Let R be a relation schema, X be a subset of the attributes of R, and let A be an attribute of R. R is in Boyce-Codd normal form if for every FD X ! A that holds over R, one of the following statements is true:

A 2 X; that is, it is a trivial FD, or

X is a super key.

Note that if we are given a set F of FDs, according to this de nition, we must consider each dependency X ! A in the closure F + to determine whether R is in

BCNF. However, we can prove that it is sufficient to check whether the left side of each dependency in F is a super key (by computing the attribute closure and seeing if it includes all attributes of R). Intuitively, in a BCNF relation the only nontrivial dependencies are those in which a key determines some attribute(s). Thus, each tuple can be thought of as an entity or relationship, identified by a key and described by the remaining attributes. Kent puts this colorfully, if a little loosely: \Each attribute must describe [an entity or relationship identified by] the key, the whole key, and nothing but the key." If we use ovals to denote attributes or sets of attributes and draw arcs to indicate FDs, a relation in BCNF has the structure illustrated in Figure, considering just one key for simplicity. (If there are several candidate keys, each candidate key can play the role of KEY in the figure, with the other attributes being the ones not in the chosen candidate key.)