Relation (Relational Algebra)

relational-algebra

Definition

Relation (Relational Algebra)

Let $D_1,D_2,\dots ,D_n$ be domains. A relation $R\subseteq D_1\times D_2\times \cdots \times D_n$ is a set of tuples based on the Cartesian product of multiple, possibly identical, domains.

Example:

$$phone\_book\subseteq \text{string}\times \text{string}\times \text{integer}$$

Schema

Definition

Schema (Relational Algebra)

A schema $\mathcal{R}$ or $\text{sch}(R)$ of a relation $R$ describes the structure of data in a relation $R$ and is denoted as $\mathrm{sch}(R)$ or as $\mathcal{R}$.

The notation is defined as follows:

$$R(A_1:D_1,A_2:D_2,\dots ,A_n:D_n)$$

where $A_i$ is an attribute of with domain $D_i$.

Example:

$$phone\_book(name:\text{string},street:\text{string},number:\text{integer})$$

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Lossless Decomposition

Lossless Decomposition

A decomposition of a relation $R$ into two relations $R_1={\pi }_{{\mathcal{R}}_1}(R)$, $R_2={\pi }_{{\mathcal{R}}_2}(R)$ is called lossless if all attributes of $\mathcal{R}$ are preserved. Thus, $R$ can be reconstructed using a natural join between $R_1$ and $R_2$:

$$R=R_1\bowtie R_2$$

Lossless decomposition can be defined in the context of functional dependencies. A decomposition is called lossless if at least one of the following functional dependencies can be derived:

1. $({\mathcal{R}}_1\cap {\mathcal{R}}_2)\to {\mathcal{R}}_1\in F_{\mathcal{R}}^{+}$ (closure).
2. $({\mathcal{R}}_1\cap {\mathcal{R}}_2)\to {\mathcal{R}}_2\in F_{\mathcal{R}}^{+}$ (closure).