How do I write $B = \left\{\left[\begin{smallmatrix} x \\ y \end{smallmatrix}\right] \in \boxed{?}| \ldots\right\}$ with proper notation

Question

Let $x \in X \subset \mathbb{R}^n$, then I define a set:

$$A = \{x \in X| 1^Tx = 0\}$$

Now supose I have another element $y \in Y \subset \mathbb{R}^n_{+}$

I concatenate $x,y$ in to a single vector and define another set, placing conditions on $y$:

\begin{equation}B = \Bigg\{\begin{bmatrix} x \\ y \end{bmatrix} \in \boxed{?}| \begin{bmatrix}1^Tx = 0 \\ 1^Ty = 1 \end{bmatrix}\Bigg\}\end{equation}

Can someone fill in the blank $\boxed{?}$. I am guessing $X \times Y$ but not confident

Is there a better way to write this set $B$?

Answer 1

The Cartesian product will work, but you want to specify how the concatenated vector is composed.

$$\left\{\vec z\in X{\times}Y ~\middle\vert~ \vec z=\vec x\Vert\vec y\,, \vec x\in X\,, \vec y\in Y\,,\mathbf 1^\top\vec x=1\,, \mathbf 1^\top\vec y=0\right\}$$

Or more elegantly: $$\left\{\vec x \Vert\vec y~\middle\vert~\vec x\in X\,, \vec y\in Y\,:\,\mathbf 1^\top\vec x=1\,, \mathbf 1^\top\vec y=0\right\}$$

Note: the conditions are not concatenated, rather the list is conjunctive.

Answer 2

Why not? Because $X\times Y=\{(x,y): x\in X,\, y\in Y\}$ so no matter what set is $X$ or what set is $Y$.