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elements_of_style_for_proofs

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elements_of_style_for_proofs [2013/09/20 07:57]
bmwoodruff [Elements of style for proofs] 		elements_of_style_for_proofs [2016/04/19 20:36] (current)
bmwoodruff
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 </html> Some complete sentences can be written purely in mathematical symbols, such as equations (e.g., $a^3=b^{-1}$), inequalities (e.g., $x<5$), and other relations (like $5\big|10$ or $7\in\mathbb{Z}$). These statements usually express a relationship between two mathematical \emph{objects}, like numbers or sets.  However, it is considered bad style to begin a sentence with symbols.  A common phrase to use to avoid starting a sentence with mathematical symbols is "We see that..." </html> Some complete sentences can be written purely in mathematical symbols, such as equations (e.g., $a^3=b^{-1}$), inequalities (e.g., $x<5$), and other relations (like $5\big|10$ or $7\in\mathbb{Z}$). These statements usually express a relationship between two mathematical \emph{objects}, like numbers or sets.  However, it is considered bad style to begin a sentence with symbols.  A common phrase to use to avoid starting a sentence with mathematical symbols is "We see that..."
-  - **Show the logical connections among your sentences.** Use phrases like "Therefore" or "because" or "if\ldots, then\ldots" or "if and only if" to connect your sentences.+  - **Show the logical connections among your sentences.** Use phrases like "Therefore" or "because" or "if$\ldots$, then$\ldots$" or "if and only if" to connect your sentences. 
 +  - **Use paragraphs to organize your work into logical chunks.** If every sentence starts a new paragraph, then you are not logical organizing your work.  Similarly, if you have a long proof and all your sentences are in a single paragraph, you are not logically organizing your work. Use paragraphs to put structure and order to your work. 
   - **Know the difference between statements and objects.** A mathematical object is a //thing//, a noun, such as a group, an element, a vector space, a number, an ordered pair, etc. Objects either exist or don't exist. Statements, on the other hand, are mathematical //sentences//:  they can be true or false.<html>   - **Know the difference between statements and objects.** A mathematical object is a //thing//, a noun, such as a group, an element, a vector space, a number, an ordered pair, etc. Objects either exist or don't exist. Statements, on the other hand, are mathematical //sentences//:  they can be true or false.<html>
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 </html> Now this isn't a bad way of //finding// a proof. Working backwards from your goal often is a good strategy //on your scratch paper//, but when it's time to //write// your proof, you have to start with the hypotheses and work to the conclusion. </html> Now this isn't a bad way of //finding// a proof. Working backwards from your goal often is a good strategy //on your scratch paper//, but when it's time to //write// your proof, you have to start with the hypotheses and work to the conclusion.
   - **Be concise.** Most students err by writing their proofs too short, so that the reader can't understand their logic. It is nevertheless quite possible to be too wordy, and if you find yourself writing a full-page essay, it's probably because you don't really have a proof, but just an intuition. When you find a way to turn that intuition into a formal proof, it will be much shorter.   - **Be concise.** Most students err by writing their proofs too short, so that the reader can't understand their logic. It is nevertheless quite possible to be too wordy, and if you find yourself writing a full-page essay, it's probably because you don't really have a proof, but just an intuition. When you find a way to turn that intuition into a formal proof, it will be much shorter.
-  - **Introduce every symbol you use.** If you use the letter "$k$," the reader should know exactly what $k$ is. Good phrases for introducing symbols include "Let $n\in \mathbb{N}$," "Let $k$ be the least integer such that\ldots," "For every real number $a$\ldots," and "Suppose that $X$ is a counterexample."+  - **Introduce every symbol you use.** If you use the letter "$k$," the reader should know exactly what $k$ is. Good phrases for introducing symbols include "Let $n\in \mathbb{N}$," "Let $k$ be the least integer such that$\ldots$," "For every real number $a\ldots$," and "Suppose that $X$ is a counterexample."
