01 February 2025
tl;dr Relational data is, contrary to popular belief, characterized not by "tables", but by sets and relational variables (also known as "relvars"), and making use of a relational algebra and predicate calculus to make it easier to do set-oriented operations.
Because most developers come to relational databases via SQL, usually either studied briefly in classes at university or as-necessary to get the work done while on the job, they often don't realize the relational model is much more academic and foundational than the flavor of SQL they've come to use and know. [Date13] 1 and [Fussell] 2 define the relational model as characterized by relation, attribute, tuple, relation value and relation variable.
A relation is, at its heart, a truth predicate about the world, a statement of facts (attributes) that provide meaning to the predicate. For example, we may define the relation "PERSON" as {SSN, Name, City}, which states that "there exists a PERSON with a Social Security Number SSN who lives in City and is called Name". Note that in a relation, attribute ordering is entirely unspecified.
A tuple is a truth statement within the context of a relation, a set of attribute values that match the required set of attributes in the relation, such as "{PERSON SSN='123-45-6789' Name='Catherine Kennedy' City='Seattle'}". Note that two tuples are considered identical if their relation and attribute values are also identical. (This is in contrast to an object system in which two objects are only identical if they have the same unique identifier--typically a memory address.)
A relation value, then, is a combination of a relation and a set of tuples that match that relation, and a relation variable is, like most variables, a placeholder for a given relation, but can change value over time. Thus, a relation variable People can be written to hold the relation {PERSON}, and consist of the relation value
{ {PERSON SSN='123-45-6789' Name='Catherine Kennedy' City='Seattle'},
{PERSON SSN='321-54-9876' Name='Charlotte Neward' City='Redmond'},
{PERSON SSN='213-45-6978' Name='Cathi Gero' City='Redmond'} }
Thanks to our familiarity with tabular storage models, we ofen write this relation variable in a tabular format like so:
| SSN | Name | City |
|---|---|---|
| 123-45-6789 | Catherine Kennedy | Seattle |
| 321-54-9876 | Charlotte Neward | Redmond |
| 213-45-6978 | Cathi Gero | Redmond |
Keep in mind, however, that in the model itself, the tuple is not exactly a row, and each tuple knows both the name and the value of each of its atoms.
We can then operate against these relation values using using a set of operators (described in some detail in Chapter 7 of [Date04] 3): restrict, project, product, join, divide, union, intersection and difference. We modify People by using the union binary operator (let's call it +) on it and a second, unnamed, relation variable containing a single tuple:
People = People + {PERSON SSN='312-54-8796' Name='Ted Neward' City='Redmond'}
After this operation, People now holds four tuples. Additional operators can be used to create derived relation values, relations that are calculated from other relation values in the database--for example, we can create a relation value that demonstrates the number of people living in individual cities by making use of the project and restrict operators across the People relation variable defined above.
The other half of relational models is the deep understanding they have of relationships between relational variables.