Consider two relational R(P, Q) and S(X, Y). Given E = {u — ∃ v ∃ w such that (u, v) ∈ R and (v, w) ∈ S} Which one of the following relational Algebra expressions is equivalent to E?

The correct option is:
π_P(S ▷◁_{S.X = R.Q} R)

Let us analyze the given tuple relational calculus (TRC) expression:
$E=\left\{u\right|\exists v\exists w\text{ such that }(u,v)\in R\text{ and }(v,w)\in S\}$

Here, the relation $R$ has attributes $(P,Q)$, and the relation $S$ has attributes $(X,Y)$.
From $(u,v)\in R$, we see that $u$ corresponds to the first attribute of $R$ (which is $P$) and $v$ corresponds to the second attribute of $R$ (which is $Q$).
From $(v,w)\in S$, we see that $v$ corresponds to the first attribute of $S$ (which is $X$) and $w$ corresponds to the second attribute of $S$ (which is $Y$).

The variable $v$ acts as the linking variable between $R$ and $S$, which means the value of the attribute $Q$ in $R$ must be equal to the value of the attribute $X$ in $S$ ($R.Q=S.X$).
This constraint represents a join operation between relations $S$ and $R$ on the condition $S.X=R.Q$.

The set $E$ consists of the values of $u$, which is the attribute $P$ of $R$. Therefore, we project attribute $P$ from the joined relation.

Putting it all together:
1. Perform the join: $S{▷◁}_{S.X=R.Q}R$
2. Project attribute $P$: ${\pi }_P\left(S{▷◁}_{S.X=R.Q}R\right)$

This matches the correct relational algebra expression:
π_P(S ▷◁_{S.X = R.Q} R)