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It is not suf cient to consider the given set of functional dependencies Rather, we need to consider all functional dependencies that hold We shall see that, given a set F of functional dependencies, we can prove that certain other functional dependencies hold We say that such functional dependencies are logically implied by F More formally, given a relational schema R, a functional dependency f on R is logically implied by a set of functional dependencies F on R if every relation instance r(R) that satis es F also satis es f Suppose we are given a relation schema R = (A, B, C, G, H, I) and the set of functional dependencies A B A C CG H CG I B H The functional dependency A H

The McGraw Hill Companies, 2001

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is logically implied That is, we can show that, whenever our given set of functional dependencies holds on a relation, A H must also hold on the relation Suppose that t1 and t2 are tuples such that t1 [A] = t2 [A] Since we are given that A B, it follows from the de nition of functional dependency that t1 [B] = t2 [B] Then, since we are given that B H, it follows from the de nition of functional dependency that t1 [H] = t2 [H] Therefore, we have shown that, whenever t1 and t2 are tuples such that t1 [A] = t2 [A], it must be that t1 [H] = t2 [H] But that is exactly the de nition of A H Let F be a set of functional dependencies The closure of F, denoted by F + , is the set of all functional dependencies logically implied by F Given F, we can compute F + directly from the formal de nition of functional dependency If F were large, this process would be lengthy and dif cult Such a computation of F + requires arguments of the type just used to show that A H is in the closure of our example set of dependencies Axioms, or rules of inference, provide a simpler technique for reasoning about functional dependencies In the rules that follow, we use Greek letters ( , , , ) for sets of attributes, and uppercase Roman letters from the beginning of the alphabet for individual attributes We use to denote We can use the following three rules to nd logically implied functional dependencies By applying these rules repeatedly, we can nd all of F + , given F This collection of rules is called Armstrong s axioms in honor of the person who rst proposed it Re exivity rule If is a set of attributes and , then holds Augmentation rule If holds and is a set of attributes, then holds Transitivity rule If holds and holds, then holds Armstrong s axioms are sound, because they do not generate any incorrect functional dependencies They are complete, because, for a given set F of functional dependencies, they allow us to generate all F + The bibliographical notes provide references for proofs of soundness and completeness Although Armstrong s axioms are complete, it is tiresome to use them directly for the computation of F + To simplify matters further, we list additional rules It is possible to use Armstrong s axioms to prove that these rules are correct (see Exercises 78, 79, and 710) Union rule If holds and holds, then holds Decomposition rule If holds, then holds and holds Pseudotransitivity rule If holds and holds, then holds.