In relational database theory, the concept of a canonical cover is related to the functional dependencies that exist between attributes in a relation. A functional dependency is a constraint that specifies the relationship between two sets of attributes in a relation.

The canonical cover is a minimal set of functional dependencies that is equivalent to the original set of functional dependencies. It means that if you have a set of functional dependencies and you find their canonical cover, you will have the same closure of attributes as you would have with the original set of functional dependencies.

To find the canonical cover, you typically start with the original set of functional dependencies and perform a series of decomposition and simplification steps. Here is a general procedure:

- Decomposition: If you have a functional dependency of the form A → B, where B is a set of attributes, and A is not a superkey (i.e., it is not a candidate key or a subset of a candidate key), then you decompose it into multiple functional dependencies, each having a single attribute on the right-hand side. For example, if you have A → BC, you would decompose it into A → B and A → C.
- Simplification: Remove any redundant or extraneous dependencies. Redundant dependencies are those that can be inferred from the remaining dependencies, while extraneous dependencies are those that are not necessary to preserve the closure of attributes. You can use Armstrong’s axioms and the closure property to check for redundancy or extraneousness.
- Repeat steps 1 and 2 until you cannot perform any further decomposition or simplification.

The resulting set of functional dependencies after these steps is the canonical cover.

The canonical cover is useful because it provides a concise representation of the functional dependencies in a relation, which can help in various database design and optimization tasks, such as normalization and query optimization.

### Extraneous attributes:

In the context of functional dependencies in a relational database, extraneous attributes refer to attributes that are not necessary to preserve the closure of attributes in a functional dependency. In other words, they are attributes that can be removed from the right-hand side of a functional dependency without affecting the information expressed by the dependency.

To determine if an attribute is extraneous in a functional dependency, you can use the concept of Armstrong’s axioms and the closure property. Here is a general procedure to identify extraneous attributes:

- Start with a functional dependency of the form A → B, where A and B are sets of attributes.
- Remove each attribute individually from the right-hand side of the dependency and calculate the closure of attributes using Armstrong’s axioms and the remaining dependencies. If the closure remains the same, the removed attribute is extraneous and can be safely eliminated.
- Repeat step 2 for each attribute on the right-hand side of the dependency.
- If you find any extraneous attributes, remove them from the right-hand side of the functional dependency. The resulting dependency will express the same information as the original dependency but without the extraneous attributes.

The process of identifying extraneous attributes helps in simplifying functional dependencies and obtaining a more concise and minimal set of dependencies, such as the canonical cover. By eliminating extraneous attributes, you can reduce redundancy and improve the efficiency of database operations and query processing.

**Calculating closure of all FD { B → A, AD → B, AD → C, C → A, C → B, C → D }:**

To calculate the closure of all functional dependencies { B → A, AD → B, AD → C, C → A, C → B, C → D }, you can use Armstrong’s axioms and the closure property. Here’s a step-by-step procedure:

- Start with the given set of functional dependencies.
- Create an initial closure set, which includes all the given functional dependencies.

Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D }

- Apply the closure property iteratively to add more attributes to the closure set until it no longer changes.

Iteration 1:

- Check if B → A implies any new functional dependencies. Since there are no dependencies with B on the left-hand side, no new dependencies are added in this step.

Iteration 2:

- Check if AD → B implies any new functional dependencies.
- Using AD → B, we can infer AD → AB (by augmentation).
- Update the closure set with the inferred functional dependency: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB }

Iteration 3:

- Check if AD → C implies any new functional dependencies.
- Using AD → C, we can infer AD → AC (by augmentation).
- Update the closure set with the inferred functional dependency: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB, AD → AC }

Iteration 4:

- Check if C → A implies any new functional dependencies.
- Using C → A, we can infer C → AB (by transitivity, since we have AD → AB from Iteration 2).
- Update the closure set with the inferred functional dependency: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB, AD → AC, C → AB }

Iteration 5:

- Check if C → B implies any new functional dependencies.
- Using C → B, we can infer C → AB (by transitivity, since we have C → A and B → A from Closure).
- Update the closure set with the inferred functional dependency: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB, AD → AC, C → AB }

Iteration 6:

- Check if C → D implies any new functional dependencies. No new dependencies are inferred in this step.

