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About the exercise on abelian groups, I hope you noticed how many people started tinkering with it. Can we skip/weaken one or the other assumption? Will it work for non abelian groups? Some related the problem to other mathematical structures.
As I often do, I learned from the thread, both in language ( @benjamindickman raised a question on closure) and in maths ( @ProfKinyon knows what a quasigroup is, I didn& #39;t — I still don& #39;t but at least I googled it).
More importantly for this thread, @yet_so_far and @thewordninja_bk discussed whether having a definition with more assumptions is better or worse than a streamlined one. Which leads us to the heart of this thread:
It is very, very useful to know as many equivalent definitions as possible. In research papers, I am often tempted to have Theorems-Definitions "The following properties are equivalent (...); if any <=> all hold, the object is called ADJECTIVE".
Good mathematical knowledge isn& #39;t necessarily, or even often, good exposition or good pedagogy. As a student, I benefitted greatly from having ONE definition of group, and a small list of exercises showing what could be shaven off.
But later, for research? The more, the better. In particular, it might be you& #39;re interested in a property you can& #39;t prove, but you can prove all the assumptions of a (non trivially equivalent) definition. The more equivalences you know (or can google), the easier your life.
As a beginner, feel free to tinker with definitions! Build examples, counterexamples (check the answers! there are plenty of cool ideas) and feel free to see if, by changing something a bit, you still get a useful notion... but here I stop, we& #39;ll go on in thread #3
TL;DR while it is often good pedagogy to start with ONE definition, ultimately the collection of all the equivalent ones you know is more important than any specific one.
Definitions aren& #39;t given from above: "The Book" is one we humans write. https://www.goodreads.com/book/show/696238.Proofs_from_THE_BOOK
One">https://www.goodreads.com/book/show... way you recognize a really good definition is because it has many, often not obvious* equivalent forms.
*Thank you @wtgowers for explicitly stating this.
One">https://www.goodreads.com/book/show... way you recognize a really good definition is because it has many, often not obvious* equivalent forms.
*Thank you @wtgowers for explicitly stating this.
