Consider concurrent execution of two transactions $T 1$ and $T 2$ in a DBMS, both of which access a data object $A$. For these two transactions to not conflict on $A$, which one of the following statements must be true?

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 $X \rightarrow Y$

Augmentation: If $X \rightarrow Y$, then $X Z \rightarrow Y Z$ for any $Z$

Transitivity: If $X \rightarrow Y$ and $Y \rightarrow Z$, then $X \rightarrow Z$

The additional rule of Union is defined as follows.

Union: If $X \rightarrow Y$ and $X \rightarrow Z$, then $X \rightarrow Y Z$

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 searchkey 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 $\mathrm{B}^{+}$tree index can be a sparse index

Which one of the following options is correct?

Which one of the following options is not a property of Boolean Algebra?

Note: + is OR operation, • is AND operation, and ' is NOT operation