The following functional dependencies are given:AB -> CD, AF —> D, DE --> F, C —> G, F —› E, G --> AWhich one of the following options is false?{CF}+ = {ACDEFG}{BG}+ = {ABCDG}
Question
The following functional dependencies are given:AB -> CD, AF —> D, DE --> F, C —> G, F —› E, G --> AWhich one of the following options is false?{CF}+ = {ACDEFG}{BG}+ = {ABCDG}
Solution
To answer this question, we need to calculate the closure of the given sets of attributes according to the provided functional dependencies.
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For {CF}+, we start with CF.
- From F —› E, we add E to get CEF.
- From DE --> F, there's no new attribute to add because F is already in the set.
- From AF —> D, there's no new attribute to add because D is already in the set.
- From AB -> CD, there's no new attribute to add because both C and D are already in the set.
- From C —> G, we add G to get CEFG.
- From G --> A, we add A to get ACEFG. So, {CF}+ = {ACEFG}. This does not match with the given {ACDEFG}. So, the first option is false.
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For {BG}+, we start with BG.
- From G --> A, we add A to get ABG.
- From AB -> CD, we add CD to get ABCDG.
- From AF —> D, there's no new attribute to add because D is already in the set.
- From DE --> F, there's no new attribute to add because neither D nor E is in the set.
- From F —› E, there's no new attribute to add because neither F nor E is in the set.
- From C —> G, there's no new attribute to add because G is already in the set. So, {BG}+ = {ABCDG}. This matches with the given {ABCDG}. So, the second option is true.
Therefore, the first option "{CF}+ = {ACDEFG}" is false.
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