NationMaster - Encyclopedia: Relational algebra

Relational algebra, an offshoot of first-order logic, is a set of relations closed under operators. Operators operate on one or more relations to yield a relation. Relational algebra is a part of computer science. First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In mathematics, an n-ary relation (or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ...

Relation algebra in pure mathematics is an algebraic structure, relevant to mathematical logic and set theory. In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

Introduction

Relational algebras received little attention until the publication of E.F. Codd's relational model of data in 1970. Codd proposed such an algebra as a basis for database query languages. The first query language to be based on Codd's algebra was ISBL, and this pioneering work has been acclaimed by many authorities as having shown the way to make Codd's idea into a useful language. Business System 12 was a short-lived industry-strength relational DBMS that followed the ISBL example. In 1998 Chris Date and Hugh Darwen proposed a language called Tutorial D intended for use in teaching relational database theory, and its query language also draws on ISBL's ideas. Rel is an implementation of Tutorial D. Even the query language of SQL is loosely based on a relational algebra, though the operands in SQL (tables) are not exactly relations and several useful theorems about the relational algebra do not hold in the SQL counterpart (arguably to the detriment of optimisers and/or users). Edgar Frank Ted Codd (August 23, 1923 â€“ April 18, 2003) was a British computer scientist who made seminal contributions to the theory of relational databases. ... The relational model for database management is a database model based on predicate logic and set theory. ... ISBL is the relational algebra notation that was invented for PRTV, one of the earliest database management systems to implement E.F. Codds relational model of data. ... Business System 12, or simply BS12, was one of was the first fully relational database management systems, designed and implemented by IBMs UK Bureau Service subsidiary. ... Christopher J. Date is an independent author, lecturer, researcher, and consultant, specializing in relational database technology. ... Hugh Darwen, employee of IBM UK from 1967 to 2004, has been involved in the history of the relational model since the beginning. ... Rel is an Open Source true relational database management system that implements a significant portion of Chris Date and Hugh Darwens Tutorial D query language. ... SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...

Because a relation is interpreted as the extension of some predicate, each operator of a relational algebra has a counterpart in predicate calculus. For example, the natural join is a counterpart of logical AND ( $land$ ). If relations R and S represent the extensions of predicates p1 and p2, respectively, then the natural join of R and S (R $bowtie$ S) is a relation representing the extension of the predicate p1 $land$ p2. The extension of a predicate is the set of true propositions that can be formed by substituting a term for each of its free variables. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

The exact set of operators may differ per definition and also depends on whether the unlabeled relational model (that uses mathematical relations) or the labeled relational model (that uses the labeled generalization of mathematical relations) is used. We will assume the labeled case here as this was the kind that Codd proposed and is thought by some to have been his most important innovation, as it eliminates dependence on an ordering to the attributes of a relation. Under this model we assume that tuples are partial functions from attribute names to values. The attribute a of a tuple t is denoted in this article as t(a). In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

It is important to realise that Codd's algebra is not in fact complete with respect to first-order logic. Had it been so, certain insurmountable computational difficulties would have arisen for any implementation of it. To overcome these difficulties, he restricted the operands to finite relations only and also proposed restricted support for negation (NOT) and disjunction (OR). Analogous restrictions are found in many other logic-based computer languages. Codd defined the term relational completeness to refer to a language that is complete with respect to first-order predicate calculus apart from the restrictions he proposed. In practice the restrictions have no adverse effect on the applicability of his relational algebra for database purposes. First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

Primitive operations

As in any algebra, some operators are primitive and the others, being definable in terms of the primitive ones, are derived. It is useful if the choice of primitive operators parallels the usual choice of primitive logical operators. Although it is well known that the usual choice in logic of AND, OR and NOT is somewhat arbitrary, Codd made a similar arbitrary choice for his algebra.

The six primitive operators of Codd's algebra are the selection, the projection, the Cartesian product (also called the cross product or cross join), the set union, the set difference, and the rename. (Actually, Codd omitted the rename, but the compelling case for its inclusion was shown by the inventors of ISBL.) These six operators are fundamental in the sense that none of them can be omitted without losing expressive power. Many other operators have been defined in terms of these six. Among the most important are set intersection, division, and the natural join. In fact ISBL made a compelling case for replacing the Cartesian product by the natural join, of which the Cartesian product is a degenerate case. In relational algebra, a selection is a unary operation written as or where: and are attribute names is a binary operation in the set is a value constant is a relation The selection selects all those tuples in for which holds between the and the attribute. ... In relational algebra, a projection is a unary operation written as where is a set of attribute names. ... In mathematics, the Cartesian product is a direct product of sets. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... Wikipedia does not yet have an article with this exact name. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...

