4 ms·
Even in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will
by dxbydt 10y ago
Even in commutative algebra, there is a fair amount of guesswork. If I tell you that 2+3=4 in an abelian group with {2,3,4}, then ask you what is 2+4, you will immediately guess 2+4=2. Its just what the millennials term "stupid obvious". Yeah you can do a lengthy proof on why 2+4=2. Proof: Its 2 because 4 must be the additive identity, and 4 is the additive identity because 2 & 3 are not, and they aren't because if 2 was then 2+3=3, and if 3 was then 2+3=2, but because I've told you 2+3=4, by closure 4 must be identity, which implies 2+4=2. That's the whole story. Literally no mathematician I know will sit down & write that lengthy proof I wrote. They'll just tell you 2+4=2, and if you pester them with "But why?" they'll say "Because" and excuse themselves :)
- yequalsx 10y agoIn an elementary group theory course this is the type of problem that would be given when groups are first introduced. It's a good homework/test question and just writing a*c = a as the answer would not suffice (depending on the level of the course and aim of the problem). The point at that level is for the student to learn how to justify their beliefs. This is especially so if one were in a basic mathematical logic course. Of course, in an algebraic topology course where this group showed up it would be assumed that everyone knows how to find the answer and why. No justification would be needed.