets, such as $$h=\Big\{ \begin{bmatrix}x & y\\y & x\end{bmatrix}:  x^2-y^2=1 \Big\};$$$$g=\Big\{ \begin{bmatrix}x & y\\-y & -x\end{bmatrix}:  x^2-y^2=1 \Big\}.$$ 
I'm not sure if putting the union sign inside the set is correct. Would it be $$\phi = \Big\{ \begin{bmatrix}x & y\\y & x\end{bmatrix}\cup \begin{bmatrix}x & y\\-y & -x\end{bmatrix}\ :  x^2-y^2=1 \Big\}?$$
 A: You can 
$(1)$: Provided you've already defined $h, g$, as you did in this post, the simplest route is to express $\phi$ as the union of $h, g$, as in $$\phi = h\cup g$$
$(2)$ Write $$\phi =  \left\{ \begin{bmatrix}x & y\\y & x\end{bmatrix}\text{ or }\; \begin{bmatrix}x & y\\-y & -x\end{bmatrix}\ :  x^2-y^2=1 \right\}$$
The union operation is an operation on sets; for example, $h, g$ are sets.
Instead of "or", you can use the notation $\lor$ in $(2)$.
Better yet, if you really like set-builder notation, you can write:
$$\phi = \{A: A \in g \lor A \in h\}$$ which means: "The set of all matrices A that belong to $h$ or belong to $g$.
You might also want to include the field to which $x, y$ belong. E.g., $x, y \in \mathbb R$.
A: Maybe 
$$ \left\{\,\begin{bmatrix}x&y\\ex&ey\end{bmatrix}\in \mathbb R^{2\times 2}: x^2-y^2=1, e=\pm1\,\right\}$$
This matches the set-builder syntax $\{\,x\in X: \phi(x)\,\}$ as per Axiom schema of Specification (or Comprehension). Depending on context, the class-builder syntax $\{\,x:\phi(x)\,\}$ may be ok, but at any rate having a "complex expression" to the left of the colon is at least ambiguous as it may suggest a replacement $\{\,f(x):x\in X\,\}$. (Actually, the matrix itseld is already such a "complex expression", so this is arguably an instance of replacement anyway).
