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Unformatted text preview: How to write proofs: a quick guide Eugenia Cheng Department of Mathematics, University of Chicago Email: [email protected] Web: http://www.math.uchicago.edu/ ∼ eugenia October 2004 A proof is like a poem, or a painting, or a building, or a bridge, or a novel, or a symphony. “Help! I don’t know how to write a proof!” Well, did anyone ever tell you what a proof is , and how to go about writing one? Maybe not. In which case it’s no wonder you’re perplexed. Writing a good proof is not supposed to be something we can just sit down and do. It’s like writing a poem in a foreign language. First you have to learn the language. And then you have to know it well enough to write poetry in it, not just say “Which way is it to the train station please?” Even when you know how to do it, writing a proof takes planning, effort and inspira tion. Great artists do make sketches before starting a painting for real; great architects make plans before building a building; great engineers make plans before building a bridge; great authors plan their novels before writing them; great musicians plan their symphonies before composing them. And yes, great mathematicians plan their proofs in advance as well. 1 Contents 1 What does a proof look like? 3 2 Why is writing a proof hard? 3 3 What sort of things do we try and prove? 4 4 The general shape of a proof 4 5 What doesn’t a proof look like? 6 6 Practicalities: how to think up a proof 9 7 Some more specific shapes of proofs 10 8 Proof by contradiction 15 9 Exercises: What is wrong with the following “proofs”? 16 2 1 What does a proof look like? A proof is a series of statements , each of which follows logically from what has gone before. It starts with things we are assuming to be true. It ends with the thing we are trying to prove. So, like a good story, a proof has a beginning, a middle and an end. • Beginning: things we are assuming to be true, including the definitions of the things we’re talking about • Middle: statements, each following logically from the stuff before it • End: the thing we’re trying to prove The point is that we’re given the beginning and the end, and somehow we have to fill in the middle. But we can’t just fill it in randomly – we have to fill it in in a way that “gets us to the end”. It’s like putting in stepping stones to cross a river. If we put them too far apart, we’re in danger of falling in when we try to cross. It might be okay, but it might not . . . and it’s probably better to be safe than sorry. 2 Why is writing a proof hard? One of the difficult things about writing a proof is that the order in which we write it is often not the order in which we thought it up. In fact, we often think up the proof backwards ....
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