Given a relation scheme \(R(x,y,z,w)\) with functional dependencies set \(…

Correct answer: A and C only.

Explanation:

• First Normal Form: All attributes are given to be single and atomic, so the relation satisfies First Normal Form.

• Determine keys: Using the functional dependencies x → y and z → w, the closure of xz is xz+ = {x, y, z, w}, so xz determines all attributes and is a candidate key. Neither x alone nor z alone determines all attributes (x+ = {x,y}, z+ = {z,w}), so the only candidate key is xz; therefore xz is the primary key.

• Second Normal Form check: 2NF requires that no non-prime attribute is partially dependent on a part of a composite primary key. Here x → y is a dependency where x is part of the composite key xz and y is a non-prime attribute (y is not part of any key). This is a partial dependency, so the relation violates Second Normal Form.

• Conclusion: The relation is in First Normal Form and xz is the primary key, but it is not in Second Normal Form. Hence the correct statement set is the one that asserts First Normal Form and that the primary key is xz.