Consider a relation R(A, B, C, D) and functional dependencies of X → Y. Which of the following statement are correct?

The correct statements are:
1. If P → R and Q → S, then PQ → RS.
2. If P → R, then PQ → R.

Let us analyze the correctness of each statement step-by-step using the formal inference rules for functional dependencies (also known as Armstrong's Axioms).

Statement 1: If $P\to R$ and $Q\to S$, then $PQ\to RS$.
This statement is correct and is known as the Composition Rule. We can derive it using Armstrong's Axioms:
1. Given $P\to R$, by applying the augmentation rule (augmenting both sides with $Q$), we get:

$PQ\to RQ$

2. Given $Q\to S$, by applying the augmentation rule (augmenting both sides with $R$), we get:

$RQ\to RS$ (since $RQ$ is same as $QR$)

3. By applying the transitivity rule to $PQ\to RQ$ and $RQ\to RS$, we obtain:

$PQ\to RS$

Thus, this statement is valid.

Statement 2: If $P\to R$, then $PQ\to R$.
This statement is correct and is known as the Augmentation Rule (specifically, a direct consequence of it).
According to the augmentation rule, if $P\to R$ holds, we can append any set of attributes $Q$ to both the left and right sides to get:

$PQ\to RQ$

Since $RQ\to R$ is a trivial functional dependency (decomposition rule), we can apply transitivity to get:

$PQ\to R$

This means that adding extra attributes to the left-hand side of a functional dependency (which is called making the determinant super-key larger) does not invalidate the dependency. Thus, this statement is valid.

Let us also verify why the other statements are incorrect to complete the explanation:

Statement 3 (Incorrect): If $PQ\to R$, then $P\to R$ or $Q\to R$.
This is incorrect because $P$ and $Q$ might jointly determine $R$, but neither of them can determine $R$ individually. For example, in a relation $R(A,B,C)$ where a customer $A$ buys a product $B$ at price $C$, the price $C$ depends on both the customer and the product combined, but not on the customer alone or the product alone.

Statement 4 (Incorrect): If $PQ\to R$ and $P\to R$, then $Q\to R$.
This is incorrect. Knowing that $P$ independently determines $R$ does not imply that $Q$ also determines $R$. For example, if we have $AB\to C$ and $A\to C$, it does not mean $B\to C$ must hold. Here, $B$ could be an entirely irrelevant attribute.