Paloma (University of the Balearic Islands, Spain) Fresno-Calleja & Janet M. (University of Northampton, UK) Wilson 
An Elementary Transition to Abstract Mathematics [PDF ebook] 

An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role.

The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part introduces key objects from abstract algebra; and the final part focuses on polynomials.

Features:

• The material is presented in many short chapters, so that one concept at a time can be absorbed by the student.
• Two ‘looking back’ chapters at the outset (pre-calculus and calculus) are designed to start the student’s transition by working with familiar concepts.
• Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns.
• A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student.