Altogether, the operators of relational algebra have identical expressive power to that of domain relational calculus or tuple relational calculus. However, for the reasons given in the Introduction above, relational algebra has strictly less expressive power than that of first-order predicate calculus without function symbols. Relational algebra actually corresponds to a subset of first-order logic that is Horn clauses without recursion and negation. In computer science, domain relational calculus (DRC) is a calculus that was introduced by Edgar F. Codd as a declarative database query language for the relational data model. ... The tuple calculus is a calculus that was introduced by Edgar F. Codd as part of the relational model in order to give a declarative database query language for this data model. ... First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In logic, and in particular in propositional calculus, a Horn clause is a proposition of the general type (p and q and . ...

Set operators

Although three of the six basic operators are taken from set theory, there are additional constraints that are present in their relational algebra counterparts: For set union and set difference, the two relations involved must be union-compatible—that is, the two relations must have the same set of attributes. As set intersection can be defined in terms of set difference, the two relations involved in set intersection must also be union-compatible. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

The Cartesian product is defined differently from the one defined in set theory in the sense that tuples are considered to be 'shallow' for the purposes of the operation. That is, unlike in set theory, where the Cartesian product of a n-tuple by an m-tuple is a set of 2-tuples, the Cartesian product in relational algebra has the 2-tuple "flattened" into an n+m-tuple. More formally, R × S is defined as follows: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

R $times$ S = {r $cup$ s| r $in$ R, s $in$ S}

In addition, for the Cartesian product to be defined, the two relations involved must have disjoint headers — that is, they must not have a common attribute name. Look up header in Wiktionary, the free dictionary. ...

Projection

Main article: Projection (relational algebra)

A projection is a unary operation written as $pi_{a_1, ...,a_n}( R )$ where $a 1,..., a n$ is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in $R$ are restricted to the set ${a 1,..., a n}$ . In relational algebra, a projection is a unary operation written as where is a set of attribute names. ... In mathematics, a unary operation is an operation with only one operand. ... An attribute is the following: Generally, an attribute is an abstraction characteristic of an entity In database management, an attribute is a property inherent in an entity or associated with that entity for database purposes. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

Selection

Main article: Selection (relational algebra)

A generalized selection is a unary operation written as $sigma_varphi(R)$ where $varphi$ is a propositional formula that consists of atoms as allowed in the normal selection and the logical operators $and$ (and), $or$ (or) and $lnot$ (negation). This selection selects all those tuples in $R$ for which $varphi$ holds. In relational algebra, a selection is a unary operation written as or where: and are attribute names is a binary operation in the set is a value constant is a relation The selection selects all those tuples in for which holds between the and the attribute. ... In mathematics, a unary operation is an operation with only one operand. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... Properties In chemistry and physics, an atom (Greek á¼„Ï„Î¿Î¼Î¿Ï‚ or Ã¡tomos meaning indivisible) is the smallest particle still characterizing a chemical element. ... In relational algebra, a selection is a unary operation written as or where: and are attribute names is a binary operation in the set is a value constant is a relation The selection selects all those tuples in for which holds between the and the attribute. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... Negation (i. ... In mathematics, a tuple is a finite sequence (also known as an ordered list) of objects, each of a specified type. ...

Rename

Main article: Rename (relational algebra)

A rename is a unary operation written as $ρ a / b (R)$ where the result is identical to $R$ except that the $b$ field in all tuples is renamed to an $a$ field.This simply used to rename the attribute of a relation or the relation itself. In relational algebra, a rename is a unary operation written as where: and are attribute names is a relation The result is identical to except that the field in all tuples is renamed to an field. ... In mathematics, a unary operation is an operation with only one operand. ...

Joins and join-like operators

Natural join

Natural join is a dyadic operator that is written as R $bowtie$ S where R and S are relations. The result of the natural join is the set of all combinations of tuples in R and S that are equal on their common attribute names. For an example consider the tables Employee and Dept and their natural join: In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales

*Dept*
DeptName	Manager
Finance	George
Sales	Harriet
Production	Charles

*Employee* $bowtie$ *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	George
Sally	2241	Sales	Harriet
George	3401	Finance	George
Harriet	2202	Sales	Harriet

Join is another term for relation composition; in category theory, the join is precisely the fiber product. In logic and mathematics, the composition of relations is the generalization of the composition of functions. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...

