Let $S$ and $E$ be two algebraically independent sets over $\Bbb Q$ such that $S\cup E$ is algebraically independent. Consider $$ A:=\{1/s\colon s\in S\} \ \ \ \text{and} \ \ B:=\{1/e\colon e\in E\}$$ I know $A$ and $B$ algebraically independent. Moreover, $A\cup E$ and $S\cup B$ are both algebraically independent. I am not quite sure about the following
Is $A\cup B$ algebraically independent?
Any help will be appreciated?