Notation: how to unzip a set?

Question

I have a set of triple $S_z=\{(x_1,y_2,z),(x_2,y_2,z),(x_3,y_1,z),\ldots\}$ where the last element of each triple is the same.

I want to create the set $S=\{x_1,y_2,x_2,x_3,y_1\}$ of all elements of $S_z$ not including $z$. What is the notation used to do this?

Something like $$S=\{i \text{ or } j: (i,j,z)\in S_z\}?$$

Answer 1

I would go for:$$S=\left\{ u\mid\exists\left(a,b,c\right)\in S_{z}\left[u\in\left\{ a,b\right\} \right]\right\} $$

Answer 2

Consider two projection maps defined on the set $S_z$:

$pr_1 ((x,y,z))=x$ and $pr_2 ((x,y,z))=y$.

Then $$ S=Im( pr_1) \cup Im( pr_2). $$

Answer 3

$$\{x\mid(x,y,z)\in S_z\}\cup\{y\mid(x,y,z)\in S_z\}$$