The closure set does not change in Iteration 6, so the process terminates.

The resulting closure of all functional dependencies is: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB, AD → AC, C → AB }

This closure set represents all the attributes that can be functionally determined by the given set of functional dependencies.

**Calculating closure of all FD { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z }:**

To calculate the closure of all functional dependencies { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z }, we can use Armstrong’s axioms and the closure property. Here’s a step-by-step procedure:

- Start with the given set of functional dependencies.
- Create an initial closure set, which includes all the given functional dependencies.

Closure: { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z }

- Apply the closure property iteratively to add more attributes to the closure set until it no longer changes.

Iteration 1:

- Check if W → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 2:

- Check if Y → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 3:

- Check if Z → W implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 4:

- Check if Z → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 5:

- Check if Z → Y implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 6:

- Check if WY → Z implies any new functional dependencies.
- Using WY → Z, we can infer WY → X (by augmentation) and then WY → XY (by transitivity, since we have Y → X from Closure).
- Update the closure set with the inferred functional dependencies: Closure: { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z, WY → X, WY → XY }

Iteration 7:

- Check if WY → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 8:

- Check if WY → XY implies any new functional dependencies. No new dependencies are inferred in this step.

The closure set does not change in Iterations 7 and 8, so the process terminates.

The resulting closure of all functional dependencies is: Closure: { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z, WY → X, WY → XY }

This closure set represents all the attributes that can be functionally determined by the given set of functional dependencies.

**Calculating closure of all FD { V → W, VW → X, Y → V, Y → X, Y → Z }:**

To calculate the closure of all functional dependencies { V → W, VW → X, Y → V, Y → X, Y → Z }, we can use Armstrong’s axioms and the closure property. Here’s a step-by-step procedure:

- Start with the given set of functional dependencies.
- Create an initial closure set, which includes all the given functional dependencies.

Closure: { V → W, VW → X, Y → V, Y → X, Y → Z }

- Apply the closure property iteratively to add more attributes to the closure set until it no longer changes.

Iteration 1:

- Check if V → W implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 2:

- Check if VW → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 3:

- Check if Y → V implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 4:

- Check if Y → X implies any new functional dependencies. No new dependencies are inferred in this step.

Iteration 5:

- Check if Y → Z implies any new functional dependencies. No new dependencies are inferred in this step.

The closure set does not change in any of the iterations, so the process terminates.

The resulting closure of all functional dependencies is: Closure: { V → W, VW → X, Y → V, Y → X, Y → Z }

This closure set represents all the attributes that can be functionally determined by the given set of functional dependencies.

**CONCLUSION:**

In conclusion, we have discussed the concept of calculating the closure of functional dependencies. The closure of functional dependencies represents all the attributes that can be determined by the given set of dependencies. By applying Armstrong’s axioms and the closure property iteratively, we can determine the closure set. We applied this process to the given sets of functional dependencies and obtained the closure sets for each case.

For the set { B → A, AD → B, AD → C, C → A, C → B, C → D }, the closure set is: Closure: { B → A, AD → B, AD → C, C → A, C → B, C → D, AD → AB, AD → AC, C → AB }

For the set { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z }, the closure set is: Closure: { W → X, Y → X, Z → W, Z → X, Z → Y, WY → Z, WY → X, WY → XY }

For the set { V → W, VW → X, Y → V, Y → X, Y → Z }, the closure set is: Closure: { V → W, VW → X, Y → V, Y → X, Y → Z }

The closure sets provide a comprehensive understanding of the functional dependencies and the attributes that are determined by them.