Solution

The correct answer is option 2: Lossless Join but not dependency preserving 

Key Points

Given : functional dependencies are :

• P → Q
• Q → R
• R → S
• S → Q

Decomposition: R1(P, Q), R2(Q, R), R3(Q, S)

Step 1: Check Dependency Preservation

• P → Q: Present in R1
• Q → R: Present in R2
• S → Q: Present in R3
• R → S: Not preserved, because no relation contains both R and S together

Step 2: Check for Lossless Join

Even though R → S is not preserved, the join is still lossless due to the presence of a candidate key P and correct attribute overlaps.

• P → Q → R → S, so P is a candidate key
• The decomposition allows recovery of original data using joins, hence lossless

Final Answer: Option 2) Lossless Join but not Dependency Preserving ✅