I'm reading a very verbose textbook on database design, but I suspect that much of the book could be condensed into a few pages if the authors were not trying to avoid mathematical language.
What is the mathematical definition of a relational database? Stated in a way that a pure mathematician would be satisfied with.
I will take your question as a reference request. Your first stop can be https://en.wikipedia.org/wiki/Relational_model and the books by Codd and Date cited there.
To add to the earlier answer, from the preface of the book The relational model for database management: version 2 by Codd:
The relational model is solidly based on two parts of mathematics: first-order predicate logic and the theory of relations.
For more on first-order predicate logic and the theory of relations, I refer you to the following links:
and
Edit: Let me try, as per the OP's request, to define a relational database in a few sentences.
A relational database is a collection of tables, which mathematically are called relations. Each table consists of columns and rows, which are represented in tabular format as follows:
Say your relation is $R = \{(0,0), (0,1),(1,2),(1,3),(2,3),(3,4)\}$. In tabular format, you can have it as:
I come to this from a different angle... I am teaching relations, ostensibly from an abstract perspective, to computer science students... however I want to try and relate what I am teaching to computers in some way.
This is what I do with relations. It is sound on the relations side of the house... databases... I am not so sure.
Let us start with two variables:
We can form the Cartesian product $P\times\mathbb{N}_0$ of $P$ and $\mathbb{N}_0$. This is an infinite set, formed of ALL ordered pairs $(p,a)$, where $p$ is a person, and $a$ is a natural number:
$$P\times\mathbb{N}_0=\{(p,a)\,\colon\,p\in P,\,a\in\mathbb{N}_0\}.$$
A relation $R$ between $P$ and $\mathbb{N}_0$ is a subset, $R\subset (P\times \mathbb{N}_0)$, that is a relation is SOME ordered pairs. And this is a database. For example, suppose the relation is:
$$R=\{(\text{Alice},34),(\text{Bob},38),(\text{Carol},33)\}.$$
As a database this would look like:
| person | age |
|---|---|
| Alice | 34 |
| Bob | 38 |
| Carol | 33 |
Now what we can do is extend the notion of relation to multiple sets. Let us consider two more variables:
We can consider the full Cartesian product:
$$P\times \mathbb{N}_0\times N\times C.$$
A relation between these sets is a subset of this, e.g.
$$R=\{(\text{Alice},34,\text{American},\text{Canada}),(\text{Bob},38,\text{Irish},\text{USA}),(\text{Carol},33,\text{English},\text{England})\}.$$
This is a database:
| person | age | nationality | residence |
|---|---|---|---|
| Alice | 34 | American | Canada |
| Bob | 38 | Irish | USA |
| Carol | 33 | English | England |
Perhaps a lot of relations do not give databases that arise. For example, the following is also a relation on $P\times \mathbb{N}_0\times N\times C$:
$$S=\{(\text{Alice},34,\text{American},\text{Canada}),(\text{Alice},38,\text{Irish},\text{USA}),(\text{Carol},33,\text{English},\text{England})\}.$$
But is it correctly a database?
I think the databases we have here are more like partial functions:
$$P\to(\mathbb{N}_0\times N\times C),$$
so no person appears twice in the database.
I think for databases, there is an implied extra variable. For example, consider the following relation on $\{a,b\}\times \{a,b\}\times\{a,b\}$ (think of these as values of variables $v_1$, $v_2$, $v_3$:
$$(a,a,b),\,(a,a,a),\,(b,a,b).$$
The database looks like:
| $v_1$ | $v_2$ | $v_3$ |
|---|---|---|
| $a$ | $a$ | $b$ |
| $a$ | $a$ | $a$ |
| $b$ | $a$ | $b$ |
I would argue that the rows give a natural fourth variable, the row number, $r\in\mathbb{N}$ and so we have a partial function:
$$\mathbb{N}\to (\{a,b\}\times \{a,b\}\times \{a,b\}):$$
| $r$ | $v_1$ | $v_2$ | $v_3$ |
|---|---|---|---|
| 1 | $a$ | $a$ | $b$ |
| 2 | $a$ | $a$ | $a$ |
| 3 | $b$ | $a$ | $b$ |
