List of top Computer Science and IT Engineering Questions on Functional Dependency

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, π‘Œ, and 𝑍 as sets of attributes over a relational schema. The three
rules of Armstrong’s Axioms are described as follows.
Reflexivity: If π‘ŒβŠ†π‘‹ , then π‘‹β†’π‘Œ
Augmentation: If π‘‹β†’π‘Œ, then π‘‹π‘β†’π‘Œπ‘ for any Z
Transitivity: If π‘‹β†’π‘Œ and π‘Œβ†’π‘, then 𝑋→𝑍
The additional rule of Union is defined as follows.
Union: If π‘‹β†’π‘Œ and 𝑋→𝑍, then π‘‹β†’π‘Œπ‘
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 ?

Consider a relational database schema with a relation 𝑅(𝐴, 𝐡, 𝐢, 𝐷). If {𝐴, 𝐡} and
{𝐴, 𝐢} are the only two candidate keys of the relation 𝑅, then the number of superkeys
of relation 𝑅 is ______. (answer in integer)

Let 𝑃, 𝑄, 𝑅 and 𝑆 be the attributes of a relation in a relational schema. Let 𝑋 βŸΆπ‘Œ
indicate functional dependency in the context of a relational database, where
𝑋, π‘Œ βŠ†{𝑃, 𝑄, 𝑅, 𝑆}.
Which of the following options is/are always true?