Some More Logic
This chapter is slightly shorter than the other chapters in the Mathematics section of the book, that however does not make it any easier. It continues on from where Chapter 1 finished. You can have a look at it, again via google books.
Starts off with a discussion of the various properties of the logical connectives. Then moves on to talking about Quantifiers. One way of turning a predicate into a proposition is by the use of quantification to say for how many possible values a parameter of a predicate actually meet a specific requirement.
The two fundamental quantifiers are universal quantification (for all) and exitenstial quantification (there exists). There follows some examples of these in action. An example might help clarify this:
The above can be read as “For all x that are members of the set N of natural numbers 2 times that value is also a member of the set of natural numbers”
Interesting point that when working with finite sets can be treated as an iterated OR while the can be treated as an iterated AND.
Quantification over the empty set is discussed. I fear that universal quantification over the empty set being always TRUE completely and utterly baffles me. Interestingly, wikipedia have this as a vacuous truth.
Moves on to discuss various properties of quantifiers, nesting, negation, distributive and various rewrite rules. I found this really quite challenging.
Finishes the chapter talking about normal forms (canonical form) is just a standard way of representing an object. In logic normal forms allow you to more easily compare two predicates. conjunctive normal form (CNF) is a conjunction (and) of clauses where a clause is a disjunction (or) of literals. disjunctive normal form (DNF).
As often seems to be the case as the mathematics section is building up a toolset for practical application later, normal forms are utilised later in the book in relation to data integrity constraints.