Definability of Cartesian Products

Question

Let $\mathcal{M}$ be an $\mathcal{L}$-structure. Suppose $X\subseteq \vert \mathcal{M}\vert^n$, $Y\subseteq \vert \mathcal{M}\vert^k$ are definable in $\mathcal{M}$ by formulas $\varphi_X$ and $\varphi_Y$, respectively. How can I show that $X\times Y\subseteq \vert \mathcal{M}\vert^{n+k}$ is also definable in $\mathcal{M}$? My first guess was by $\varphi(\bar{x}, \bar{y}) = \varphi_X \land \varphi_Y$, but I think that defines $X\cap Y$ instead. Thanks in advance.

Answer 1

Your expression $$\varphi(\overline{x},\overline{y})\equiv\varphi_X\wedge\varphi_Y$$ has the right idea, but it's missing a crucial component: what are the variables involved in each of $\varphi_X$ and $\varphi_Y$? Think, for example, of the difference between $\psi(a)\wedge\theta(a)$ and $\psi(a)\wedge\theta(b)$ ...

Specifically, you want the formula $\varphi_X(\overline{x})\wedge\varphi_Y(\overline{y})$.