Functional Dependency (Relational Algebra)

relational-algebra

Definition

Functional Dependency (Relational Algebra)

Let $\mathcal{R}$ be a schema of a relation $R$ and $\alpha ,\beta \subseteq \mathcal{R}$ (subsets of attributes of $\mathcal{R}$).

A functional dependency $\alpha \to \beta$ in $\mathcal{R}$ exists iff for all legal tuples $R$ of $\mathcal{R}$ the following holds:

$$\forall r,s\in R:r.\alpha =s.\alpha ⟹r.\beta =s.\beta$$

Thus, the $\alpha$-values uniquely identify the $\beta$-values. In other words: The $\alpha$-values determine the $\beta$-values functionally.

Trivial

Definition

Trivial Functional Dependency (Relational Algebra)

A functional dependency $\alpha \to \beta$ is called trivial iff $\beta \subseteq \alpha$.

Example: $\{a,b,c\}\to \{a\}$ is trivial. See Armstrong’s axioms.

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Set Equivalence

Two sets of functional dependencies $F$ and $G$ are equivalent if their closures equal:

$$F\equiv G:⟺F^{+}=G^{+}$$