## Description of database usage

** Algebra ** is the original set A with operations defined on it of the form f: An → A, where n is the dimension.

** Calculus ** is a set of rules for handling any symbols.

RI is inherently simpler, but its application is limited by the following circumstances:

• Practically it is not possible - within the framework of the RI - to prove the completeness of the transformation operations performed;

• The large combinatorial complexity of relational schemes is not opened by RI.

All this is possible with the help of RA, the program implementation of which, as can be seen from the very formulation of the problem, are * procedural * programming languages.

** Relational Algebra. ** Let Cl (L) be the set of all closed formulas of the system L.

If the formula φ e CI (L), then we say that the model M satisfies φ (φ • M) if φ is true on M.

Let γ & Icirc; Cl (L). The formula ψ is called the consequence of y (* is deducible * from γ) if ψ • M implies γ = M for any model M.

Any relation constructed correctly using the adopted system of operators and mappings is called an * algebraic expression. * Suppose, as before, that U is a universe (a set of attributes); D is the set of domains; dom is a complete function from U (dom: U → D); R = {Ri, i = 1, p} is the set of relations schemes; d = {ri, i = I, p} is the set of all ratios ri (Ri); θ = {≠, ≤, ≥, & lt ;, & gt;} is the set of binary relations (conditions over domains in D); O is the set of operators (operations) that use attributes from U and relations from θ.

The following operations are distinguished in relational algebra: projection (denoted by π or P in different sources), selection (o or S), connection (J), union (U), difference (DF), division, intersection, Cartesian product ( CP). Let there be two relations R (A, B, C) and P (D, E, F). Unification, intersection and subtraction (difference) are performed over relations of the same arity.

** 1. Unification operation ** U (R, P) - without repeating lines:

** 2. The difference ** (DF (R, P)) - the rows in P are deleted from R:

** 3. Intersection ** R & Ccedil; P are common elements of sets:

** 4. Cartesian product ** (CP (R, P)): each record of the relation R is added to each record of the relation P:

** 5. Projection ** πs (A) (R), where S (A) is the list of domains of the resulting relation from the number of domains of the R relation: columns are selected and ordered and the redundancy from the rows is deleted

** 6. Selection ** (choice) σF (R), where F (Ai, θ, "constant") is the initial ratio * n * -ariness; - attribute of the relation R; θ is a logical condition (& lt ;, & gt ;, =, * ≠, * ≤, ≥, & Cced ;, & Egrave ;, & ugrave;).

** 7. Connection ** JAmB (R, P) = Q = sAmB:

** 8. ** If the compared fields whose names are best made the same, in the resulting relation are considered only once, then talk about the natural connection of the ** (merger) NJ: **

- a list of matching attributes in the original relationship; 1, ..., m is an ordered list of all components of the Cartesian product R x P, except for

If the relation consists of one tuple, then the natural connection results in selection.

9. The division (X, Y) + Y = X.

Operations go to a binary (divisor) and unary (divisible) relations, and the result (private) is obtained by a unary relation. The element x appears in the resulting relation if the pair & lt; x, y & gt; is present in the value of the element y that is present in the divisor for all values. Private - those left-handed components of the divisible whose right-side elements include any component of the divisor.

Let there be

The most commonly used operations are selection (S), projection (P) and connection (J), called SPJ operations.

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