Indatabase theory, arelation, as originally defined byE. F. Codd,[1] is aset oftuples (d1,d2,...,dn), where each element dj is a member of Dj, adata domain. Codd's original definition notwithstanding, and contrary to the usual definition in mathematics, there is no ordering to the elements of the tuples of a relation.[2][3] Instead, each element is termed anattribute value. Anattribute is a name paired with a domain (nowadays more commonly referred to as atype ordata type). Anattribute value is an attribute name paired with an element of that attribute's domain, and a tuple is aset of attribute values in which no two distinct elements have the same name. Thus, in some accounts, a tuple is described as afunction, mapping names to values.
A set of attributes in which no two distinct elements have the same name is called aheading. It follows from the above definitions that to every tuple there corresponds a unique heading, being the set of names from the tuple, paired with the domains from which the tuple elements' domains are taken. A set oftuples that all correspond to the same heading is called abody. A relation is thus a heading paired with a body, the heading of the relation being also the heading of each tuple in its body. The number of attributes constituting a heading is called thedegree, which term also applies to tuples and relations. The termn-tuple refers to a tuple of degreen (n ≥ 0).
E. F. Codd used the term "relation" in its mathematical sense of afinitary relation, a set of tuples on some set ofn setsS1,S2,....,Sn.[4] Thus, ann-ary relation is interpreted, under theClosed-World Assumption, as the extension of somen-adicpredicate: all and only thosen-tuples whose values, substituted for corresponding free variables in the predicate, yield propositions that hold true, appear in the relation.
A heading paired with a set of constraints defined in terms of that heading is called arelation schema. A relation can thus be seen as an instantiation of a relation schema if it has the heading of that schema and it satisfies the applicable constraints.
Sometimes a relation schema is taken to include a name.[5][6] A relational database definition (database schema, sometimes referred to as a relational schema) can thus be thought of as a collection of namedrelation schemas.[7][8]
In implementations, the domain of each attribute is effectively adata type[9] and a named relation schema is effectively arelation variable (relvar for short).
InSQL, adatabase language for relational databases, relations are represented bytables, where each row of a table represents a single tuple, and where the values of each attribute form a column.
Below is an example of a relation having three named attributes: 'ID' from the domain ofintegers, and 'Name' and 'Address' from the domain ofstrings:
| ID (Integer) | Name (String) | Address (String) |
|---|---|---|
| 102 | Yon | Naha, Okinawa |
| 202 | Nil | Sendai, Miyagi |
| 104 | Murata Makoto | Kumamoto, Kumamoto |
| 152 | Matsumoto Yukihiro | Okinawa, Okinawa |
A predicate for this relation, using the attribute names to denote free variables, might be "Employee numberID is known asName and lives atAddress". Examination of the relation tells us that there are just four tuples for which the predicate holds true. So, for example, employee 102 is known only by that name, Akin, and does not live anywhere else but in Naha, Okinawa. Also, apart from the four employees shown, there is no other employee who has both a name and an address.
Under the definition ofbody, the tuples of a body do not appear in any particular order - one cannot say "The tuple of 'Murata Makoto' is above the tuple of 'Matsumoto Yukihiro'", nor can one say "The tuple of is the first tuple." A similar comment applies to the rows of an SQL table.
Under the definition ofheading, the attributes of an element do not appear in any particular order either, nor, therefore do the elements of a tuple. A similar comment doesnot apply here to SQL, which does define an ordering to the columns of a table.
A relational database consists of namedrelation variables (relvars) for the purposes of updating the database in response to changes in the real world. An update to a single relvar causes the body of the relation assigned to that variable to be replaced by a different set of tuples. Relvars are classified into two classes:base relation variables andderived relation variables, the latter also known asvirtual relvars but usually referred to by the short termview.
Abase relation variable is a relation variable which is not derived from any other relation variables. InSQL the termbasetable equates approximately to base relation variable.
A view can be defined by an expression using the operators of therelational algebra or therelational calculus. Such an expression operates on one or more relations and when evaluated yields another relation. The result is sometimes referred to as a "derived" relation when the operands are relations assigned to database variables. A view is defined by giving a name to such an expression, such that the name can subsequently be used as a variable name. (Note that the expression must then mention at least one base relation variable.)
By using aData Definition Language (DDL), it is able to define base relation variables. In SQL,CREATE TABLE syntax is used to define base tables. The following is an example.
CREATETABLEList_of_people(IDINTEGER,NameCHAR(40),AddressCHAR(200),PRIMARYKEY(ID))
The Data Definition Language (DDL) is also used to define derived relation variables. In SQL,CREATE VIEW syntax is used to define a derived relation variable. The following is an example.
CREATEVIEWList_of_Okinawa_peopleAS(SELECTID,Name,AddressFROMList_of_peopleWHEREAddressLIKE'%, Okinawa')
R is arelation on these n domains if it is a set of elements of the form (d1, d2, ..., dn) where dj ∈ Dj for each j=1,2,...,n.
... tuples have no left-to-right ordering to their attributes ...
One reason for abandoning positional concepts altogether in the relations of the relational model is that it is not at all unusual to find database relations, each of which has as many as 50, 100, or even 150 columns.
The termrelation is used here in its accepted mathematical sense
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