Functional dependency

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In relational database theory, a functional dependency is the following constraint between two attribute sets in a relation: Given a relation R and attribute sets ${\displaystyle X,Y\subseteq R}$, X is said to functionally determine Y (written X → Y) if each X value is associated with precisely one Y value. R is then said to satisfy the functional dependency X → Y. Equivalently, the projection ${\displaystyle \Pi _{X,Y}R}$ is a function, that is, Y is a function of X.^[1][2] In simple words, if the values for the X attributes are known (say they are x), then the values for the Y attributes corresponding to x can be determined by looking them up in any tuple of R containing x. Customarily X is called the determinant set and Y the dependent set. A functional dependency FD: X → Y is called trivial if Y is a subset of X.

In other words, a dependency FD: X → Y means that the values of Y are determined by the values of X. Two tuples sharing the same values of X will necessarily have the same values of Y.

The determination of functional dependencies is an important part of designing databases in the relational model, and in database normalization and denormalization. A simple application of functional dependencies is Heath's theorem; it says that a relation R over an attribute set U and satisfying a functional dependency X → Y can be safely split in two relations having the lossless-join decomposition property, namely into ${\displaystyle \Pi _{XY}(R)\bowtie \Pi _{XZ}(R)=R}$ where Z = U − XY are the rest of the attributes. (Unions of attribute sets are customarily denoted by there juxtapositions in database theory.) An important notion in this context is a candidate key, defined as a minimal set of attributes that functionally determine all of the attributes in a relation. The functional dependencies, along with the attribute domains, are selected so as to generate constraints that would exclude as much data inappropriate to the user domain from the system as possible.

A notion of logical implication is defined for functional dependencies in the following way: a set of functional dependencies ${\displaystyle \Sigma }$ logically implies another set of dependencies ${\displaystyle \Gamma }$, if any relation R satisfying all dependencies from ${\displaystyle \Sigma }$ also satisfies all dependencies from ${\displaystyle \Gamma }$; this is usually written ${\displaystyle \Sigma \models \Gamma }$. The notion of logical implication for functional dependencies admits a sound and complete finite axiomatization, known as Armstrong's axioms.