Consider a relational database schema with two relations $R(P, Q)$ and $S(X, Y)$. Let $E = \{\langle u \rangle \mid \exists v \,\exists w \,\langle u, v \rangle \in R \land \langle v, w \rangle \in S\}$ be a tuple relational calculus expression. Which one of the following relational algebraic expressions is equivalent to $E$ ?

Given the schema $R(P, Q)$ and $S(X, Y)$:

1. In the tuple relational calculus (TRC) expression:

   $$E = \{\langle u \rangle \mid \exists v \,\exists w \,(\langle u, v \rangle \in R \land \langle v, w \rangle \in S)\}$$

   • A tuple $\langle u, v \rangle \in R$ assigns $u$ to attribute $P$ and $v$ to attribute $Q$.

   • A tuple $\langle v, w \rangle \in S$ assigns $v$ to attribute $X$ and $w$ to attribute $Y$.

2. The shared variable $v$ represents the equality condition between the second attribute of $R$ ($Q$) and the first attribute of $S$ ($X$):

   $$R.Q = S.X$$

3. The target tuple is $\langle u \rangle$, which corresponds to attribute $P$ of relation $R$.

4. Translating this into relational algebra gives:

   $$\Pi_P (S \bowtie_{S.X = R.Q} R) \quad \text{or} \quad \Pi_P (R \bowtie_{R.Q = S.X} S)$$

Hence, option (B) is the correct equivalent relational algebraic expression.