The natural join is arguably one of the most important operators since it is the relational counterpart of logical AND. Note carefully that if the same variable appears in each of two predicates that are connected by AND, then that variable stands for the same thing and both appearances must always be substituted by the same value. In particular, natural join allows the combination of relations that are associated by a foreign key. For example, in the above example a foreign key probably holds from Employee.DeptName to Dept.DeptName and then the natural join of Employee and Dept combines all employees with their departments. Note that this works because the foreign key holds between attributes with the same name. If this is not the case such as in the foreign key from Dept.manager to Emp.emp-number then we have to rename these columns before we take the natural join. Such a join is sometimes also referred to as an equijoin (see θ-join). In the context of relational databases, a foreign key is a referential constraint between two tables[1]. The foreign key identifies a column or a set of columns in one (referencing) table that refers to a column or set of columns in another (referenced) table. ...

More formally the semantics of the natural join is defined as follows:

R $bowtie$ S = { t $cup$ s : t $in$ R, s $in$ S, fun (t $cup$ s) }

where fun(r) is a predicate that is true for a binary relation r iff r is a functional binary relation. It is usually required that R and S must have at least one common attribute, but if this constraint is omitted then in that special case the natural join becomes exactly the Cartesian product as defined above. In mathematics, a predicate is either a relation or the boolean-valued function that amounts to the characteristic function or the indicator function of such a relation. ... When someone sincerely agrees with an assertion, they might claim that it is the truth. ... In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

The natural join can be simulated with Codd's primitives as follows. Assume that b₁,...,b_m are the attribute names common to R, S, a₁,...,a_n are the attribute names unique to R and c₁,...,c_k are the attribute unique to S. Furthermore assume that the attribute names d₁,...,d_m are neither in R nor in S. In a first step we can now rename the common attribute names in S: : S' := ρ_d₁/b₁(...ρ_{d_m/b_m}( S)...) Then we take the Cartesian product and select the tuples that are to be joined: : T := σ_b₁=d₁(...σ_{b_m=d_m}(R × S')...) Finally we take a projection to get rid of the renamed attributes: : U := π_{a₁,...,a_n,b₁,...,b_m,c₁,...,c_k}(T)

θ-join and equijoin

Consider tables Car and Boat which list models of cars and boats and their respective prices. Suppose a customer wants to buy a car and a boat, but she doesn't want to spend more money for the boat than for the car. The θ-join on the relation CarPrice ≥ BoatPrice produces a table with all the possible options.

*Car*
CarModel	CarPrice
CarA	20'000
CarB	30'000
CarC	50'000

*Boat*
BoatModel	BoatPrice
Boat1	10'000
Boat2	40'000
Boat3	60'000

$begin{matrix} Car bowtie Boat scriptstyle CarPrice geq BoatPrice end{matrix}$
CarModel	CarPrice	BoatModel	BoatPrice
CarA	20'000	Boat1	10'000
CarB	30'000	Boat1	10'000
CarC	50'000	Boat1	10'000
CarC	50'000	Boat2	40'000

If we want to combine tuples from two relations where the combination condition is not simply the equality of shared attributes then it is convenient to have a more general form of join operator, which is the θ-join (or theta-join). The θ-join is a dyadic operator that is written as $begin{matrix} R bowtie S a theta bend{matrix}$ or $begin{matrix} R bowtie S a theta vend{matrix}$ where a and b are attribute names, θ is a binary relation in the set {<, ≤, =, >, ≥}, v is a value constant, and R and S are relations. The result of this operation consists of all combinations of tuples in R and S that satisfy the relation θ. The result of the θ-join is defined only if the headers of S and R are disjoint, that is, do not contain a common attribute. In mathematics, a binary relation (or a dyadic relation) is an arbitrary association of elements of one set with elements of another (perhaps the same) set. ...

The simulation of this operation in the fundamental operations is therefore as follows:

R $bowtie$ _φ S = σ_φ(R × S)

In case the operator θ is the equality operator (=) then this join is also called an equijoin.

Note, however, that a computer language that supports the natural join and rename operators does not need θ-join as well, as this can be achieved by selection from the result of a natural join (which degenerates to Cartesian product when there are no shared attributes).

