The most important part of this chapter is the section having to do with predicates (along with the subsequent section, which discusses the related issue of relations versus types). Basically, every relvar R has an associated predicate P (the relvar predicate for R); P is the intended interpretation or intension for R, and it doesn't change over time. And if the current value of R is relation r, then r represents the current extension of P; it consists of a set of tuples, each of which represents a true proposition, or in other words a true instantiation of P. The extension does change over time. Thus, a database together with its operators can be seen as a logical system.
Here are some other important points from the body of the chapter:
Only relvars are updatable; talk of "updating a tuple" or "updating an attribute" is convenient but sloppy. Updating is always set-at-a-time.
Every relvar has at least one (candidate) key. Keys have the properties of uniqueness and irreducibility. Key values are tuples.
Some relvars have foreign keys. SQL supports certain associated "referential actions," such as CASCADE; such actions might well be useful in practice, but they aren't part of the relational model. What's more, foreign keys themselves aren't truly fundamental, either.
Operations on views are implemented by mapping them into appropriate operations on the underlying base relvars. (The mapping process, which basically works because of closure, is straightforward for read-only operations but less so for update operations.) The Principle of Interchangeability states that there must be no arbitrary and unnecessary distinctions between base and virtual relvars.
"Types are to relations as nouns are to sentences."
Types and relations are both necessary and sufficient to represent any data we like. (At the logical level, that is; of course, other constructs are useful at the physical level, as we all know, but that's because the objectives are different at that level. The physical level is deliberately beyond the purview of the relational model.)