Binary Relations, Lattices, and Infinity

Abstract

Section 5.1 introduces the two most popular flavors of binary relations on a set, namely equivalence relations and partial orders. Modular arithmetic is seen as a special case of forming equivalence classes, and we see that a variety of objects of interest to computer scientists can be related to a partial order. Particularly important are those partial orders for which we can speak of maxima and minima. We discuss these objects — called lattices — in Section 5.2. There we pay particular attention to the Boolean algebras, an important class of lattices which include the truth values equipped with the operations of disjunction and conjunction, and the set of subsets of a set equipped with union and intersection. Then, in Section 5.3, we show that some infinities are bigger than others ! This may seem irrelevant to computer science, but in fact quite the opposite is true — the technique used in this proof, Cantor’s diagonal argument, is one of the most important techniques in the theory of computability, and can be used to demonstrate that certain problems cannot be solved by any computer program. Finally, we conclude this chapter with a general, abstract look at trees. Tree structures have shown up in a number of important contexts so far in this book, and the treatment they are given in Section 5.4 highlights some of the algebraic features these structures have in common.