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Examples on Dependency Preserving

Q. 

R(ABCD)

F = {A → B, B → C, C → D}

D = {AB, BC, CD}

So, here all FDs of F present at F1 ∪ F2. So, it is dependency preserving.

Q. 

R (ABCD)

F = {A → B, B → C, C → D, D → A, A → C A → D}

D = {AD, BC, CD}

Only left A → B, which can easily derive from F1 ∪ F2.

Or

So, F1 ∪ F2 ⊇ F is true and F1 ∪ F2 ⊆ F  always true then F1 ∪ F2 = F  (true), it is dependency preserving.

Don’t think that at F1 ∪ F2 number of FDs is larger compare to the given F, but actually, both are the same F1 ∪ F2 contain some redundant FDs, 

Both are contained the same number of FDs.

Q. 

R (ABCDE)

F = {AB → CD, C → D, D → E}

D = {ABC, CD, DE}

It is dependency preserving.

Q.

R (ABCDEF)

F = {AB → CD, C → D, D → E, E → F} and D = {AB, CDE, EF}

No Functional Dependencies derive or possible.

But it better to write minimize the way 

Here AB → CD FD not present at F1 ∪ F2 so, it is not dependency preserving.

Q. 

R (ABCDEG)

F = {AB → C, AC → B, AD → E, B → D, BC → A, E → G}

D = {ABC, ACDE, ADG}

Here B → D and E → G FDs are lost so, it is not dependency preserving.