mathscr{C}$ on $K^n \times Y$.Then $R^2_\mathscr{C}, R^{1,2}_\mathscr{C} $ are also definable.We also note that  if  $b \in R^2_\mathscr{C}$ then $V_b$ is precisely the set $\big\{ a \in K^n : (a, b) \in R^{1,2}_\mathscr{C} \big\}$.For each $b \in Y$, there is a choice $\mathscr{C}$ as above such that $b \in R^2_{\mathscr{C}}$.By a standard compactness argument, there are finitely many choices $\mathscr{C}_1, \ldots, \mathscr{C}_k$ as above such that for any $b \in Y$, there is $i \in \{ 1, \ldots, k\} $ such that $b \in R^2_{\mathscr{C}_i}$.Then the family $\{V_b\}_{b \in Y}$ as a subset of $K^n \times Y$ consists of $(a, b)$ such that for some $i \in \{1, \ldots k\}$, $(a, b) \in R^{1,2}_{\mathscr{C}_i}$.Thus $\{V_b\}_{b \in Y}$ is definable.\end{proof}\begin{cor} \label{Definability}Suppose $\{T_b\}_{b \in Y}, \{T'_{b'}\}_{b' \in Y'}$ are definable families of pc-subsets of $K^n$.The following sets are definable:\begin{enumerate}\item for $k \in \mathbb{N}$, the set $\big\{ b \in Y : \mathrm{r}_K(T_b) = k \big\}$;\item for $k,l \in \mathbb{N}$, the set $\big\{ b \in Y : \mathrm{r}_K(T_b) =