Functional Dependencies#

A functional dependence betweeen two attributes \(X\) and \(Y\) are notated as \(X \rightarrow Y\). This implies for any two rows \(r_1\) and \(r_2\) in the table,

\[ \pi_X(r_1) = \pi_X(r_2) \implies \pi_Y(r_1) = \pi_Y(r_2) \]

Keys#

Functional dependency plays a role on the definition of keys.

  • Superkey: A set of keys \(K\) that determine the rest of the columns in the table.

\[ K \rightarrow \text{all attributes} \]

  • Candidate Key A minimal set of keys \(K\) that such that for \(L \subset K\)

\[ L \not \rightarrow \text{all attributes} \]

  • Primary Key A single candidate key.

Armstrong’s Axiom#

  1. Reflectivity: \(X \supseteq Y \implies X \rightarrow Y\)

  2. Augmentation: \(X \rightarrow Y \implies XZ \rightarrow YZ\)

  3. Transitivity: \(X \rightarrow Y \land Y \rightarrow Z \implies X \rightarrow Z\)

In corollarly,

  • Union and Decomposition: \(X \rightarrow Y \land X \rightarrow Z \iff X \rightarrow YZ\)