Relational algebra
The relational algebra is a set of operations that manipulate relations as they are defined in the relational model and as such describes part of the data manipulation aspect of this data model. Because of their algebraic properties these operations are often used in database query optimization as an intermediate representation of a query to which certain rewrite rules can be applied to obtain a more efficient version of the query.
The exact set of operations 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 is the most common way the relational model is defined. That means that 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).
The Fundamental Operations
The six basic operations of the algebra are the selection, the projection, the cartesian product, the set union, the set difference, and the rename. These six operations are fundamental in the sense that (1) all other operations can be expressed as combinations of these operations and (2) none of these six operations can be omitted without losing expressive power.
The Selection
The selection is a unary operation that is written as σaθb(R) or σaθv(R) where a and b are attribute names, θ a binary operation in the set {<, ≤, =, >, ≥}, v is a value constant and R is a relation. The selection σaθb(R) selects all those tupels in R for which θ holds between the a and the b attribute. The selection σaθv(R) selects all those tuples in R for which θ holds between the a attribute and the value v.
For an example consider the following tables where the first table gives the relation Person, the second table gives the result of σAge≥34(Person) and the third table gives the result of σAge=Weight(Person).
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More formally the semantics of the selection is defined as follows:
- σaθb(R) = { t : t ∈ R, t(a) θ t(b) }
- σaθv(R) = { t : t ∈ R, t(a) θ v }
The result of the selection is only defined if the attribute names that it mentions are in the header of the relation that it operates upon.
The Projection
The projection is a unary operation that is written as πa1,..,an(R) where a1,..,an is a set of attribute names. The result of such a projection is defined as the set that is obtained when all tuples in R are restricted to the set {a1,..,an}. For an example consider the following two tables which are the relation Person and its projection on the attributes Age and Weight:
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Note that Harry and Peter have the same age and weight, but since the result is a relation and therefore a set this combination only appears once in the result.
More formally the semantics of the projection is defined as follows:
- πa1,..,an(R) = { t[a1,..,an] : t ∈ R }
where t[a1,..,an] is the restriction of the tuple t to the set {a1,..,an}, i.e., t[a1,..,an] = { (a' , v) | (a' , v) ∈ t, a' ∈ {a1,..,an} }.
The result of a projection πa1,..,an(R) is only defined if {a1,..,an} is a subset of the header of R.
The Cartesian Product
The cartesian product (also called cross product or cross join) is a binary operation that is very similar to the Cartesian product in set theory. It is written as R × S where R and S are relations. The result of the cartesian product is the set of all combinations of tupels in R and S. For an example consider the tables Employee and Dept and their Cartesian product:
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More formally the semantics of the Cartesian product is defined as follows:
- R × S = { t ∪ s : t ∈ R, s ∈ S }
To ensure that the result of the cartesian product is again a relation it is required that the headers of R and S are disjoint, i.e., do not contain the same attribute.
The Set Union
The set union is a binary operation that is written as R ∪ S and is defined as the set union in set theory. For an example consider the tables Engineer, Manager and their union:
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Note that Harriet only appears once in the result because the result is a set.
Formally, the union is defined as follows:
- R ∪ S = { t : t ∈ R ∨ t ∈ S }
The result of the set union is only defined when the two relations have the same headers.
The Set Difference
The set difference is a binary operation that is written as R - S and is defined as the usual set difference in set theory. For an example consider the relations Engineer, Manager and their difference:
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Formally, the semantics of the union is defined as follows:
- R - S = { t : t ∈ R, ¬ t ∈ S }
The result of the set difference is only defined when the two relations have the same headers.
The Rename
The rename operation is a unary operation that is written as ρa/b(R) where a and b are attribute names and R is a relation. The result is identical to R except that the b field in all tuples is renamed to an a field. For an example consider the following Employee relation and its renamed version:
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Formally the semantics of the rename operator is defined as follows:
- ρa/b(R) = { t[a/b] : t ∈ R }
where t[a/b] is defined as the tuple t with the b attribute renamed to a, i.e., t[a/b] = { (c, v) | (c, v) ∈ t, c ≠ b } ∪ { (a, t(b)) }. The result of the rename is only defined when the attribute a did not appear already in the header of the operand.
The Expressive Power
These operations are enough to express all queries that can be expressed in tuple calculus and domain calculus which is essentially the same as first-order logic.
- insert a sketch of proof here
Other Operations expressible with the Basic Operations
Next to the six basic operations some other operations are also often included even though they can be expressed by combinations of the basic operations. This is because they either have interesting algebraic properties or because they can be implemented more efficiently than their simulations. These operations are the set intersection, the natural join and the division.
The Set Intersection
The set intersection is a binary operation that is written as R ∩ S and is defined as the usual set intersection in set theory. For an example consider the tables Engineer, Manager and their intersection:
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Formally, the union is defined as follows:
- R ∩ S = { t : t ∈ R, t ∈ S }
The result of the set intersection is only defined when the two relations have the same headers. This operation can be simulated in the basic operations as follows:
- R ∩ S = R - (R - S)
The Natural Join
The natural join is a binary operation that is written as R (×) 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:
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The natural join is arguably one of the most important operators since it allows the combination of relations that are associated by a foreign key. For example, in the above example there holds probably a foreign key 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 columns 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. (See also equi-join.)
More formally the semantics of the natural join is defined as follows:
- R (×) S = { t ∪ s : t ∈ R, s ∈ S, fun(t ∪ 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.
The natural join can be simulated with the basic operations as follows. Assume that a1,...,an are the attribute names unique to R, b1,...,bm are the attribute names common to R and S and c1,...,cm are the attribute unique to S. Furthermore assume that the attribute names d1,...,dm are neither in R nor in S. In a first step we can now rename the common attribute names in S:
- S' := ρd1/b1(...ρdm/bm( S)...)
Then we take the Cartesion product and select the tuples that are to be joined:
- T := σb1=d1(...σbm=dm(R × S')...)
Finally we take a projection to get rid of the renamed attributes:
- U := πa1,...,an,b1,...,bm,c1,...,cm(T)
The 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. More formally:
- R ÷ S = { t|{a1,...,an} : ∀ s ∈ S ( (t|{a1,...,an} ∪ s) ∈ R) }
where a1,...,an is the set of attribute names unique to R. 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,...,an are the attribute names unique to R and b1,...,bm are the attribute names of S. In the first step we project R on its unique attribute names and construction all combinations with tuples in S:
- T := πa1,...,an(R) × S
In the next step we subtract R from this relation:
- U := T - R
Note that in U we have the combinations that "should 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 := πa1,...,an(U)
So what remains to be done is take the projection of R on its unique attribute names and subtract those in V:
- W := πa1,...,an(R) - V
Generalized Operations
The General Selection
- Allows general propositional formulas and other comparison operators
The θ-join
- Allow other comparison operators
Operations for null values
The null is like zero in normal algebra. Like 0x1 or 0x2, the null value absorbs any value when joining tables.