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In a schema with attributes A, B, C, D and E following set of functional dependencies are given
A → B A → C CD → E B → D E → A
Q. Which of the following functional dependencies is NOT implied by the above set?
- a)CD → AC
- b)BD → CD
- c)BC → CD
- d)AC → BC
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In a schema with attributes A, B, C, D and E following set of function...
option (b)
For every options given, find the closure set of left side of each FD. If the closure set of left side contains the right side of the FD, then the particular FD is implied by the given set.
Option (a): Closure set of CD = CDEAB. Therefore CD->AC can be derived from the given set of FDs.
Option (c): Closure set of BC = BCDEA. Therefore BC->CD can be derived from the given set of FDs.
Option (d): Closure set of AC = ACBDE. Therefore AC->BC can be derived from the given set of FDs.
Option (b): Closure set of BD = BD. Therefore BD->CD cannot be derived from the given set of FDs.
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In a schema with attributes A, B, C, D and E following set of function...
Given Information
In a schema with attributes A, B, C, D, and E, the following set of functional dependencies are given:
A B
A C
CD E
B D
E A
The question asks to identify the functional dependency that is NOT implied by the above set.
Solution
To identify the functional dependency that is NOT implied by the given set, we need to check each option one by one and see if it can be derived from the given set of functional dependencies.
Option A: CD AC
To derive CD AC, we can use the transitive rule of functional dependency. Since we have the functional dependency A C and CD E, we can derive CD AC as follows:
CD E (given)
A C (given)
Therefore, CD AC (by transitive rule)
Option B: BD CD
To derive BD CD, we need to have a functional dependency between B and C. However, there is no such dependency given in the set. Therefore, option B is NOT implied by the given set.
Option C: BC CD
To derive BC CD, we can use the transitive rule of functional dependency. Since we have the functional dependency CD E and E A, we can derive BC AD as follows:
CD E (given)
E A (given)
Therefore, CD A (by transitive rule)
A B (given)
Therefore, BC CD (by transitive rule)
Option D: AC BC
To derive AC BC, we can use the transitive rule of functional dependency. Since we have the functional dependency A C and C D, we can derive AC BD as follows:
A C (given)
C D (given)
Therefore, A D (by transitive rule)
B D (given)
Therefore, AC BD (by transitive rule)
Therefore, the correct answer is option B: BD CD.
In a schema with attributes A, B, C, D, and E, the following set of functional dependencies are given:
A B
A C
CD E
B D
E A
The question asks to identify the functional dependency that is NOT implied by the above set.
Solution
To identify the functional dependency that is NOT implied by the given set, we need to check each option one by one and see if it can be derived from the given set of functional dependencies.
Option A: CD AC
To derive CD AC, we can use the transitive rule of functional dependency. Since we have the functional dependency A C and CD E, we can derive CD AC as follows:
CD E (given)
A C (given)
Therefore, CD AC (by transitive rule)
Option B: BD CD
To derive BD CD, we need to have a functional dependency between B and C. However, there is no such dependency given in the set. Therefore, option B is NOT implied by the given set.
Option C: BC CD
To derive BC CD, we can use the transitive rule of functional dependency. Since we have the functional dependency CD E and E A, we can derive BC AD as follows:
CD E (given)
E A (given)
Therefore, CD A (by transitive rule)
A B (given)
Therefore, BC CD (by transitive rule)
Option D: AC BC
To derive AC BC, we can use the transitive rule of functional dependency. Since we have the functional dependency A C and C D, we can derive AC BD as follows:
A C (given)
C D (given)
Therefore, A D (by transitive rule)
B D (given)
Therefore, AC BD (by transitive rule)
Therefore, the correct answer is option B: BD CD.
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