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understanding - how it helps you follow later statements If...

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Some Thoughts on How to Understand Proofs and to Write Them Most people find it difficult to learn what constitutes a valid proof. Often ability in this area comes about by some form of “osmosis” in which eventually it either sinks in — or it doesn’t! I believe it is possible, by expending some effort, to build a feeling for this essential area of mathematics. Some thoughts are presented next. 1. As you are examining a proof written by someone else, do the following: For each statement, figure out what is its job. If it is defining some notation, notice how the notation relates to objects in the theorem statement. For example, if a statement “Let F be a field” appears, note that this probably is giving a specific name to the generic concept of “field” which appears in the theorem statement. Often this allows the proof to be more concise than it otherwise would be. If the statement is a clarifying one, attempt to understand
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Unformatted text preview: how it helps you follow later statements. If the statement is advancing the proof by creating a new result, look for and find the justification even if it is not given explicitly in the proof. Determine where each hypothesis of the theorem is used, making sure all hypotheses are. If a hypothesis truly is not used, it need not be in the theorem statement and a stronger theorem (because it has fewer restrictions) is the result. If a proof says that other parts of the conclusion can be obtained by similar arguments, do those arguments. 2. Practice the scrutiny outlined above on All theorems presented in class All theorems we cover in the text Theorems you run into in other courses 3. Once you have written a proof, examine it with the same sharp eye using the above suggestions. 4. If possible, show your proof to some knowledgeable friend and ask him or her to harshly criticize it....
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