Semijoin

The semijoin is joining similar to the natural join and written as R $ltimes$ S where R and S are relations. The result of the semijoin is only the set of all tuples in R for which there is a tuple in S that is equal on their common attribute names. For an example consider the tables Employee and Dept and their semi join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Production

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* $ltimes$ *Dept*
Name	EmpId	DeptName
Sally	2241	Sales
Harriet	2202	Production

More formally the semantics of the semijoin is defined as follows:

R $ltimes$ S = { t : t $in$ R, s $in$ S, fun (t $cup$ s) }

where fun(r) is as in the definition of natural join.

The semijoin can be simulated using the natural join as follows. Assume that a₁,...,a_n are the attribute names of R, then it holds that:

R $ltimes$ S =

Π

_{a₁,..,a_n}(R $bowtie$ S)

Since we can simulate the natural join with the basic operators it follows that this also holds for the semijoin.

Antijoin

The antijoin, written as R $triangleright$ S where R and S are relations, is similar to the natural join, but the result of an antijoin is only those tuples in R for which there is NOT a tuple in S that is equal on their common attribute names.

For an example consider the tables Employee and Dept and their antijoin:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* $triangleright$ *Dept*
Name	EmpId	DeptName
Harry	3415	Finance
George	3401	Finance

The antijoin is formally defined as follows:

R $triangleright$ S = { t : t $in$ R $and$ $negexists$ s $in$ S : fun (t $cup$ s) }

R $triangleright$ S = { t : t $in$ R, there is no tuple s of S that satisfies fun (t $cup$ s) }

where fun(r) is as in the definition of natural join.

The antijoin can also be defined as the complement of the semijoin, as follows: In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

R $triangleright$ S = R - R $ltimes$ S

Given this, the antijoin is sometimes called the anti-semijoin, and the antijoin operator is sometimes written as semijoin symbol with a bar above it, instead of $triangleright$ .

Division

The division is a binary operation that is written as R ÷ S. The result consists of the restrictions of tuples in R to the attribute names unique to R, i.e., in the header of R but not in the header of S, for which it holds that all their combinations with tuples in S are present in R. For an example see the tables CompletedTask, DBProject and their division:

*Completed*
Student	Task
Fred	Database1
Fred	Database2
Fred	Compiler1
Eugene	Database1
Eugene	Compiler1
Sara	Database1
Sara	Database2

*DBProject*
Task
Database1
Database2

*Completed* ÷ *DBProject*
Student
Fred
Sara

If DBProject contains all the tasks of the Database project then the result of the division above contains exactly all the students that have completed the Database project.

More formally the semantics of the division is defined as follows:

R ÷ S = { t[a₁,...,a_n] : t $in$ R $wedge$ $forall$ s $in$ S ( (t[a₁,...,a_n] $cup$ s) $in$ R) }

where {a₁,...,a_n} is the set of attribute names unique to R and t[a₁,...,a_n] is the restriction of t to this set. It is usually required that the attribute names in the header of S are a subset of those of R because otherwise the result of the operation will always be empty.

The simulation of the division with the basic operations is as follows. We assume that a₁,...,a_n are the attribute names unique to R and b₁,...,b_m are the attribute names of S. In the first step we project R on its unique attribute names and construct all combinations with tuples in S:

T := π_{a₁,...,a_n}(R) × S

In the prior example, T would represent a table such that every Student (because Student is the unique key / attribute of the Completed table) is combined with every given Task. So Eugene, for instance, would have two rows, Eugene -> Database1 and Eugene -> Database2 in T.

In the next step we subtract R from this relation:

U := T - R

Note that in U we have the possible combinations that "could have" been in R, but weren't. So if we now take the projection on the attribute names unique to R then we have the restrictions of the tuples in R for which not all combinations with tuples in S were present in R:

V := π_{a₁,...,a_n}(U)

So what remains to be done is take the projection of R on its unique attribute names and subtract those in V:

W := π_{a₁,...,a_n}(R) - V

Outer joins

It has been suggested that this section be split into a new article entitled outer join. (Discuss)

Whereas the result of a join (or inner join) consists of tuples formed by combining matching tuples in the two operands, an outer join contains those tuples and additionally some tuples formed by extending an unmatched tuple in one of the operands by "fill" values for each of the attributes of the other operand. Image File history File links Splitsection. ... A join combines records from two or more tables in a relational database. ...

The operators defined in this section assume the existence of a null value, ω, which we do not define, to be used for the fill values. It should not be assumed that this is the NULL defined for the database language SQL, nor should it be assumed that ω is a mark rather than a value, nor should it be assumed that the controversial three-valued logic is introduced by it. Look up null in Wiktionary, the free dictionary. ... SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ...

