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Unformatted text preview: Notes by David Groisser, Copyright c 1998 What is a proof? When asked to “show” or “prove” something, you are being asked to supply an airtight logical argument leading from the hypothesis to the conclusion. A written proof is a one-way conversation between the writer and the reader. This conversation takes place in English. To save space and time, mathematical symbols may be used to stand for words, but each mathematical symbol has a fixed, conventional word (or a small set of words) that it is allowed to substitute for; you are not free to make up your own (unless you explicitly state what you are defining your symbols). For example, “=” stands for “equals”, “which equals”, or “is equal to”; it does not stand for “Doing the next step in this problem, I arrive at the expression I’m writing to the right of the equals sign”. Your written work should have the property that, when the conventional meanings of your symbols are substituted for the symbols themselves, the result is a collection of sentences, with correct grammar and punctuation, detailing the logical flow of the argument. Some useful mathematical abbreviations are the following: • “ ∀ ” stands for “for all”, “for every”, or “for each” • “ ∃ ” stands for “there exists” or “there exist” • “ ⇒ ” stands for “implies” or “which implies” • “ ⇐ ” stands for “which is implied by” (this can also be read “implies”, if you read from right to left or from the bottom of a page up) • “ ⇐⇒ ” stands for “if and only if” or “which is equivalent to” When you see someone else’s final proof, it may appear that he or she has pulled something out of thin air. This is a common misunderstanding of what’s being asked in “show” and “prove” problems. Your approach to any such problem involves some thought process , which you hope will lead you to an answer (i.e. a proof). As far as the thought process goes, anything is valid—you can work backwards, make leaps of faith, make mistakes, etc. This is all okay because at this stage you are not claiming to have an answer. All you are doing is trying to collect facts and ideas that you will later assemble into an answer. This thought process is not what you are being asked to write down; you are only being asked to write down the final proof. When you write this down correctly, you are showing two things: (i) that you recognize what a valid proof is, and (ii) in all likelihood, you came upon your proof by an intelligent thought process, because the chance that you stumbled onto a correct proof by pure luck is very small. There is no single method guaranteed always to lead you to a proof, but here are a few methods that work well in certain problems: 1 • If you’re instructed “Show that this thing here is a widget”, usually what you have to do is write down the definition of widget and check that this thing here meets all the criteria of the definition. If the problem reads “Show that this thing here isthe criteria of the definition....
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- Spring '09
- Logic, Natural number, Mathematical logic, Universal quantification, Gators