Question

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

• A. \(\Pi_P(R \bowtie_{R.P = S.X} S)\)
• B. \(\Pi_P(S \bowtie_{S.X = R.Q} R)\)
• C. \(\Pi_P(R \bowtie_{R.P = S.Y} S)\)
• D. \(\Pi_P(S \bowtie_{S.Y = R.Q} R)\)

Hint:

Match each shared/free variable in the tuple calculus to the attribute position it occupies in R and S, then translate the shared existential variable into an equi-join condition before projecting.

Answer 1

The Correct Option is B

Step 1: Understand the relation schemas and the calculus expression.
We are given two relations \(R(P, Q)\) and \(S(X, Y)\). A tuple written as \(\langle u, v \rangle \in R\) means that in relation \(R\), the value of attribute \(P\) is \(u\) and the value of attribute \(Q\) is \(v\), since tuple components follow the declared attribute order. Similarly, \(\langle v, w \rangle \in S\) means the value of attribute \(X\) is \(v\) and the value of attribute \(Y\) is \(w\).

Step 2: Interpret the tuple relational calculus expression \(E\).
\(E = \{\langle u \rangle \mid \exists v\, \exists w\; \langle u, v \rangle \in R \wedge \langle v, w \rangle \in S\}\)
This collects every value \(u\) such that there is some \(v\) with \((u, v)\) a tuple of \(R\) (so \(u = R.P\), \(v = R.Q\)), and that same \(v\) also occurs as the \(X\)-value of some tuple of \(S\) (so \(v = S.X\)), for some \(w = S.Y\). The existential quantifier on \(w\) adds no real restriction: for any tuple of \(S\) with \(X\)-value \(v\), some \(Y\)-value \(w\) automatically exists. So the only real binding condition is \(R.Q = S.X\).

Step 3: Translate to relational algebra.
This is a classic join-then-project pattern. Join \(R\) and \(S\) on the condition \(R.Q = S.X\) (the shared variable \(v\)), then project only the attribute \(P\) (the free variable \(u\)): \[ E = \Pi_P\big(R \bowtie_{R.Q = S.X} S\big) \]
Step 4: Match against the options.
Since a theta-join is commutative, \(R \bowtie_{R.Q=S.X} S\) is the same set of tuples as \(S \bowtie_{S.X=R.Q} R\), and projecting on \(P\) afterward gives an identical result either way.
Option (A) uses condition \(R.P = S.X\) -- wrong attribute of \(R\) (uses \(P\) instead of \(Q\)).
Option (B) uses condition \(S.X = R.Q\) -- exactly matches the join condition derived in Step 3.
Option (C) uses condition \(R.P = S.Y\) -- wrong attributes on both sides.
Option (D) uses condition \(S.Y = R.Q\) -- wrong attribute of \(S\) (uses \(Y\) instead of \(X\)).

Step 5: Conclusion.
Only option (B), \(\Pi_P(S \bowtie_{S.X = R.Q} R)\), reproduces exactly the join condition \(R.Q = S.X\) required by \(E\). \[ \boxed{\text{Option (B): } \Pi_P(S \bowtie_{S.X = R.Q} R)} \]