   - **Use appropriate quantifiers (once).** When you introduce a variable $x\in S$, it must be clear to your reader whether you mean "for all $x\in S$" or just "for some $x\in S$." If you just say something like "$y=x^2$ where $x\in S$," the word "where" doesn't indicate whether you mean "for all" or "some".<html>   - **Use appropriate quantifiers (once).** When you introduce a variable $x\in S$, it must be clear to your reader whether you mean "for all $x\in S$" or just "for some $x\in S$." If you just say something like "$y=x^2$ where $x\in S$," the word "where" doesn't indicate whether you mean "for all" or "some".<html>
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 </html> If you were to write it backwards, saying "Let $a^{2}=b$," then your startled reader would ask, "What if $a^{2}\neq b$?" </html> If you were to write it backwards, saying "Let $a^{2}=b$," then your startled reader would ask, "What if $a^{2}\neq b$?"
   - **Make your counterexamples concrete and specific.** Proofs need to be entirely general, but counterexamples should be absolutely concrete. When you provide an example or counterexample, make it as specific as possible. For a set, for example, you must name its elements, and for a function you must give its rule. Do not say things like "$\theta$ could be one-to-one but not onto"; instead, provide an actual function $\theta$ that //is// one-to-one but not onto.   - **Make your counterexamples concrete and specific.** Proofs need to be entirely general, but counterexamples should be absolutely concrete. When you provide an example or counterexample, make it as specific as possible. For a set, for example, you must name its elements, and for a function you must give its rule. Do not say things like "$\theta$ could be one-to-one but not onto"; instead, provide an actual function $\theta$ that //is// one-to-one but not onto.
-  - **Don't include examples in proofs.** Including an example very rarely adds anything to your proof. If your logic is sound, then it doesn't need an example to back it up. If your logic is bad, a dozen examples won't help it (see **proof_by_example**). There are only two valid reasons to include an example in a proof: if it is a **counterexample** disproving something, or if you are performing complicated manipulations in a general setting and the example is just to help the reader understand what you are saying.+  - **Don't include examples in proofs.** Including an example very rarely adds anything to your proof. If your logic is sound, then it doesn't need an example to back it up. If your logic is bad, a dozen examples won't help it (see 19, **Don't "prove by example."**). There are only two valid reasons to include an example in a proof: if it is a **counterexample** disproving something, or if you are performing complicated manipulations in a general setting and the example is just to help the reader understand what you are saying.
   - **Use scratch paper.** Finding your proof will be a long, potentially messy process, full of false starts and dead ends. Do all that on scratch paper until you find a real proof, and only then break out your clean paper to write your final proof carefully. //Do not hand in your scratch work!//<html>   - **Use scratch paper.** Finding your proof will be a long, potentially messy process, full of false starts and dead ends. Do all that on scratch paper until you find a real proof, and only then break out your clean paper to write your final proof carefully. //Do not hand in your scratch work!//<html>
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 \item \textbf{Show the logical connections among your sentences.} Use phrases like ``Therefore'' or ``because'' or ``if\ldots, then\ldots'' or ``if and only if'' to connect your sentences. \item \textbf{Show the logical connections among your sentences.} Use phrases like ``Therefore'' or ``because'' or ``if\ldots, then\ldots'' or ``if and only if'' to connect your sentences.
-  + 
 +\item \textbf{Use paragraphs to organize your work into logical chunks.} If every sentence starts a new paragraph, then you are not logical organizing your work.  Similarly, if you have a long proof and all your sentences are in a single paragraph, you are not logically organizing your work. Use paragraphs to put structure and order to your work.  
 \item \textbf{Know the difference between statements and objects.} A mathematical object is a \emph{thing}, a noun, such as a group, an element, a vector space, a number, an ordered pair, etc. Objects either exist or don't exist. Statements, on the other hand, are mathematical \emph{sentences}:  they can be true or false. \item \textbf{Know the difference between statements and objects.} A mathematical object is a \emph{thing}, a noun, such as a group, an element, a vector space, a number, an ordered pair, etc. Objects either exist or don't exist. Statements, on the other hand, are mathematical \emph{sentences}:  they can be true or false.
  
elements_of_style_for_proofs.1379678278.txt.gz · Last modified: 2013/09/20 07:57 by bmwoodruff

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