Three outer join operators are defined: left outer join, right outer join, and full outer join. (The word "outer" is sometimes omitted.)

Left outer join

The left outer join is written as R =X S where R and S are relations. The result of the left outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition (loosely speaking) to tuples in R that have no matching tuples in S.

For an example consider the tables Employee and Dept and their left outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* =X *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	ω
Sally	2241	Sales	Harriet
George	3401	Finance	ω
Harriet	2202	Sales	Harriet
Tim	1123	Executive	ω

In the resulting relation, tuples in S which have no common values in common attribute names with tuples in R take a null value, ω.

Since there are no tuples in Dept with a DeptName of Finance or Executive, ωs occur in the resulting relation where tuples in DeptName have tuples of Finance or Executive.

R $cup$ (R $bowtie$ S) means T $cup$ (R $bowtie$ S)

where T = RxN

where N = "A relation with the sames attributes of S minus the attributes of R, with an only tuple with all its attributes set to NULL"

The left outer join can be simulated using the natural join and set union as follows:

R =X S = R $cup$ (R $bowtie$ S)

Right outer join

The right outer join behaves almost identically to the left outer join, above, with the exception that all the values from the "other" relation appear in the resulting relation.

The right outer join is written as R X= S where R and S are relations. The result of the right outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R.

For example consider the tables Employee and Dept and their right outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* X= *Dept*
Name	EmpId	DeptName	Manager
Sally	2241	Sales	Harriet
Harriet	2202	Sales	Harriet
ω	ω	Production	Charles

In the resulting relation, tuples in R which have no common values in common attribute names with tuples in S take a null value, ω.

Since there are no tuples in Employee with a DeptName of Production, ωs occur in the Name attribute of the resulting relation where tuples in DeptName had tuples of Production.

The right outer join can be simulated using the natural join and set union as follows:

R X= S = S $cup$ (R $bowtie$ S)

Outer join

The outer join or full outer join in effect combines the results of the left and right outer joins.

The full outer join is written as R =X= S where R and S are relations. The result of the full outer join is the set of all combinations of tuples in R and S that are equal on their common attribute names, in addition to tuples in S that have no matching tuples in R and tuples in R that have no matching tuples in S in their common attribute names.

For an example consider the tables Employee and Dept and their full outer join:

*Employee*
Name	EmpId	DeptName
Harry	3415	Finance
Sally	2241	Sales
George	3401	Finance
Harriet	2202	Sales
Tim	1123	Executive

*Dept*
DeptName	Manager
Sales	Harriet
Production	Charles

*Employee* =X= *Dept*
Name	EmpId	DeptName	Manager
Harry	3415	Finance	ω
Sally	2241	Sales	Harriet
George	3401	Finance	ω
Harriet	2202	Sales	Harriet
Tim	1123	Executive	ω
ω	ω	Production	Charles

In the resulting relation, tuples in R which have no common values in common attribute names with tuples in S take a null value, ω. Tuples in S which have no common values in common attribute names with tuples in R also take a null value, ω.

The full outer join can be simulated using the left and right outer joins (and hence the natural join and set union) as follows:

R=X=S = (R=XS) $cup$ (RX=S)

R=X=S = R $cup$ S $cup$ (R $bowtie$ S)

Operations for domain computations

The aggregation operation

There are five aggregate functions that are included with most databases. These operations are Sum, Count, Average, Maximum and Minimum. In relational algebra, it is written as _{Exp1,Exp2,Exp3...}G_{func1,func2,func3...}(Relation). While one must specify the function to use, the expressions, however, are optional. Let's assume that we have a table named Account with two columns, namely Branch_Name and Balance defined, and we wish to find the branch name with the highest balance, we would write _{Branch_Name}G_Max(Balance)(Account). To find the highest balance, we could simply write G_Max(Balance)(Account). In computer science, an aggregate function is a function that compute a single result value from a collection of input values such as a set, a bag or a list. ...

The extend operation

Limitation of relational algebra

Although relational algebra seems powerful enough for most practical purposes, there are some simple and natural operators on relations which cannot be expressed by relational algebra. The transitive closure of a binary relation is one of them. Given a domain D, let binary relation R be a subset of DxD. The transitive closure R⁺ of R is the smallest subset of DxD containing R which satifies the following condition: In mathematics, the transitive closure of a binary relation R on a set X is the smallest transitive relation on X that contains R. For any relation R the transitive closure of R always exists. ...

