In the context of DBMS, consider the two sets T and S given below.

T	S
I: Logical schema	L: Views
II: Physical schema	M: File organization and indexes
III: External schema	N: Relations

Which one of the following is the correct match from T to S ?

Consider a relational database schema with two relations $R(P,Q)$ and $S(X,Y)$.

Let $E=\{⟨u⟩\mid \exists v\exists w⟨u,v⟩\in R\wedge ⟨v,w⟩\in S\}$ be a tuple relational calculus expression.

Which one of the following relational algebraic expressions is equivalent to $E$?

In the context of schema normalization in relational DBMS, consider a set F of functional dependencies. The set of all functional dependencies implied by F is called the closure of F. To compute the closure of F, Armstrong’s Axioms can be applied. Consider $X$, $Y$, and $Z$ as sets of attributes over a relational schema. The three rules of Armstrong’s Axioms are described as follows.

Reflexivity: If $Y\subseteq X$, then $X\to Y$
Augmentation: If $X\to Y$, then $XZ\to YZ$ for any Z
Transitivity: If $X\to Y$ and $Y\to Z$, then $X\to Z$

The additional rule of Union is defined as follows.

Union: If $X\to Y$ and $X\to Z$, then $X\to YZ$

It can be proved that the additional rule of Union is also implied by the three rules of Armstrong’s Axioms. Listed below are four combinations of these three rules. Which one of these combinations is both necessary and sufficient for the proof ?

An index in a DBMS is said to be dense if an index entry appears for every search-key value in the indexed file. Otherwise it is called a sparse index. Consider the following two statements.

S1: A hash index must be a dense index
S2: A $B^{+}$ tree index can be a sparse index

Which one of the following options is correct?