Extensions

Some years after publication of his 1970 model, Codd proposed a three-valued logic (True, False, Missing/NULL) version of it to deal with missing information, and in his The Relational Model for Database Management Version 2 (1990) he went a step further with a four-valued logic (True, False, Missing but Applicable, Missing but Inapplicable) version.^[4]

Basic Concepts

A relation with 5 attributes (its degree) and 4 tuples (its cardinality) can be visualized as a table with 5 columns and 4 rows. However, unlike rows and columns in a table, a relation's attributes and tuples are unordered.

A relation consists of a heading and a body. The heading defines a set of attributes, each with a name and data type (sometimes called a domain). The number of attributes in this set is the relation's degree or arity. The body is a set of tuples. A tuple is a collection of n values, where n is the relation's degree, and each value in the tuple corresponds to a unique attribute.^{[nb 1]} The number of tuples in this set is the relation's cardinality.^{[5]: 17–22}

Relations are represented by relational variables or relvars, which can be reassigned.^{[5]: 22–24} A database is a collection of relvars.^{[5]: 112–113}

In this model, databases follow the Information Principle: At any given time, all information in the database is represented solely by values within tuples, corresponding to attributes, in relations identified by relvars.^[5]: 111

Constraints

A database may define arbitrary boolean expressions as constraints. If all constraints evaluate as true, the database is consistent; otherwise, it is inconsistent. If a change to a database's relvars would leave the database in an inconsistent state, that change is illegal and must not succeed.^[5]: 91

In general, constraints are expressed using relational comparison operators, of which just one, "is subset of" (⊆), is theoretically sufficient.

Two special cases of constraints are expressed as keys and foreign keys:

Relational operations

Users (or programs) request data from a relational database by sending it a query. In response to a query, the database returns a result set.

Often, data from multiple tables are combined into one, by doing a join. Conceptually, this is done by taking all possible combinations of rows (the Cartesian product), and then filtering out everything except the answer.

There are a number of relational operations in addition to join. These include project (the process of eliminating some of the columns), restrict (the process of eliminating some of the rows), union (a way of combining two tables with similar structures), difference (that lists the rows in one table that are not found in the other), intersect (that lists the rows found in both tables), and product (mentioned above, which combines each row of one table with each row of the other). Depending on which other sources you consult, there are a number of other operators – many of which can be defined in terms of those listed above. These include semi-join, outer operators such as outer join and outer union, and various forms of division. Then there are operators to rename columns, and summarizing or aggregating operators, and if you permit relation values as attributes (relation-valued attribute), then operators such as group and ungroup.

The flexibility of relational databases allows programmers to write queries that were not anticipated by the database designers. As a result, relational databases can be used by multiple applications in ways the original designers did not foresee, which is especially important for databases that might be used for a long time (perhaps several decades). This has made the idea and implementation of relational databases very popular with businesses.

Database normalization

Relations are classified based upon the types of anomalies to which they're vulnerable. A database that is in the first normal form is vulnerable to all types of anomalies, while a database that is in the domain/key normal form has no modification anomalies. Normal forms are hierarchical in nature. That is, the lowest level is the first normal form, and the database cannot meet the requirements for higher level normal forms without first having met all the requirements of the lesser normal forms.^[6]

Example

If a relation of Employees contains the attributes {Name, ID}, then the tuple {Alice, 1} represents the proposition: "There exists an employee named Alice with ID 1". This proposition may be true or false. If this tuple exists in the relation's body, the proposition is true (there is such an employee). If this tuple is not in the relation's body, the proposition is false (there is no such employee).^{[5]: 96–97}

Furthermore, if {ID} is a key, then a relation containing the tuples {Alice, 1} and {Bob, 1} would represent the following contradiction:

1. There exists an employee with the name Alice and the ID 1.
2. There exists an employee with the name Bob and the ID 1.
3. There do not exist multiple employees with the same ID.

Under the principle of explosion, this contradiction would allow the system to prove that any arbitrary proposition is true. The database must enforce the key constraint to prevent this.^[5]: 104

Database

An idealized, very simple example of a description of some relvars (relation variables) and their attributes:

• Customer (Customer ID, Name)
• Order (Order ID, Customer ID, Invoice ID, Date)
• Invoice (Invoice ID, Customer ID, Order ID, Status)

In this design we have three relvars: Customer, Order, and Invoice. The bold, underlined attributes are candidate keys. The non-bold, underlined attributes are foreign keys.

