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?

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

Consider the following relational schema along with all the functional dependencies that hold on them.

$$\begin{aligned} & R 1(A, B, C, D, E):\{D \rightarrow E, E A \rightarrow B, E B \rightarrow C\} \\ & R 2(A, B, C, D):\{A \rightarrow D, A \rightarrow B, C \rightarrow A\} \end{aligned}$$

Which of the following statement(s) is/are TRUE?

Consider a relational schema team(name, city, owner), with functional dependencies \{name $\rightarrow$ city, name $\rightarrow$ owner}.

The relation team is decomposed into two relations, $t 1$ (name, city) and $t 2$ (name, owner). Which of the following statement(s) is/are TRUE?