$forall$ r $forall$ s $exist$ t (r[x,y] $in$ R⁺ $wedge$ s[y,z] $in$ R⁺ $Rightarrow$ t[x,z] $in$ R⁺)

It can be proven that there is no relational algebra expression E(R) taking R as a variable argument which produces R⁺. The proof is based on the fact that, given a relational expression E for which it is claimed that E(R) = R⁺, where R is a variable, we can always find an instance r of R (and a corresponding domain d) such that E(r) ≠ r⁺.^[1]

Use of algebraic properties for query optimization

Queries can be represented as a tree, where

the internal nodes are operators,
leaves are relations,
subtrees are subexpressions.

Our primary goal is to transform expression trees into equivalent expression trees, where the average size of the relations yielded by subexpressions in the tree are smaller than they were before the optimization. Our secondary goal is to try to form common subexpressions within a single query, or if there are more than one queries being evaluated at the same time, in all of those queries. The rationale behind that second goal is that it is enough to compute common subexpressions once, and the results can be used in all queries that contain that subexpression.

Here we present a set of rules, that can be used in such transformations.

Selection

Rules about selection operators play the most important role in query optimization. Selection is an operator that very effectively decreases the number of rows in its operand, so if we manage to move the selections in an expression tree towards the leaves, the internal relations (yielded by subexpressions) will likely shrink.

Basic selection properties

$σ A (R) = σ A σ A (R)$
$σ A σ B (R) = σ B σ A (R)$

Breaking up selections with complex conditions

The first two rules are used to split/merge consecutive selection operators. It is useful to merge in some cases, because only one operator needs to be evaluated instead of two. It also makes sense to split selections with complex conditions, because it may be possible to move the individual selection components where the whole selection can not be moved.

$sigma_{A land B}(R)=sigma_{A}(sigma_{B}(R))=sigma_{B}(sigma_{A}(R))$
$sigma_{A lor B}(R)=sigma_{A}(R)cupsigma_{B}(R)$

Selection and cross product

Cross product is the costliest operator to evaluate. If the input relations have $N$ and $M$ rows, the result will contain $N M$ rows. Therefore it is very important to do our best to decrease the size of both operands before applying the cross product operator.

This can be effectively done, if the cross product is followed by a selection operator, e.g. $σ A$ ( $R$ × $P$ ). Considering the definition of join, this is the most likely case. If the cross product is not followed by a selection operator, we can try to push down a selection from higher levels of the expression tree using the other selection rules.

In the above case we break up condition $A$ into conditions $B$ , $C$ and $D$ using the split rules about complex selection conditions, so that $A$ = $B$ $wedge$ $C$ $wedge$ $D$ and $B$ only contains attributes from $R$ , $C$ contains attributes only from $P$ and $D$ contains the part of $A$ that contains attributes from both $R$ and $P$ . Note, that $B$ , $C$ or $D$ are possibly empty. Then the following holds:

$sigma_{A}(R times P) = sigma_{B wedge C wedge D}(R times P) = sigma_{D}(sigma_{B}(R) times sigma_{C}(P))$

Selection and set operators

The following three rules are used to push selection below set operations in the expression tree. Note, that in the setminus and the intersection operators it is possible to apply the selection operator to only one of the operands after the transformation. This can make sense in cases, where one of the operands is small, and the overhead of evaluating the selection operator outweighs the benefits of using a smaller relation as an operand.

$sigma_{A}(Rsetminus P)=sigma_{A}(R)setminus sigma_{A}(P) =sigma_{A}(R)setminus P$
$sigma_{A}(Rcup P)=sigma_{A}(R)cupsigma_{A}(P)$
$sigma_{A}(Rcap P)=sigma_{A}(R)capsigma_{A}(P)=sigma_{A}(R)cap P=Rcap sigma_{A}(P)$

Selection and projection

Projection

• Project – Yields all values for selected attributes – Yields a vertical subset of a table

Project is a unary operation. It defines a relation that contains a vertical subset of R extracting the values of specified attributes and eliminating duplicates.

