1) Let r be a relation instance with schema R = (A, B, C, D). We define r1 = ‘select A,B,C from r’ and r2 = ‘select A, D from r’. Let s = r1 * r2 where * denotes natural join. Given that the decomposition of r into r1 and r2 is lossy, which one of the following is TRUE? (a) s is subset of r (b) r U s = r (c) r is a subset of s (d) r * s = s Answer (c) Consider the following example with lossy decomposition of r into r1 and r2. We can see that r is a subset of s. 

Table r A 1 1 1 Table r1 A 1 1 1 1 B 10 20 20 1 Table r2 A D 1 1 1 B 20 1000 1001 10 20 C 10 20 100 200 200 C Tables (natural join of r1 and r2) A B C D 100 200 D 100 200 100 200 1000 1000 1001 1000 1000 1001 1001