Correct : c,d

Option A — False
The claim is that Q→R follows when both PQ→R and P→R hold. This is not a valid inference rule. Consider a case where P alone determines R but Q has no independent connection to R — Q could take different values that map to different R values depending on other rows. The presence of P→R tells us P is doing all the work, not Q. Q cannot be concluded to determine R on its own. A is false.
Option B — False
The claim is that if a combined attribute set PQ determines R, then at least one of P or Q individually must also determine R. This is also not valid. PQ→R simply means the combination is sufficient — neither attribute needs to be sufficient alone. A classic example is a composite key where neither component alone is a key but together they uniquely identify R. B is false.
Option C — True (Union rule)
Given P→R, by the Augmentation axiom we can add Q to both sides: PQ→RQ. Given Q→S, by Augmentation we can add P to both sides: PQ→PS. Now PQ determines both R (from the first) and S (from the second). By the Union rule, PQ→RS. This always holds — if we know P and Q, we can determine R using P and determine S using Q, so together PQ determines both R and S. C is always true.
Option D — True (Augmentation axiom)
Given P→R, adding any extra attribute Q to the left side never breaks the dependency. If P alone is enough to determine R, then having both P and Q is certainly enough to determine R — knowing more attributes can only help, never hurt. This is the Augmentation rule: P→R implies PQ→R for any Q. D is always true.
Correct answer: C and D ✓