A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof. For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn's lemma and the axiom of choice are included. More challenging problems are marked with a star. All these materials are optional, depending on the instructor and the goals of the course.

Von Geheimrat Professor M. Rub n e r. Die Getreidearten liefem fUr die menscWiche Emiihrung einen wesentlichen Teil der Volksnahrung, bei den Europaem zwischen 40-60%, bei vielen asia schen Volkem noch viel mehr. Die Mehrzahl der \Veltbevolkerung gehort allerdings noch zu den Brei essern, welche Brot nicht herstellen, wie die Reisesser. Bei uns war der Brotverbrauch noch Ende des 18. und zu Beginn des 19. J ahrhunderts viel groBer wie heutzutage. Das Brot ist in Deutschland allmiihlich durch die Kartoffel verdrangt worden. Heute bestehen bei uns 40,8% des Konsums aus Brot- und Mehl verbrauch und 12,0% der Kost aus Kartoffeln, wiihrend der Ita liener kaum 2% seiner Nahrung durch die letztere deckt. In Deutschland wurden vor dem Krieg 7,6 Millionen Tonnen Roggen und 3,7 Millionen Tonnen Wei zen geerntet und 2 Millionen Tonnen Wei zen eingefiihrt, woraus sich ein Verhaltnis des Konsums von I Teil Weizen zu 1,6 Teilen Roggen ergibt. In anderen Liindern tritt der Roggen als Brotfrucl, J.t gegentiber dem Weizen sehr oder vollig zurtick. Die Miillerei stellt aus dem Getreide nicht ei ne Sorte Mehl her, sondern seit Einfiihrung der Hochmiillerei je nach Bedarf und Preisverhaltnissen der Ernten sehr verschiedene Mehlsorten, be sonders aus Weizen. Einfacher ist die Vermahlung beim Roggen. DaB Getreide ganz mitseinen Hiilsen zu Mehl verarbeitet wird, ist selten, noch seltener das einfache Schroten. Die tiberwiegende Masse des MeWes wird unter AbfaH von Kleie gewonnen. Letztere hat mit Rticksicht auf die verschiedene Beimengung von MeW eine ganz verschiedene Zusammensetzung."

A Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next is the construction of integers including some elementary number theory. The notions of finite and infinite sets, cardinality of counting techniques and combinatorics illustrate more techniques of proof. For more advanced readers, the text concludes with sets of rational numbers, the set of reals and the set of complex numbers. Topics, like Zorn's lemma and the axiom of choice are included. More challenging problems are marked with a star. All these materials are optional, depending on the instructor and the goals of the course.