Question

Consider a relation $R(A,B,C,D,E)$ with the following dependencies: \[ A B \rightarrow C, C D \rightarrow E, D E \rightarrow B \] Is $AB$ a candidate key of this relation? If not, is $ABD?$ Explain your answer.

Short answer

'AB' is not a candidate key because it cannot identify every attribute in the relation. However, 'ABD' is a candidate key because it can uniquely identify every possible tuple in the relation.

Step-by-step solution

Test 'AB' as Candidate Key

First let's check whether 'AB' is a candidate key. Doing this involves determining the closure of AB, denoted as $A{B}^{+}$. We start by adding the attributes that are part of the candidate key itself, so $A{B}^{+}$ initially is AB. We then apply the functional dependencies. By applying the dependency $AB\to C$, we get $A{B}^{+}=ABC$. There are no more dependencies that we can apply with the attributes in $A{B}^{+}$, so we stop here.

Analyze Results

We look at the resulting $A{B}^{+}$ and compare it to the original relation $R(ABDE)$. Since AB does not uniquely identify D and E in every possible tuple of the relation, 'AB' is not a candidate key.

Test 'ABD' as Candidate Key

Now let's test whether 'ABD' is a candidate key. Using the same process, we start with $AB{D}^{+}=ABD$. Applying the functional dependencies, the dependency $CD\to E$ can be added because we now have $D$ in $AB{D}^{+}$. So, $AB{D}^{+}=ABDE$. We also apply the dependency $AB\to C$, which doesn't add a new attribute as $C$ is already in $AB{D}^{+}$. No other dependencies can be added.

Analyze Results

We again compare the resulting $AB{D}^{+}$ to the original relation $R(ABCDE)$. This time, $AB{D}^{+}$ contains all the attributes of $R$, so 'ABD' can uniquely identify every possible tuple of the relation and therefore, 'ABD' is a candidate key.

Key concepts

Functional Dependency

Functional dependencies are essential concepts in database normalization. They describe relationships between different attributes in a relation. Functional dependencies can be understood as a form of constraint that determines how one attribute's value is determined by another. The relation "depends" on attributes based on these associations, a critical aspect when understanding data structure integrity.
In the given exercise, we have a set of functional dependencies:

• **AB → C**: This suggests that attribute "C" is determined by the combination of attributes "A" and "B".
• **CD → E**: Attribute "E" depends on the combined values of "C" and "D".
• **DE → B**: Indicates that "B" is determined by "D" and "E".

Understanding these dependencies helps in identifying which attributes are needed to derive others, essential in designing efficient databases. Recognizing these relationships allows us to reduce redundancy and improve data integrity.

Candidate Key

A candidate key is a minimal set of attributes that can uniquely identify a tuple in a relation. It's crucial in defining relationships in databases in order to enforce uniqueness. When assessing whether a set of attributes forms a candidate key, you must determine if it can uniquely identify every tuple without redundancy.
In the exercise, we first considered whether "AB" could be a candidate key. By discussing the closure of "AB", we realized it didn't cover all attributes of the relation "R". Thus, "AB" alone couldn't uniquely identify every tuple, ruling it out as a candidate key.
Next, "ABD" was tested as a potential candidate key. By expanding its closure, "ABD+" successfully captured all attributes in the relation "R". Hence, "ABD" serves as a candidate key. For database design, finding a candidate key is fundamental to correctly structuring the schema and maintaining data integrity.

Closure of Attributes

The closure of a set of attributes, denoted as $X^{+}$ for a set $X$, is a critical concept in database design as it reflects all attributes functionally determined by $X$. Calculating the closure helps us verify if a set of attributes can potentially be a candidate key. It involves applying all possible functional dependencies until no new attributes can be added.
For instance, when examining "AB" in the provided exercise, we calculated $A{B}^{+}$ by starting with {A, B}. By expanding through the given dependencies, $AB\to C$ was applied, resulting in $A{B}^{+}=ABC$. However, this set did not encompass all relation attributes, indicating "AB" was insufficient as a candidate key.
With "ABD", we computed the closure "ABD+" using similar logic. Starting with {A, B, D}, applying $CD\to E$, and recognizing existing enhancements produced $AB{D}^{+}=ABCDE$. This result illustrated total attribute coverage matching the relation, supporting that "ABD" could uniquely identify the data, affirming its status as a candidate key.