I have the following question and given answer:
Question: List the names of managers who have at least one dependant.
Answer:
{e.Fname, e.Lname | EMPLOYEE(e) AND (∃d)(∃t)(DEPARTMENT(d) AND DEPENDANT(t) AND e.Ssn = d.Mgrssn AND t.Essn = e.Ssn) }
My answer was exactly the same except that instead of the (∃d)(∃t)(...) section I said (∃d)(DEPARTMENT(d) AND e.Ssn = d.Mgrssn) AND (∃t)(DEPENDANT(t) AND t.Essn = e.Ssn).
Are these equivalent?
Let me concentrate, as you asked, on section $(\exists d)(\exists t)(\cdots)$.
In case when one part of conjuction is "not depend" on letter under existence quantifier, holds $$(\exists d)(A \text{ and } B)\Leftrightarrow (\exists d)(A) \text{ and } B$$ Where we assume $B$ does not containt letter $d$. $$$$ In your case, for simplicity, let make following designations: $$\begin{array}{l} A(d) = \text{ DEPARTMENT(d) }\\ B(t) = \text{ DEPENDANT(t) } \\ C(d) = (\text{e.Ssn = d.Mgrssn})\\ E(t) = (\text{t.Essn = e.Ssn}) \end{array}$$ and assume, that $A,C$ doesn't depend on $t$ and $B,E$ doesn't depend on $d$. Then we have
$$(\exists d)(\exists t)\big(A(d) \text{ and } B(t)\text{ and } C(d) \text{ and } E(t)\big) \Leftrightarrow \\ \Leftrightarrow (\exists d)\big(A(d) \text{ and } C(d)\big) \text{ and } (\exists t) \big(B(t)\text{ and } E(t)\big)$$ So, you are right, they are equivalent in reviewed case.
Generally we cannot distribute existence quantifier with respect to $\text{AND}$(conjunction) operation as shows example: take $A=$"$d$ is even" and $B=$"$d$ is odd". Then, of course, sentence "$(\exists d)(A) \text{ and } (\exists d)(B)$" is true, but sentence "$(\exists d)(A \text{ and } B)$" is false.
