Basics

The basic notational conventions used in this book are described. Composition of mappings is defined the standard left-handed way: $$f\ g$$ f g means mapping g was applied first. But things are a little more complicated than that since we must also deal with both left and right operators, binary operations and monoids. For example, right operators are sometimes indicated exponentially—that is by right superscripts (as in group conjugation)—or by right multiplication (as in right R-modules). Despite this, the “ $$\circ $$ ∘ ”-notation for composition will always have its left-handed interpretation. Of course a basic discussion of sets, maps, and equivalence relations should be expected in a beginning chapter. Finally the basic arithmetic of the natural and cardinal numbers is set forth so that it can be used throughout the book without further development. (Proofs of the Schröder-Bernstein Theorem and the fact that $$\aleph _0\cdot \aleph _0 = \aleph _0$$ ℵ 0 · ℵ 0 = ℵ 0 appear in this discussion.) Clearly this chapter is only about everyone being on the same page at the start.