GATE CSE 2024 | Set 1 | Question: 34

Question

​The symbol $\rightarrow$ indicates functional dependency in the context of a relational database. Which of the following options is/are TRUE?

• A. $(X, Y) \rightarrow(Z, W)$ implies $X \rightarrow(Z, W)$ 
• B. $(X, Y) \rightarrow(Z, W)$ implies $(X, Y) \rightarrow Z$
• C. $((X, Y) \rightarrow Z$ and $W \rightarrow Y)$ implies $(X, W) \rightarrow Z$
• D. $(X \rightarrow Y$ and $Y \rightarrow Z)$ implies $X \rightarrow Z$ 

Answer 1

Correct Answer - Option B,C,D

Option B - Decompostion rule of functional dependencies. 

Option C - Apply augmentation rule with X on W -> Y, Then apply Transitivity rule to get X, W -> Z. 

Option D - Definition of Transitivity rule for functional dependencies

Answer 2

$(X,Y)\rightarrow(Z,W)\implies X\rightarrow ZW$, this is false statement because only $X^+\rightarrow X$,  cannot determine $ZW$.

$(XY)\rightarrow(ZW)\implies XY\rightarrow Z$,True it is split rule.

$(XY\rightarrow Z)$ and$(W\rightarrow Y)\implies XW \rightarrow Z$,agumentation rule true.

$(X\rightarrow Y)$ and $(Y\rightarrow Z)\implies X\rightarrow Z$, True, it is trasitive rule.

So option $(B,C,D)$ are true.

Answer 3

Option A). It is incorrect because Both X and Y together determine Z and W.

RULE: Don't separate the attributes on LHS of functional dependency.

Option B). Correct. Because Both X and Y determine Z and W. It means that X and Y determine Z  and, X and Y determine W also.

Rule: You can separate the attributes on RHS of functional dependency.

Option C). Correct.

X and Y determine Z

and W determines Y which means in functional dependency Y is a subset of W. (Axiom of Reflexivity)

hence, X and W determine Z because the subset with X determines Z then the superset also with X determines Z.

Option D). It is Transitively correct.