Usually one candidate key is chosen to be called the primary key and used in preference over the other candidate keys, which are then called alternate keys.

A candidate key is a unique identifier enforcing that no tuple will be duplicated; this would make the relation into something else, namely a bag, by violating the basic definition of a set. Both foreign keys and superkeys (that includes candidate keys) can be composite, that is, can be composed of several attributes. Below is a tabular depiction of a relation of our example Customer relvar; a relation can be thought of as a value that can be attributed to a relvar.

Customer relation

If we attempted to insert a new customer with the ID 123, this would violate the design of the relvar since Customer ID is a primary key and we already have a customer 123. The DBMS must reject a transaction such as this that would render the database inconsistent by a violation of an integrity constraint. However, it is possible to insert another customer named Alice, as long as this new customer has a unique ID, since the Name field is not part of the primary key.

Foreign keys are integrity constraints enforcing that the value of the attribute set is drawn from a candidate key in another relation. For example, in the Order relation the attribute Customer ID is a foreign key. A join is the operation that draws on information from several relations at once. By joining relvars from the example above we could query the database for all of the Customers, Orders, and Invoices. If we only wanted the tuples for a specific customer, we would specify this using a restriction condition. If we wanted to retrieve all of the Orders for Customer 123, we could query the database to return every row in the Order table with Customer ID 123 .

There is a flaw in our database design above. The Invoice relvar contains an Order ID attribute. So, each tuple in the Invoice relvar will have one Order ID, which implies that there is precisely one Order for each Invoice. But in reality an invoice can be created against many orders, or indeed for no particular order. Additionally the Order relvar contains an Invoice ID attribute, implying that each Order has a corresponding Invoice. But again this is not always true in the real world. An order is sometimes paid through several invoices, and sometimes paid without an invoice. In other words, there can be many Invoices per Order and many Orders per Invoice. This is a many-to-many relationship between Order and Invoice (also called a non-specific relationship). To represent this relationship in the database a new relvar should be introduced whose role is to specify the correspondence between Orders and Invoices:

OrderInvoice (Order ID, Invoice ID)

Now, the Order relvar has a one-to-many relationship to the OrderInvoice table, as does the Invoice relvar. If we want to retrieve every Invoice for a particular Order, we can query for all orders where Order ID in the Order relation equals the Order ID in OrderInvoice, and where Invoice ID in OrderInvoice equals the Invoice ID in Invoice.

SQL and the relational model

SQL, initially pushed as the standard language for relational databases, deviates from the relational model in several places. The current ISO SQL standard doesn't mention the relational model or use relational terms or concepts.

According to the relational model, a Relation's attributes and tuples are mathematical sets, meaning they are unordered and unique. In a SQL table, neither rows nor columns are proper sets. A table may contain both duplicate rows and duplicate columns, and a table's columns are explicitly ordered. SQL uses a Null value to indicate missing data, which has no analog in the relational model. Because a row can represent unknown information, SQL does not adhere to the relational model's Information Principle.^{[5]: 153–155, 162}

Key constraints and functional dependencies

One of the simplest and most important types of relation constraints is the key constraint. It tells us that in every instance of a certain relational schema the tuples can be identified by their values for certain attributes.

Superkey

A superkey is a set of column headers for which the values of those columns concatenated are unique across all rows. Formally:

A superkey is written as a finite set of attribute names.

A superkey ${\displaystyle K}$ holds in a relation ${\displaystyle (H,B)}$ if:

• ${\displaystyle K\subseteq H}$ and
• there exist no two distinct tuples ${\displaystyle t_{1},t_{2}\in B}$ such that ${\displaystyle t_{1}[K]=t_{2}[K]}$.

A superkey holds in a relation universe ${\displaystyle U}$ if it holds in all relations in ${\displaystyle U}$.

Theorem: A superkey ${\displaystyle K}$ holds in a relation universe ${\displaystyle U}$ over ${\displaystyle H}$ if and only if ${\displaystyle K\subseteq H}$ and ${\displaystyle K\rightarrow H}$ holds in ${\displaystyle U}$.

Candidate key

A candidate key is a superkey that cannot be further subdivided to form another superkey.

A superkey ${\displaystyle K}$ holds as a candidate key for a relation universe ${\displaystyle U}$ if it holds as a superkey for ${\displaystyle U}$ and there is no proper subset of ${\displaystyle K}$ that also holds as a superkey for ${\displaystyle U}$.

