+- MKLab (https://mklab.gr)
+-- Forum: [INDEX] (https://mklab.gr/forumdisplay.php?fid=1)
+--- Forum: MATHEMATICS (https://mklab.gr/forumdisplay.php?fid=3)
+---- Forum: BOOKS (https://mklab.gr/forumdisplay.php?fid=6)
+----- Forum: NEW BOOKS (https://mklab.gr/forumdisplay.php?fid=42)
+------ Forum: FOREIGN (https://mklab.gr/forumdisplay.php?fid=91)
+------- Forum: PURE AND APPLIED MATHS (https://mklab.gr/forumdisplay.php?fid=94)
+-------- Forum: FOUNDATIONS OF MATHS (https://mklab.gr/forumdisplay.php?fid=168)
+-------- Thread: Introduction to Mathematical Structures and Proofs [Gerstein] (/showthread.php?tid=1830)
Introduction to Mathematical Structures and Proofs [Gerstein] - mklabgr - 09-03-2026
Introduction to Mathematical Structures and Proofs
Author: Larry J. Gerstein
Publication date: 2012, 2nd edition
Publisher: Springer New York
Series:Undergraduate Texts in Mathematics
Introduction to Mathematical Structures and Proofs is designed as a bridge from computational undergraduate mathematics—especially calculus—to proof-based higher mathematics. Its central objective is to develop mathematical maturity: the ability to work comfortably with definitions, abstraction, logical reasoning, and rigorous proof. Gerstein emphasizes that successful mathematics requires both intuition and formal rigor, and frequently presents more than one proof of the same theorem to illustrate that mathematical arguments can be approached from different directions. The book is suitable both for a university transition course and for independent study.
The book begins with logic and proof techniques, then develops the basic language of sets, relations and functions. It proceeds to finite and infinite sets, explaining how comparison of cardinalities can be understood through functions and leading naturally to ideas associated with Cantor's theorems. The later chapters introduce combinatorics and number theory, covering counting principles, mathematical induction, divisibility, prime numbers, the Fundamental Theorem of Arithmetic and number-theoretic functions. The second edition also expands the number-theory material—including primitive roots—and introduces complex numbers and Gaussian integers.
A particularly valuable feature is its treatment of proof as a process rather than simply a finished argument. The exercises range from straightforward applications of definitions to problems requiring substantial insight, making the text useful for students preparing to study subjects such as real analysis, abstract algebra, topology or advanced number theory. It is therefore less a book about one particular branch of mathematics than a book about how mathematicians think, formulate statements and construct proofs.
Key takeaways
- Best suited for: students moving from calculus to rigorous university mathematics.
- Core goal: learning how to understand, construct and write mathematical proofs.
- Main topics: logic → sets → functions → finite/infinite sets → combinatorics → number theory → complex numbers.
- Strong point: many examples and exercises, with multiple proofs of some important results.
- Level: approximately early undergraduate, but particularly useful before courses such as real analysis, abstract algebra and topology.
- Overall: an excellent example of the classic “Introduction to Proofs / Transition to Higher Mathematics” textbook.
BOOK