Post AsuIE2INxAe2NInQ5g by mjd@mathstodon.xyz
 (DIR) More posts by mjd@mathstodon.xyz
 (DIR) Post #AsuIE2INxAe2NInQ5g by mjd@mathstodon.xyz
       2025-04-08T16:28:05Z
       
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       What does "without loss of generality" mean really?Here are a couple of simple examples:a. Consider a jar that contains red beans and blue beans.  Select a bean from the jar.  Without loss of generality, it is red.b. Let real numbers \( x\) and \(y\) be given.  Without loss of generality, \( x\le y\).c. Let \( S\) be a countable set.  Without loss of generality, \(S=\Bbb  N\).
       
 (DIR) Post #AsuIE3WFP7KMAaI0Ku by mjd@mathstodon.xyz
       2025-04-08T16:32:57Z
       
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       One is tempted to say that it captures the existence of some sort of symmetry in the problem statement, but the notion of "symmetry in the problem statement" is exactly the part I want to explicate.
       
 (DIR) Post #AsuIE4f99W2XiTScqW by mjd@mathstodon.xyz
       2025-04-08T16:38:32Z
       
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       Symmetry in the problem statement implies that there is a mapping from problem statements to problem statements and that this particular mapping is invertible.  But I don't know any formal construction of problem statements.And also, it's not enough to _just_ have an invertible mapping.  Consider a auto-bijection of the points of a square that scrambles the points around at random. This could be considered a symmetry of the square, but it normally isn't. Usually we want the mapping to respect the geometric structure of the square, for it to be an isometry. The correct understanding of symmetries of problem statements will require them to be _structure-preserving_ maps.Is there a category of mathematical problems?
       
 (DIR) Post #AsuIE5SmAzjsCOgGCe by mjd@mathstodon.xyz
       2025-04-09T06:16:13Z
       
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       I feel like there is some standard term for "autobijection" that I should have used here.
       
 (DIR) Post #AsuIE6A1aCKkMWunc8 by mjd@mathstodon.xyz
       2025-04-09T06:45:29Z
       
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       "Permutation". The word is "permutation".Sheesh.
       
 (DIR) Post #AsuIECg9Lfb0ZtzXWK by mjd@mathstodon.xyz
       2025-04-08T16:40:33Z
       
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       Going back to example (c) of the ancestor post, consider:c1. Let \(S\)  be a countable set.  Without loss of generality, \(S=\Bbb N\).c2.  Let \(S\)  be a totally ordered countable set.  Without loss of generality, \(S=\Bbb N\). *(c1) is good, but (c2) is bad, because in (c2), \(S\) could be \(\omega + 1\).  The object types are an important part of the problem statement.
       
 (DIR) Post #AsuIEKr4wv5jw7ANfM by mjd@mathstodon.xyz
       2025-04-08T18:35:04Z
       
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       Here's a supeerficially weird example:“Let \( r\) be a positive real number.  Without loss of generality, \( r=1\).”But in some contexts this makes perfect sense.  In fact I thinking on it a little more, I realize that I have seen numerous examples of exactly this.  It's just more often phrased this way: “By suitable choice of units, assume that \( A\) and \(B\) are a unit distance apart.”
       
 (DIR) Post #AsuJIc0jCtJ2FwHbg8 by mjd@mathstodon.xyz
       2025-04-09T07:14:13Z
       
       1 likes, 0 repeats
       
       @josh Automorphism implies structure-preserving. That was the opposite of what I wanted in this case.