Believe it or not, there’s still a small problem ... Consider a version of the suppliers relvar—I’ll call it SCC—with attributes SNO, CITYA, and CITYB. Let SCC be subject to the constraint that for any given supplier, the CITYA and CITYB values are identical. Result: Redundancy! Of course, this is a crazy design, but it’s a possible one, and it would be nice to extend the orthogonality principle to take care of such designs also. And the following final (?) formulation should do the trick (I’ll leave it as an exercise for you to figure out exactly how):
Definition (“final” version): Let R1 and R2 be relvars (not necessarily distinct), and let the JD
{X1,...,Xn} be irreducible with respect to R1. Let there exist some Xi (1 ≤ i ≤ n) and some possibly empty set of attribute renamings on the projection, R1X say, of R1 on Xi that maps R1X into R1Y, say, where R1Y has the same heading as some subset Y (distinct from Xi, if R1 and R2 are one and the same) of the heading of R2. Further, let the projection of R2 on Y be R2Y. Then The Principle of Orthogonal Design is violated by R1 and R2 if and only if there exist restriction conditions c1 and c2, neither of which is identically false, such that the equality dependency (R1Y WHERE c1) = (R2Y WHERE c2) holds.
This version of the principle subsumes all previous versions.