Consider the following relational schema. $\text{Students}(\underline{\text{rollno: integer}}, \text{ sname: string})$ $\text{Courses}(\underline{\text{courseno: integer}}, \text{ cname: string})$ $\text{Registration}(\underline{\text{rollno: integer}}, \underline{\text{courseno: integer}}, \text{ percent: real})$ Which of the following queries are equivalent to this query in English? “Find the distinct names of all students who score more than 90% in the course numbered 107” (I) SELECT DISTINCT S.snameFROM Students as S, Registration as RWHERE R.rollno=S.rollno AND R.courseno=107 AND R.percent >90 (II) $\Pi_{\text{sname}}(\sigma_{\text{courseno}=107 \wedge \text{percent}>90} (\text{Registration} \bowtie \text{Students}))$ (III) $\{ \text{T} \mid \exists \text{S} \in \text{Students}, \exists \text{R} \in \text{Registration} \ ( \text{S.rollno}=\text{R.rollno} \wedge \text{R.courseno}=107 \wedge \text{R.percent}>90 \wedge \text{T.sname}=\text{S.sname})\}$ (IV) $\{ \langle \text{S}_\text{N} \rangle \mid \exists \text{S}_\text{R} \exists \text{R}_\text{P} \ ( \langle \text{S}_\text{R}, \text{S}_\text{N} \rangle \in \text{Students} \wedge \langle \text{S}_\text{R}, 107, \text{R}_\text{P} \rangle \in \text{Registration} \wedge \text{R}_\text{P} > 90)\}$

Let us evaluate all four given query representations:

1. Query (I) (SQL): Performs a join of Students and Registration on rollno, filters records having courseno = 107 and percent > 90, and projects unique sname using SELECT DISTINCT. Hence, it correctly matches the English query.

2. Query (II) (Relational Algebra): Performs a natural join $\text{Registration} \bowtie \text{Students}$ on the common attribute rollno, applies the selection $\sigma_{\text{courseno}=107 \wedge \text{percent}>90}$, and projects $\Pi_{\text{sname}}$. By definition of set semantics in relational algebra, projection eliminates duplicates. Thus, this is equivalent.

3. Query (III) (Tuple Relational Calculus - TRC): Selects tuple $T$ containing sname such that there exist tuples $S \in \text{Students}$ and $R \in \text{Registration}$ satisfying $S.\text{rollno} = R.\text{rollno}$, $R.\text{courseno} = 107$, $R.\text{percent} > 90$, and $T.\text{sname} = S.\text{sname}$. Since TRC operates on sets, it returns distinct student names.

4. Query (IV) (Domain Relational Calculus - DRC): Expresses domain variable $\langle \text{S}_\text{N} \rangle$ such that there exist $\text{S}_\text{R}$ and $\text{R}_\text{P}$ with $\langle \text{S}_\text{R}, \text{S}_\text{N} \rangle \in \text{Students}$ and $\langle \text{S}_\text{R}, 107, \text{R}_\text{P} \rangle \in \text{Registration}$ with $\text{R}_\text{P} > 90$. This also correctly returns the set of distinct student names.

Therefore, all four queries (I), (II), (III), and (IV) are equivalent. The correct option is (A).