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What is the best way to prepare for the transition into proof based mathematics in college following high school. #Spring25

I've taken Calc I-III, Linear Algebra, and I'm currently studying a bit of Real Analysis, but I just want to make sure that I'm fully prepared. ?

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Chiquria’s Answer

Get ready to succeed in proof-based mathematics by focusing on building your logical reasoning and proof-writing skills. Since you already know Calculus and Linear Algebra, it's a great time to explore formal proof techniques like direct proofs, proof by contradiction, proof by induction, and using counterexamples. These skills are key, and you can improve them by working through your Real Analysis textbook or beginner books like "How to Prove It" by Daniel Velleman.

Clear and precise writing is vital in proof-based math, so practice writing detailed proofs step by step without skipping explanations. Review important definitions and theorems from your past studies, especially in Real Analysis and Linear Algebra. Regular practice is crucial—tackle problems from textbooks, online resources, or problem sets, and break down complex problems into smaller parts if needed.

Reading and analyzing well-written proofs in textbooks or academic papers can show you how experts build their arguments. Don't be afraid to ask for feedback from classmates or professors, as working with others can offer new insights into challenging problems. Be ready to think more abstractly, as proof-based math involves less calculation and more abstract thinking. By focusing on logic, writing, and abstract reasoning, and practicing often, you'll be on the path to success in your proof-based math courses.
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Teklemuz Ayenew’s Answer

Transitioning into proof-based mathematics in college can be challenging, but with the right approach, resources, and a proactive mindset, it’s entirely manageable. Focus on mastering essential proof techniques like direct proofs, contradiction, and induction. Books like How to Prove It by Daniel Velleman are excellent for building your proof-writing skills. Strengthen your understanding of formal logic and set theory with resources like Naive Set Theory by Paul Halmos. For real analysis, focus on limits and continuity, while for linear algebra, concentrate on vector spaces and eigenvalues. Texts like Principles of Mathematical Analysis by Rudin and Linear Algebra Done Right by Axler are key resources.

Regular practice is crucial—solve problems, study proofs, and understand the reasoning behind them. Use online platforms like 3Blue1Brown for intuitive visual explanations, or watch lectures on Khan Academy, MIT OpenCourseWare, and YouTube channels such as Professor Leonard. Participate in study groups or forums like Math Stack Exchange to discuss challenges and gain support. Websites like Art of Problem Solving offer challenging problems to sharpen your skills.

With consistent effort and patience, you'll be well on your way to mastering proof-based mathematics.
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