Examples include operators to filter certain attributes (columns) or tuples (rows) from an input relation.
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.
In addition, the Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be "shallow" for the purposes of the operation.
where φ is a propositional formula that consists of atoms as allowed in the normal selection and the logical operators
To obtain a listing of all friends or business associates in an address book, the selection might be written as
[a] 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:[citation needed]
Note that neither the employee named Mary nor the Production department appear in the result.
Mary does not appear in the result because Mary's Department, "Human Resources", is not listed in the Dept relation and the Production department does not appear in the result because there are no tuples in the Employee relation that have "Production" as their DeptName attribute.
For example, the composition of Employee and Dept is their join as shown above, projected on all but the common attribute DeptName.
In particular, natural join allows the combination of relations that are associated by a foreign key.
If this is not the case such as in the foreign key from Dept.Manager to Employee.Name then these columns must be renamed before taking the natural join.
More formally the semantics of the natural join are defined as follows: where Fun(t) is a predicate that is true for a relation t (in the mathematical sense) iff t is a function (that is, t does not map any attribute to multiple values).
Note, however, that a computer language that supports the natural join and selection 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).
It is important to note: forming the flattened Cartesian product then filtering the rows is conceptually correct, but an implementation would use more sophisticated data structures to speed up the join query.
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.
In practice the classical relational algebra described above is extended with various operations such as outer joins, aggregate functions and even transitive closure.
Outer joins are not considered part of the classical relational algebra discussed so far.
In order to make subsequent selection operations on the resulting table meaningful, a semantic meaning needs to be assigned to nulls; in Codd's approach the propositional logic used by the selection is extended to a three-valued logic, although we elide those details in this article.
[citation needed] For an example consider the tables Employee and Dept and their full outer join:
Practical query languages have such facilities, e.g. the SQL SELECT allows arithmetic operations to define new columns in the result SELECT unit_price * quantity AS total_price FROM t, and a similar facility is provided more explicitly by Tutorial D's EXTEND keyword.
[7]: 213 Furthermore, computing various functions on a column, like the summing up of its elements, is also not possible using the relational algebra introduced so far.
The transitive closure R+ of R is the smallest subset of D×D that contains R and satisfies the following condition: It can be proved using the fact that there is no relational algebra expression E(R) taking R as a variable argument that produces R+.
[8] SQL however officially supports such fixpoint queries since 1999, and it had vendor-specific extensions in this direction well before that.
Rules about selection operators play the most important role in query optimization.
Therefore, it is important to decrease the size of both operands before applying the cross product operator.
Then the following holds: Selection is distributive over the set difference, intersection, and union operators.
In other cases, if the selection condition is relatively expensive to compute, moving selection outside the projection may reduce the number of tuples which must be tested (since projection may produce fewer tuples due to the elimination of duplicates resulting from omitted fields).
Subsequently, ISBL was created, and this pioneering work has been acclaimed by many authorities[9] 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.