Introduction to Proofs and Proof Strategies

Emphasizing the creative nature of mathematics, this conversational textbook guides students through the process of discovering a proof. The material revolves around possible strategies to approaching a problem without classifying 'types of proofs' or providing proof templates. Instead, it helps students develop the thinking skills needed to tackle mathematics when there is no clear algorithm or recipe to follow. Beginning by discussing familiar and fundamental topics from a more theoretical perspective, the book moves on to inequalities, induction, relations, cardinality, and elementary number theory. The final supplementary chapters allow students to apply these strategies to the topics they will learn in future courses. With its focus on 'doing mathematics' through 200 worked examples, over 370 problems, illustrations, discussions, and minimal prerequisites, this course will be indispensable to first- and second-year students in mathematics, statistics, and computer science. Instructor resources include solutions to select problems.

強調數學的創造性本質，這本對話式教科書引導學生通過發現證明的過程。教材圍繞著解決問題的可能策略展開，而不是將「證明的類型」進行分類或提供證明模板。相反，它幫助學生培養解決數學問題所需的思考能力，即在沒有明確的算法或配方可循時如何應對數學問題。從從更理論的角度討論熟悉且基礎的主題開始，本書進一步涵蓋了不等式、歸納法、關係、基數和初等數論。最後的補充章節讓學生能夠將這些策略應用到他們將來學習的主題上。通過200個實例、超過370個問題、插圖、討論和最低的先備知識，這門課程對於數學、統計和計算機科學的一、二年級學生來說是不可或缺的。教師資源包括選定問題的解答。