Post AmtsRIDnzDunR1EdrE by antoinechambertloir@mathstodon.xyz
 (DIR) More posts by antoinechambertloir@mathstodon.xyz
 (DIR) Post #Amts2fg1dyAiwLh7LM by antoinechambertloir@mathstodon.xyz
       2024-10-11T14:38:25Z
       
       0 likes, 0 repeats
       
       Light poll for people who read math and sometimes grade exams — the question was to give the definition of a well ordered set. (For the other one, I copied two extracts of textbooks in the next message.) A student answers “An ordered set is well-ordered if every subset has a smallest element.” How many points would you give:
       
 (DIR) Post #Amts2gh7rMe863DVhI by ColinTheMathmo@mathstodon.xyz
       2024-10-11T14:46:47Z
       
       0 likes, 0 repeats
       
       @antoinechambertloir As always it depends on the context, and what the student is attempting to become.They missed "linear" in the "ordered set" (it might be partially ordered), and they missed non-empty (they probably assumed it).Both are important but technical.I don't know whether I should vote, but I want to see the results.  So I'm voting 1/2 because they got the broad substance, but not the technical components.
       
 (DIR) Post #Amts2h7iGVuhQVek8e by josh@squ.alid.pw
       2024-10-11T14:58:14.659371Z
       
       0 likes, 0 repeats
       
       @ColinTheMathmo @antoinechambertloir aren't both of those conditions implied by "every subset has a least element"? To show that x and y are comparable, take the least element of {x, y}, and for "non-empty" take the smallest element of ∅
       
 (DIR) Post #Amts71FRNtKt05KhOK by josh@squ.alid.pw
       2024-10-11T14:59:03.822697Z
       
       0 likes, 0 repeats
       
       @ColinTheMathmo @antoinechambertloir Practically speaking I'd probably want to see that the student understands that these important conditions follow from the basic claim, but you can't quibble that it's not an equivalent definition
       
 (DIR) Post #AmtsEKbPp74v3Q692G by josh@squ.alid.pw
       2024-10-11T15:00:23.370011Z
       
       0 likes, 0 repeats
       
       @ColinTheMathmo @antoinechambertloir actually, I suppose now I think about it maybe you don't get a least element of ∅ for free, depending on how you've specified "least element". So you probably do still need to explicitly require non-empty
       
 (DIR) Post #AmtsRIDnzDunR1EdrE by antoinechambertloir@mathstodon.xyz
       2024-10-11T15:01:02Z
       
       0 likes, 0 repeats
       
       @josh @ColinTheMathmo the empty subset has no element, so it can't have a smallest one.And Indeed, the condition implies the ordering is total (aka linear).
       
 (DIR) Post #Amtsd5jIG2RyAMabvU by josh@squ.alid.pw
       2024-10-11T15:04:51.764490Z
       
       0 likes, 0 repeats
       
       @antoinechambertloir @ColinTheMathmo you're right, as I realised in my other reply I think I was getting least element and infimum confused in my head. I think that solidifies my confidence in 1/2, since the non-empty condition is important, and you can view total ordering as a theorem about well-ordered sets.