Commutativity and associativity rules

Topics in database management systems (DBMS) ( view • talk • edit )
Concepts Database • Database models • Database storage • Relational model • Distributed DBMS • ACID • Null Relational database • Relational algebra • Relational calculus • Database normalization • Referential integrity • Relational DBMS Primary key, Foreign key, Surrogate key, Superkey, Candidate key In mathematics, the Cartesian product is a direct product of sets. ... This article does not cite any references or sources. ... Logic of relatives, short for logic of relative terms, is a term used to cover the study of relations in their logical, philosophical, or semiotic aspects, as distinguished from, though closely coordinated with, their more properly formal, mathematical, or objective aspects. ... In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ... In relational algebra, a projection is a unary operation written as where is a set of attribute names. ... In set theory, a projection is one of two closely related types of functions or operations, namely: A set-theoretic operation typified by the jth projection map, written , that takes an element of the cartesian product to the value . ... In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two... In mathematics, relation algebra (RA) is an algebraic structure, a proper extension of the two-element Boolean algebra 2 intended to capture the mathematical properties of binary relations. ... The relational calculus refers to the two calculi, the tuple calculus and the domain calculus, that are part of the relational model for databases and that provide a declarative way to specify database queries. ... In logic and mathematics, relation construction and relational constructibility have to do with the ways that one relation is determined by an indexed family or a sequence of other relations, called the relation dataset. ... In logic and mathematics, the composition of relations is the generalization of the composition of functions. ... In logic and mathematics, relation reduction and relational reducibility have to do with the extent to which a given relation is determined by an indexed family or a sequence of other relations, called the relation dataset. ... A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ... The relational model for database management is a database model based on predicate logic and set theory. ... In logic and mathematics, a tacit extension is in formal respects the simplest or the logically least committal of the several possible set operations that are inverse to the set-theoretic operation of projection. ... The theory of relations treats the subject matter of relations in its combinatorial aspect, as distinguished from, though related to, its more properly logical study on one side and its more generally mathematical study on another. ... In logic, mathematics, and semiotics, a triadic relation or a ternary relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. ... Tutorial D is an example of a truly relational database query language, developed by Christopher J. Date and Hugh Darwen and described in The Third Manifesto. ... D is a set of requirements proposed by Christopher J. Date and Hugh Darwen in The Third Manifesto for what they believe a relational language ought to be like; D is not a language itself. ... D4 is a computer language used in Dataphor, a truly Relational Database Management System. ... A database management system (DBMS) is computer software designed for the purpose of managing databases. ... This article does not cite any references or sources. ... A data model is not just a way of structuring data: it also defines a set of operations that can be performed on the data. ... Database tables/indexes are typically stored in memory or on hard disk in one of many forms, ordered/unordered Flat files, ISAM, Heaps, Hash buckets or B+ Trees. ... The relational model for database management is a database model based on predicate logic and set theory. ... According to Elmasri and Navathe (2004, p. ... Acidity redirects here. ... The Greek lowercase omega (Ï‰) character is historically used by academics to represent Null in relational databases. ... A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ... The relational calculus refers to the two calculi, the tuple calculus and the domain calculus, that are part of the relational model for databases and that provide a declarative way to specify database queries. ... Database normalization is a design technique for structuring relational database tables. ... An example of a database that has not enforced referential integrity. ... A relational database management system (RDBMS) is a database management system (DBMS) that is based on the relational model as introduced by E. F. Codd. ... In database design, a primary key is a value that can be used to identify a unique row in a table. ... In the context of relational databases, a foreign key is a referential constraint between two tables[1]. The foreign key identifies a column or a set of columns in one (referencing) table that refers to a column or set of columns in another (referenced) table. ... A surrogate key is a unique primary key generated by the relational database management system that is not derived from any data in the database and whose only significance is to act as the primary key. ... A superkey is defined in the relational model as a set of attributes of a relation variable (relvar) for which it holds that in all relations assigned to that variable there are no two distinct tuples (rows) that have the same values for the attributes in this set. ... In the relational model a candidate key of a relation variable (relvar) is a set of attributes of that relvar such that (1) at all times it holds in the relation assigned to that variable that there are no two distinct tuples with the same values for these attributes and...
