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.
On
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 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.