Successfully addressing the frustration many students feel as they make the transition from beginning calculus to a more rigorous level of mathematics, A Transition to Advanced Mathematics provides a firm foundation in the major ideas needed for continued work in the discipline. The authors guide students to think and to express themselves mathematically--to analyze a situation, extract pertinent facts, and draw appropriate conclusions. With their proven approach, Smith, Eggen, and St. Andre introduce students to rigorous thinking about sets, relations, functions and cardinality. The text also includes introductions to modern algebra and analysis with sufficient depth to capture some of their spirit and characteristics.

Benefits:

• The section in the Fourt Edition on basic proof techniques has been divided into a section on direct proofs and a section on indirect proofs. This facilitates understanding the differences in those approaches to proofs and when each may be applied.
• Worked examples and exercises throughout the text, ranging from the routine to the challenging, reinforce the text.
• Proofs To Grade exercises help students learn mathematical thinking by observing, in a non-threatening way, instances of flawed reasoning and misunderstandings of definitions and theorems. Both routine and novel correct proofs are included among these exercises.
• Chapters introducing modern algebra (from groups to group homomorphisms) and analysis (equivalents of the completeness of the reals).
• Discussions of sequences and real analysis concepts are tied to students' experience in elementary calculus.
• A section on orders and partial orders is included.
• Additional early examples of proofs, both direct and indirect.
• Basic number theory concepts have been gathered together in the section that introduces direct proofs.
• Improved explanations and additional examples for topics that students often find difficult.
• Improved exercises, especially those of the Proofs to Grade variety.
• Optimally organized. Introduces enough logic to justify basic proof techniques, and builds from the study of sets to relations, to functions, and then to cardinality.
• There are extensive explanations of: working with quantifers, operations with families of sets, induction, equivalence relations, injective and surjective functions, and finite, denumerable, and uncountable sets.
• A summary of approaches to writing proofs.
• In addition to teaching how to do proofs, the text provides the student with an introduction to fundamental mathematical concepts that are useful in all upper level mathematics courses.