My question might be very simple for those that have a deep understanding for algebraically independent sets.
Definition 1.1. Let $F$ be an extension field of $K$ and $S$ a subset of $F$. $S$ is algebraically dependent over $K$ if for some positive integer $n$ there exists a nonzero polynomial $f\in K[x_1, . . . , X_n]$ such that $f(s_1, . . . , s_n) = 0$ for some distinct $S_1, . . . , s_n\in S$. $S$ is algebraically independent over $K$ if $S$ is not algebraically dependent over $K$.
Let $P_1\cup P_2$ are two algebraically independent sets of $\mathbb R$ over $\mathbb Q$ and $P_1\cap P_2=\emptyset.$ Assume there exist $a,b\in\mathbb Q\setminus\{0\}$ such that $P^*=aP_1+b\subset J$ for some open interval $J$ subset of $\mathbb R$ Does the following hold ?
I. $P^*$ is algebraically independent ?
II. $P^*\cup P_2$ is algebraically ?
Clearly, I is correct since $q P_1$ is algebraic and shifting algebraically independent by rational number is still algebraically independent set.
About, II I do not find an example that shows it is incorrect or proof it. I am tried to find algebraic depending relation between elements of $P^*\cup P_1$
Any help will be appreciated greatly.