The sixteen chapters of this book provide the core material for the Elements of Mathematics Program, a secondary sequence developed for highly motivated students with strong verbal abilities. The sequence is based on a functional-relational approach to mathematics teaching, and emphasizes teaching by analysis of real-life situations. This text is intended to provide students with a general background in the analytical methods of mathematics and practice in reasoning with mathematical objects; other texts (Books 1-12) prepared for use with book O explore particular areas in depth. Beginning with everyday examples of ordered pairs, this chapter introduces the idea of ordered set and develops the notion of the solution of an open sentence in n variables as a set of ordered n-tuples. Vector addition is introduced through grid diagrams. Identities for subsets of Cartesian products are developed. (SD)