Functional dependency

Functional dependency is the property that a value in a tuple may be derived from another value in that tuple.

A functional dependency (FD for short) is written as ${\displaystyle X\rightarrow Y}$ for ${\displaystyle X,Y}$ finite sets of attribute names.

A functional dependency ${\displaystyle X\rightarrow Y}$ holds in a relation ${\displaystyle (H,B)}$ if:

• ${\displaystyle X,Y\subseteq H}$ and
• ${\displaystyle \forall }$ tuples ${\displaystyle t_{1},t_{2}\in B}$, ${\displaystyle t_{1}[X]=t_{2}[X]~\Rightarrow ~t_{1}[Y]=t_{2}[Y]}$

A functional dependency ${\displaystyle X\rightarrow Y}$ holds in a relation universe ${\displaystyle U}$ if it holds in all relations in ${\displaystyle U}$.

Trivial functional dependency

A functional dependency is trivial under a header ${\displaystyle H}$ if it holds in all relation universes over ${\displaystyle H}$.

Theorem: An FD ${\displaystyle X\rightarrow Y}$ is trivial under a header ${\displaystyle H}$ if and only if ${\displaystyle Y\subseteq X\subseteq H}$.

Closure

Armstrong's axioms: The closure of a set of FDs ${\displaystyle S}$ under a header ${\displaystyle H}$, written as ${\displaystyle S^{+}}$, is the smallest superset of ${\displaystyle S}$ such that:

• ${\displaystyle Y\subseteq X\subseteq H~\Rightarrow ~X\rightarrow Y\in S^{+}}$ (reflexivity)
• ${\displaystyle X\rightarrow Y\in S^{+}\land Y\rightarrow Z\in S^{+}~\Rightarrow ~X\rightarrow Z\in S^{+}}$ (transitivity) and
• ${\displaystyle X\rightarrow Y\in S^{+}\land Z\subseteq H~\Rightarrow ~(X\cup Z)\rightarrow (Y\cup Z)\in S^{+}}$ (augmentation)

Theorem: Armstrong's axioms are sound and complete; given a header ${\displaystyle H}$ and a set ${\displaystyle S}$ of FDs that only contain subsets of ${\displaystyle H}$, ${\displaystyle X\rightarrow Y\in S^{+}}$ if and only if ${\displaystyle X\rightarrow Y}$ holds in all relation universes over ${\displaystyle H}$ in which all FDs in ${\displaystyle S}$ hold.

Completion

The completion of a finite set of attributes ${\displaystyle X}$ under a finite set of FDs ${\displaystyle S}$, written as ${\displaystyle X^{+}}$, is the smallest superset of ${\displaystyle X}$ such that:

• ${\displaystyle Y\rightarrow Z\in S\land Y\subseteq X^{+}~\Rightarrow ~Z\subseteq X^{+}}$

The completion of an attribute set can be used to compute if a certain dependency is in the closure of a set of FDs.

Theorem: Given a set ${\displaystyle S}$ of FDs, ${\displaystyle X\rightarrow Y\in S^{+}}$ if and only if ${\displaystyle Y\subseteq X^{+}}$.

Irreducible cover

An irreducible cover of a set ${\displaystyle S}$ of FDs is a set ${\displaystyle T}$ of FDs such that:

• ${\displaystyle S^{+}=T^{+}}$
• there exists no ${\displaystyle U\subset T}$ such that ${\displaystyle S^{+}=U^{+}}$
• ${\displaystyle X\rightarrow Y\in T~\Rightarrow Y}$ is a singleton set and
• ${\displaystyle X\rightarrow Y\in T\land Z\subset X~\Rightarrow ~Z\rightarrow Y\notin S^{+}}$.

Algorithm to derive candidate keys from functional dependencies

algorithm derive candidate keys from functional dependencies is
    input: a set S of FDs that contain only subsets of a header H
    output: the set C of superkeys that hold as candidate keys in
            all relation universes over H in which all FDs in S hold

    C := ∅         // found candidate keys
    Q := { H }      // superkeys that contain candidate keys
    while Q <> ∅ do
        let K be some element from Q
        Q := Q – { K }
        minimal := true
        for each X->Y in S do
            K' := (K – Y) ∪ X   // derive new superkey
            if K' ⊂ K then
                minimal := false
                Q := Q ∪ { K' }
            end if
        end for
        if minimal and there is not a subset of K in C then
            remove all supersets of K from C
            C := C ∪ { K }
        end if
    end while