This book is essentially self-contained. The only previous knowledge required is high school algebra and the understanding that usual algebraic rules for transforming expressions, and solving equations and inequalities, can actually be derived from the properties of addition, multiplication, exponentiation and order in the real number system. Another prerequisite is of psychological nature: the reader is expected to find delight in rigorous thinking. This is absolutely necessary to enjoy the book. We warn the reader that due to the fact that matters are treated with complete rigor, the reading of arguments quite often may turn painful.
It is important to remark that a deliberate effort was made in presenting algebraic manipulations by what they are, i.e. logical transformations of statements. Let us consider for example the solution process of the equation 5x − 2 = 2x + 7 in the real number system. In high school this process is explained as follows:
“Let us solve the equation 5x − 2 = 2x + 7. The −2 passes to the +2, and the 2x passes to the other side as −2x, obtaining 5x − 2x = 7 + 2, which is 3x = 9. Now since 3 is multiplying on the left hand side, it passes to divide to the right hand side, and so
This is not an explanation at all! This is the application of an algorithm which indeed solves the equation. A good explanation would be as follows:
“Let us solve the equation 5x−2 = 2x+7 in the real number system. This means that we want to determine all the possible real numbers x such that five times x minus 2 equals twice x plus 7. Properties of addition and multiplication among real numbers, imply the validity of all the following assertions.
The sentence
“x is a real number such that 5x − 2 = 2x + 7”
is logically equivalent to the sentence
“x is a real number such that (5x − 2) + 2 = (2x + 7) + 2”
(Logical equivalence means that the first sentence implies the second sentence, and that the second sentence implies the first sentence).
Likewise, the sentence
“x is a real number such that (5x − 2) + 2 = (2x + 7) + 2”
is logically equivalent to the sentence
“x is a real number such that 5x = 2x + 9”.
Now, the sentence
“x is a real number such that 5x = 2x + 9”
is logically equivalent to the sentence
“x is a real number such that 5x − 2x = (2x + 9) − 2x”
and this last sentence is logically equivalent to the sentence
“x is a real number such that 3x = 9”.
Finally, the sentence
“x is a real number such that 3x = 9”
is logically equivalent to the sentence
“x is a real number such that
which is logically equivalent to the sentence
“x is a real number such that x = 3”.
In conclusion, the sentence
“x is a real number such that 5x − 2 = 2x + 7”
is logically equivalent to the sentence
“x is a real number such that x = 3”.
But determining all the possible objects x satisfying the latter condition is trivial; only the real number 3 satisfies it. One concludes that x = 3 and no other real number, is such that 5x − 2 = 2x + 7.”
We make some remarks about the exercises proposed in the book. They vary in several respects. Some exercises are proposed as the theory is developed. These type of exercises are meant to help understanding the ideas that are being developed. At the end of some sections or strings of sections there are sets of exercises. These exercises are intended to expose the reader to variations of the situations treated in the corresponding section or string of sections. Many times in reading a proof, the reader will find indications like (?), (do it!), (why?), (check!), inviting the reader to reflect or take the corresponding action, about what has just been claimed. Also, some parts of some proofs and examples, are explicitly left as exercises for the reader.
Finally, the book has many, many pictures, for illustrating concepts, steps of proofs, etc. There is a constant effort in presenting two pictures of the same concept, an abstract one and a concrete one, where the plane is taken to be the usual euclidean plane.
Chapter 2 is a preliminary chapter, necessary for the understanding of the rest of the book. It starts with a review of the methods for proving statements of the form “P implies Q”, and also of methods for proving other types of statements, with particular emphasis on the Induction Method, used for proving statements of the form “P(n) for n ≥ n0”. Then there is a rapid introduction to the symbolism used in logic. After this the basics of the elementary theory of sets are reviewed, including a discussion of the notion of equivalence relation, and of the important fact that an equivalence relation defined on a set, determines a partition of the set into equivalence classes.
Chapter 3 introduces the notion of incidence geometry as a set together with a collection formed by some of its subsets, having three properties called axioms of incidence. Then examples of incidence geometries of various kinds are presented. Then the “main” examples of incidence geometries, namely the real cartesian plane, the hyperbolic plane and the elliptic geometry, are presented in complete detail. In particular, complete proofs, based only on the properties of addition and multiplication in the real number system, that the three axioms of incidence hold in these examples, are supplied. After this the subject of parallelism of lines is discussed, and an additional axiom, called Playfair’s axiom is studied. Playfair’s axiom is a refined version of Euclid’s fifth postulate. The chapter ends with a long discussion of how paralllelism behaves in all the examples of incidence geometries previously given. It is of particular importance the discussion of the behaviour of parallelism in the real cartesian plane, the hyperbolic plane and the elliptic geometry. It is shown that in the real cartesian plane, given any point A and any line l, with A not in l, there exists a unique line passing through A and being parallel to l; that in the hyperbolic plane, given any point A and any line l, with A not in l, there exists an infinite number of lines passing through A and being parallel to l; and that in the elliptic geometry, given any point A and any line l, with A not in l, there is no line passing through A and being parallel to l.
Chapter 4 treats the formalization of the notion that one point is between two other points, i.e., the concept of a betweenness structure for an incidence geometry. A betweenness structure is defined as a collection of ordered triples of points of the plane, having four properties called the betweenness axioms. We remark that the realization