R = {A, B, C, D, E} into

R1 = {A, B, C} and

R2 = {A, D, E}

Show that this decomposition is lossless join decomposition if the following set of functional dependencies hold

I. The decomposition of R into R1 and R2 can be called as lossless join decomposition if any one of the following conditions holds true:

1. R1 ∩ R2 → R1

2. R1 ∩ R2 → R2

II. R1 = (A, B, C), R2 = (A, D, E) ….Given

Hence, R1 ∩ R2 = A

III. It can also be proved that A is a candidate key as follows:

1. Since A → BC, we can conclude: A → B and A → C.
2. Since A → B and B → D, A → D ……. (Decomposition, transitive)
3. Since A → CD and CD → E, A → E …… (Union, decomposition, transitive)
4. Since A → A… (Reflexive)
5. A → ABCDE from the above steps … (Union) Hence, R1 ∩ R2 → R1 Thus, the decomposition mentioned is a lossless decomposition.