Given the relational schema $R = (U,V,W, X, Y, Z)$ and the set of functional dependencies: $$\{U \rightarrow V, U \rightarrow W, WX \rightarrow Y, WX \rightarrow Z, V \rightarrow X\}$$ Which of the following functional dependencies can be derived from the above set?
Functional Dependency Derivation for R=(U,V,W,X,Y,Z)
We are given a relational schema $R = (U,V,W, X, Y, Z)$ and a set of functional dependencies $F = \{U \rightarrow V, U \rightarrow W, WX \rightarrow Y, WX \rightarrow Z, V \rightarrow X\}$. We need to determine which of the given options can be derived using Armstrong's axioms.
Deriving VW -> YZ (Option A)
This functional dependency can be derived using transitivity and augmentation.
- Start with the given dependency $V \rightarrow X$.
- Using augmentation with $W$ on both sides of $V \rightarrow X$, we get $VW \rightarrow WX$.
- Axiom Used: Augmentation
- We are given $WX \rightarrow Y$ and $WX \rightarrow Z$. Using the union rule, we can combine these into $WX \rightarrow YZ$.
- Axiom Used: Union
- Now, using transitivity on $VW \rightarrow WX$ and $WX \rightarrow YZ$, we derive $VW \rightarrow YZ$.
- Axiom Used: Transitivity
Therefore, $VW \rightarrow YZ$ is derivable.
Deriving WX -> YZ (Option B)
This functional dependency is directly derived using the union rule.
- We are given two functional dependencies: $WX \rightarrow Y$ and $WX \rightarrow Z$.
- Applying the union rule, which states that if $X \rightarrow Y$ and $X \rightarrow Z$, then $X \rightarrow YZ$, we get $WX \rightarrow YZ$.
- Axiom Used: Union
Therefore, $WX \rightarrow YZ$ is derivable.
Deriving VW -> Y (Option D)
This functional dependency can be derived using transitivity and augmentation.
- As shown in the derivation for Option A, we start with $V \rightarrow X$.
- Using augmentation with $W$, we get $VW \rightarrow WX$.
- Axiom Used: Augmentation
- We are given the functional dependency $WX \rightarrow Y$.
- Using transitivity on $VW \rightarrow WX$ and $WX \rightarrow Y$, we derive $VW \rightarrow Y$.
- Axiom Used: Transitivity
Therefore, $VW \rightarrow Y$ is derivable.
Important Questions from Relational Model
- The ‘12 Rules for Relational Database’ were given by
- Which of the following statement(s) is/are FALSE in the context of Relational DBMS ?
I. Views in a database system are important because they help with access control by allowing users to see only a particular subset of the data in the database.
II. E-R diagrams are useful to logically model concepts.
III. An update anomaly is when it is not possible to store information unless some other, unrelated information is stored as well.
IV. SQL is a procedural language. - Let $R(A,B,C,D,E)$ be a relational schema with functional dependency set $F=\{A\rightarrow BC,\ CD\rightarrow E,\ E\rightarrow A\}$.
Which of the following statements is correct? Consider R (A,B,C,D,E) be relation with following dependencies:
C- >F, E- > A, EC- >D, A- >B. Which of the following is a key for R?Consider the relation T1(A,B,C,D,E) with the dependencies {EB ->C,D ->E, EA ->B} and T2(A,B,C,D) with the dependencies {C->A,A->B, A->D}. Which of the following is TRUE?
