Decompostion with loss of information shown by junction F={A → B; B → C; C → D}


R is decomposed in R¹(A,B) &amp; R²(A,D) &amp; R³(C, D)

Is it without loss of information?

I would have said yes as far as with the following array



A → B x²²= b

B → C x¹³=x²³=c

Therfore we should see that it is without loss of information (we should have a straight line of determinated data). However, the answer says that it clearly lacks informations and in order to show that it is with loss of information, one should find an instance r such that 

r ≠ r¹⋈ r²⋈ r³

I know that I have then to do some arrays and some junctions but I don't even know how to start the arrays...
The trick to solve this exercise it to repeat as much as possible the elements of the table. Since A is the key, you cannot repeat A, but you can, for instance, repeat two times B and C and three times D. Here is a very simple and short example (note that the functional dependencies are respected):

r =
+---+---+---+---+
| A | B | C | D |
+---+---+---+---+
|a1 |b1 |c1 |d1 |
|a2 |b1 |c1 |d1 |
|a3 |b2 |c2 |d1 |
+---+---+---+---+


And since this is an exercise, I will leave to you the task of projecting r over R1, R2 and R3, and discovering how the natural join of the three relations differs from the original relation r.

Finally a terminological note: a relation instance is not an array! There is no order in the rows or the columns of a relation, while in an array the order is essential.