Objects Trigger • View • Table • Cursor • Log • Transaction • Index Stored procedure • Partition A database trigger is procedural code that is automatically executed in response to certain events on a particular table in a database. ... In database theory, a view is a virtual or logical table composed of the result set of a query. ... In relational databases, SQL databases, and flat file databases, a table is a set of data elements (values) that is organized using a model of horizontal rows and vertical columns. ... In database packages, the term cursor refers to a control structure for the successive traversal (and potential processing) of records in a result set as returned by a query. ... In in the field of databases in computer science, a transaction log (also database log or binary log) is a history of actions executed by a database management system to guarantee ACID properties over crashes or hardware failures. ... A database transaction is a unit of interaction with a database management system or similar system that is treated in a coherent and reliable way independent of other transactions that must be either entirely completed or aborted. ... It has been suggested that Bitmap index be merged into this article or section. ... A stored procedure is a subroutine available to applications accessing a relational database system. ... A partition is a division of a logical database or its constituting elements into distinct independent parts. ...	Topics in SQL Select • Insert • Update • Merge • Delete • Join • Union • Create • Drop Begin work • Commit • Rollback • Truncate • Alter SQL (IPA: or IPA: ), commonly expanded as Structured Query Language, is a computer language designed for the retrieval and management of data in relational database management systems, database schema creation and modification, and database object access control management. ... A SELECT statement in SQL returns a result set of records from one or more tables. ... An SQL INSERT statement adds one or more records to a table in a relational database. ... An UPDATE statement in SQL changes data in one or more records in a relational database management system. ... Wikipedia does not have an article with this exact name. ... It has been suggested that this article or section be merged into Data Manipulation Language. ... A JOIN clause in SQL combines records from two tables in a relational database and results in a new (temporary) table, also called joined table. ... In SQL the UNION operator combines the results of two SQL queries into a single table of all matching rows. ... A CREATE statement in SQL creates an object inside of a relational database management system (RDBMS). ... A DROP statement in SQL removes an object from a relational database management system (RDBMS). ... It has been suggested that this article or section be merged into Database transaction. ... A COMMIT statement in SQL ends a transaction within a relational database management system (RDBMS) and makes all changes visible to other users. ... In database technologies, a rollback is an operation which returns the database to some previous state. ... The Truncate statement removes all the data from a table. ... An ALTER statement in SQL changes the properties of an object inside of a relational database management system (RDBMS). ...
Implementations of database management systems
Types of implementations Relational • Flat file • Deductive • Dimensional • Hierarchical • Object oriented • Object relational • Temporal • XML data stores A relational database is a database that conforms to the relational model, and refers to a databases data and schema (the databases structure of how that data is arranged). ... A simple diagram depicting conversion of a CSV-format flat file database table into a relational database table. ... A deductive database system is a database system which can make deductions (ie: infer additional rules or facts) based on rules and facts stored in the (deductive) database. ... A dimensional database is one which, rather than representing data in multiple relations (as a relational database does), represents key data entities as different dimensions. ... In a hierarchical data model, data are organized into a tree-like structure. ... In an object oriented database, information is represented in the form of objects as used in Object-Oriented Programming. ... An object-relational database (ORD) or object-relational database management system (ORDBMS) is a relational database management system that allows developers to integrate the database with their own custom data types and methods. ... A temporal database is a database management system with built-in time aspects, e. ... In Software engineering, an XML database is a data persistence software system that allows data to be imported, accessed and exported in the XML format. ...
Database products Object-oriented (comparison) • Relational (comparison) The following is a list of object-oriented database management systems. ... This article or section is not written in the formal tone expected of an encyclopedia article. ... See DBMS for a shorter list of â€œtypicalâ€, representative database management systems. ... The following tables compare general and technical information for a number of relational database management systems. ...	Components Query language • Query optimizer • Query plan • ODBC • JDBC Query languages are computer languages used to make queries into databases and information systems. ... The query optimizer is a component of database management system that is used to analyzes queries submitted to database server for execution, and then determines the optimal way to execute the query. ... A query plan (or query execution plan) is an set of steps used to access information in a SQL relational database management system. ... In computing, Open Database Connectivity (ODBC) provides a standard software API method for using database management systems (DBMS). ... JDBC is an API for the Java programming language that defines how a client may access a database. ...

References

^ Aho, Alfred V.; Jeffrey D. Ullman (1979). "Universality of data retrieval languages". Proceedings of the 6th ACM SIGACT-SIGPLAN symposium on Principles of programming languages: 110-119.

External links

LEAP - An implementation of the relational algebra
Query Optimization This paper is an excellent introduction into the use of the relational algebra in optimizing queries, and includes numerous citations for more in-depth study.

Categories: Articles to be split | Articles with sections needing expansion | Data modeling | Relational algebra | Relational model

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Contents

Introduction

Primitive operations

Set operators

Projection

Selection

Rename

Joins and join-like operators

Natural join

θ-join and equijoin

Semijoin

Antijoin

Division

Outer joins

Left outer join

Right outer join

Outer join

Operations for domain computations

The aggregation operation

The extend operation

Limitation of relational algebra

Use of algebraic properties for query optimization

Selection

Basic selection properties

Breaking up selections with complex conditions

Selection and cross product

Selection and set operators

Selection and projection

Projection

Commutativity and associativity rules

See also

References

External links