Course Material : Introduction to Automated Reasoning , Course given at the 18 th European Summerschool in Logic , Language and Information , Malaga , Spain , July 31 st until August 4 th , 2006

Preliminaries In this chapter we give some preliminaries. The main intention is to introduce some notation. The reader can take the risk and skip this chapter if he is in a hurry. In this section we define the concepts of order, well-order, and ordinal number. An order is a relation that specifies how to line up the elements in a set: Definition 1.1.1 An order ≺ is a relation with the following properties: O1 Never d ≺ d. It is possible to obtain a relation from ≺ as follows: d 1 d 2 iff d 1 ≺ d 2 , or d 1 = d 2. This relation satisfies: O1' Always d d. O2' If d 1 d 2 , and d 2 d 3 , then d 1 d 3. O3' If d 1 d 2 , and d 2 d 1 , then d 1 = d 2. It is also possible to reverse the process: If is a relation that satisfies O1', O2', and O3', then a relation ≺ can be extracted by d 1 ≺ d 2 iff d 1 d 2 , and d 1 = d 2. It can be shown that ≺ satisfies O1 and O2. As a consequence we will call both ≺ and an order, and use the one that is most convenient, dependent on the situation. In the same way we will use the pair <, ≤, and <,. 5 6 CHAPTER 1. PRELIMINARIES Definition 1.1.2 If ≺ is an order, and D is a set, and d ∈ D, then d is maximal in D if for no d ∈ D it is the case that d ≺ d. d is minimal in D if for no d ∈ D, it is the case that d ≺ d. If a minimal element is guaranteed to exist and unique, we will write Min(D) for the minimal element of D. In the same way we write Max(D) for the maximal element of D. Minimal and maximal elements do not necessarily exist in a set D, and they are not necessarily unique if they exist. If however ≺ is total, then maximal and minimal elements are unique if they exist. An order for which minimal elements do always exist, is called well-founded: Definition 1.1.3 An order ≺ is well-founded on a set D, if every nonempty X ⊆ D contains minimal elements. …