Let P,Q,R and S be the attributes of a relation in a relational schema. Let X → Y indicate functional dependency in the context of a relational database, where X,Y ⊆{P,Q,R,S}. Which of the following options is/are always true?

The correct options are:
1. If $\left\{P\right\}\to \left\{R\right\}$ and $\left\{Q\right\}\to \left\{S\right\}$, then $\{P,Q\}\to \{R,S\}$
2. If $\left\{P\right\}\to \left\{R\right\}$, then $\{P,Q\}\to \left\{R\right\}$

Let's analyze the validity of each statement step-by-step using the standard rules of functional dependencies in relational databases, known as Armstrong's Axioms.

1. Analyzing the statement: If $\left\{P\right\}\to \left\{R\right\}$ and $\left\{Q\right\}\to \left\{S\right\}$, then $\{P,Q\}\to \{R,S\}$
This is the composition rule (sometimes referred to as the additive rule), which can be derived from Armstrong's Axioms as follows:
- Given $\left\{P\right\}\to \left\{R\right\}$, we can apply the rule of augmentation by adding $Q$ to both sides. This gives us:
$\{P,Q\}\to \{R,Q\}$ (Step 1)
- Similarly, given $\left\{Q\right\}\to \left\{S\right\}$, we can apply the rule of augmentation by adding $R$ to both sides. This gives us:
$\{Q,R\}\to \{S,R\}$ (which is equivalent to $\{R,Q\}\to \{R,S\}$) (Step 2)
- Using the transitivity rule on Step 1 ($\{P,Q\}\to \{R,Q\}$) and Step 2 ($\{R,Q\}���\{R,S\}$), we get:
$\{P,Q\}\to \{R,S\}$
Therefore, this option is always true.

2. Analyzing the statement: If $\left\{P\right\}\to \left\{R\right\}$, then $\{P,Q\}\to \left\{R\right\}$
This is a direct application of the augmentation rule in Armstrong's Axioms:
- If we have a functional dependency $X\to Y$, we can augment the left-hand side with any attribute set $Z$ to yield $XZ\to Y$ (or $XZ\to YZ$).
- Applying this rule to $\left\{P\right\}\to \left\{R\right\}$ by augmenting the left-hand side with $\left\{Q\right\}$, we obtain:
$\{P,Q\}\to \{R,Q\}$
- By decomposition (or reflectivity since $\left\{R\right\}\subseteq \{R,Q\}$), we get:
$\{P,Q\}\to \left\{R\right\}$
Therefore, this option is also always true.

Why the other choices are incorrect / not always true:
- If $\{P,Q\}\to \left\{R\right\}$, then ($\left\{P\right\}\to \left\{R\right\}$ or $\left\{Q\right\}\to \left\{R\right\}$): This is invalid. For example, if we have a relation schema with attributes $P$, $Q$, and $R$ representing (StudentID, CourseID) → Grade, neither StudentID → Grade nor CourseID → Grade holds individually.
- If ($\{P,Q\}\to \left\{R\right\}$ and $\left\{P\right\}\to \left\{R\right\}$), then $\left\{Q\right\}\to \left\{R\right\}$: This does not necessarily follow. The fact that the combination of $P$ and $Q$ determines $R$, and $P$ alone also determines $R$, tells us nothing about whether $Q$ alone can determine $R$. Therefore, $\left\{Q\right\}\to \left\{R\right\}$ is not guaranteed.