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For example I have a proof where I can only use positive numbers less than or equal to 4 for a variable. The equation just off looking is pretty obvious, if I plugged any number from 1-4 into it then I'd prove the proposition is true. I wanted to give one example, where I assume the variable is 1 show the proposition is true then say WLOG the other 3 cases are true. Is that too much of an over reach in solving proofs?

Ahmed Kidwai
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  • There is a more general question here, which is "How much detail can I leave to the reader when I write proofs?" How much you can skip by simply writing "WLOG" is a special case of this. – Arthur Feb 07 '18 at 09:46
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    See this. – Pedro Feb 07 '18 at 10:00
  • This sort of things depend a lot on who you expect to read your proof. – Arnaud D. Feb 07 '18 at 10:17
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    I think "w.l.o.g." point to a symmetry in the setup. If the calculations are wastly different then I think this phrase is misplaced. But one can say "we exemplarily show this for $x=1$. The calculations for $x=2,3,4$ are comparatively easy." – M. Winter Feb 07 '18 at 10:35

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You are probably ok showing one example and assuming the rest. However where you seem to be going wrong is in saying 'WLOG'. This expression has a specific meaning where symmetry or some similar consideration mean that it is logically valid to skip a case. For example, if there are two interchangeable numbers in your proof, one must be greater than or equal to the other. So we can assume that $a \geq b$, since $b \geq a$ is symmetric.

This only applies when the symmetric cases are interchangeable. It certainly doesn't seem to be the case for 1, 2, 3 or 4. So you should use different language if you are skipping one or more of these.

However, isn't it possible to write a more general proof which works for any number smaller or equal to 4? This is probably a good, readable, approach if possible.

jwg
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