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elements_of_style_for_proofs

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elements_of_style_for_proofs [2013/09/20 08:00]
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> 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.1379678446.txt.gz · Last modified: 2013/09/20 08:00 by bmwoodruff

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