Set Theory: Proofs by Detachment, Contraposition, and Contradiction

Abstract

This chapter introduces set theory from two parallel perspectives: as an intuitive mathematical theory, and as a simple applied predicate calculus of first order. Starting from first-order logic and some of the Zermelo-Fraenkel axioms (extensionality, empty set, pairing, power set, separation, and union), where all objects under consideration are sets, the chapter first derives relations between sets, subsets, supersets, unions, intersections, and Cartesian products of sets of sets. Subsequent sections introduce relations, functions, injections, surjections, bijections, composite functions, and inverse functions. Another section focuses on the duality between partitions and equivalence relations. The last section deals with pre-orders, partial orders, linear or total orders, and well-orders. Many proofs begin with an informal intuitive proof, then demonstrate how to design a more formal proof, and finally present a detailed outline of such a formal proof in first-order logic. The other Zermelo-Fraenkel axioms (choice and infinity or substitution) are only mentioned here, because they form the topic of subsequent chapters. The prerequisites for this chapter consist of a working knowledge of first-order logic, for instance, as described in chapters 1 and 2, which contain all the logical theorems cited in this chapter.