ABSTRACT
The three central axioms of the theory are introduced in this chapter: unions, power set, replacement. These axioms guarantee that the simple operations with classes seen so far (like unions, intersections, cartesian products, differences, domains, etc.) produce sets when they are applied to sets. This is mainly dealt with in the last section, which studies sets that are relations, functions or operations. In that section, bijective functions are considered, as well as functions and operations that are compatible with an equivalence.
The axiom of replacement is an axiom scheme, that is, there is an axiom (an instance of the axiom) for each concrete class. Each instance of the axiom of replacement entails an instance of the principle of separation, a principle which has a much more intuitive meaning and is far more useful than the restricted form of replacement. In fact, the study of maps and relations in sets of the last section is made possible by the use of separation, the main tool in proving that certain collections are sets. Separation is not taken as an independent axiom because having only one axiom scheme (replacement) instead of two has some technical advantages.
