/sci/ - Science & Math

>>10275715
Yes, that proof is correct.
You don't really have to go by contradiction though. For any y in B, let x in A such that f(x) = y. Then g(y) = g(f(y)) = g'(f(y)) = g'(x).
Thus g = g' on all of B.
It's best practice to try to pick the most simple, direct proof you can. Sometimes contradiction will be simpler. But it's always useful to see how to prove something in different looking ways.

Looks good to me, I'm learning them too right now

>>10275732
Thank you! And yes I always try direct proofs first but this just clicked in my head when going for contradiction.

>>10275732
I'm glad to see that the first post encourages a direct proof. In this case, the direct proof makes it very clear why surjectivity of f is the critical condition in the lemma.

>>10275715
Phi is almost exclusively used to denote maps, not elements of a set, so avoid that. I think using the lowercase of the set name is clearer, eg. "a in A".

>>10275751
Good point. I didn't even think to mention notation. There are ways it can be "standardized" here.
I'd choose b or y in B, a or x in A. We typically either use the lowercase letter of the set, or x and y when there are a clear domain and codomain (and typically z for a 3rd set chained on, so z might have been in C if we needed it).
Otherwise, good choices, formatting, and such. Eventually you'll want to be peppering in some \forall and \exists just to reduce some word bloat, but some people don't even do that. Don't worry about it til you get sick of typing a lot.

>assuming LEM

Lots of good advice in this thread. Just seconding the point about using a direct proof wherever you can. It's good to try a direct proof first, and if it's too difficult or complicated, then try contradiction or contrapositive. It's common for students to fall in love with proof by contradiction and try to use it all the time.

>>10275779
Agreed, when I took real analysis I often found that when I truly understood a proof by contradiction, I would be able to rewrite it as a direct proof.

>>10275757
Yeah same here on the x,y,z in A,B,C, I was just too lazy to elaborate on it.
What a wholesome thread.

>>10275787
The most fun I had in early proof classes was figuring out a meh way to do a problem and then looking back and seeing a way to massively improve or clarify my proof. That happened all the time in real analysis, especially with counterexamples. So satisfying.

>>10275790
I know that feel, there's something both infuriating and extremely satisfying about those pathological counter-examples in analysis. I'm taking analysis II with Pugh next semester at Berkeley, hopefully it'll be cool. Apparently his analysis textbook is real good.

>>10275804
what the fuck is going on
i feel like i'm talking to 2 fucking people on /sci/, tuba jam and you
i'm the one in the uni thread who had 202a last semester and said 105 is a great class
lmao

>>10275808
Top kek yeah that with me in the uni thread, late night /sci/

>>10275808
So what year are you? Are you planning on applying to any graduate programs?

>>10275787
>I often found that when I truly understood a proof by contradiction, I would be able to rewrite it as a direct proof
This happens to me a lot too in number theory and algebra. Doing a proof in a couple different ways really helps me understand why something is true.

>>10275822
yes, i am. the more information i give in a thread the more likely i dox myself (ie cutting down the group in 202a last semester by a third by saying my year) so i'd rather not. sorry.
i'll leave it at, based on my /sci/ posting history after the last 6 months chances are relatively high that we already know of each other in some capacity.

>>10275840
Fair enough, cntrl F Berkeley if you want some late night chuckles
>>10222177

>>10275849
Piece of shit link didn't work
4chan . org/sci/thread/10222177

>>10275849
I remember that thread, I decided not to post anything cause I figured I'd be shat on by jealous Ivy trustfund brainlets. Good to know that didn't happen.
We should be 1, obviously.

>>10275808

another cal student reporting in, sophomore baby taking math 113 spring semester. kind of amazed at how much ppl here go on sci though.

>>10275882
Pro-tip, Paulin teaches 113 in the summers and he's the shit. I'd also recommend spending at least 1 summer at Berkeley, it's quite nice and the summer sessions are very relaxed, particularly if you have a good teacher.
Other than that, for the love of god don't take Math 128A with Strain, and the applied math major is a meme.

>>10275715
The proof is fine, I'd just like to make some suggestions.

1. Mixing English and math notation is generally frowned upon. Instead, write out only with logic or only English.

2. Even though it's implied by your explanation, consider writing out your assumptions explicitly.

It would be like this:
Assume the functions exist
Assume g o f. Assume g_not o f. Assume f is surjective.

Now you can write the proof more compactly. Again, the proof is valid but I would not have gotten full credit for it in my upper division classes.

>>10276073
>Mixing English and math notation is generally frowned upon. Instead, write out only with logic or only English.
Retard

>>10276073
>assume the functions exist
>>10276075
No, that's true.

OP here. Thank you all for your advices! I'm going to take them. My style and formatting is still a mess lol.

>>10276073
Upper class as in Real Analysis?

>>10275715
Look up the left and right inversion theorem. If you know that, then it is obvious.

>>10275715
Assuming the axiom of choice, [math] f [/math] is a split epimorphism in [math] \textbf{Set} [/math], so the result follows trivially.

>>10275882
113 is a great class, and this is coming from someone who despises algebra
lovejoy is also really good, and a couple of the other 113 profs next semester. you pretty much can't go wrong as long as you're not taking wodzicki or givental.
it's pretty wild, but at least we're not as bad as Colorado Mines.

>>10276073
>mixing english and math is frowned upon
except this is exactly the OPPOSITE of what is true you moron
>consider writing out your assumptions explicitly
holy shit how fucking dumb are you
i hope you're not being serious that you'd mark ANYONE off for a valid and readable proof in an upper division class. people who do that are fucking awful. and VERY VERY few do. you're punishing students for minutia instead of rewarding the thought process, when exams are ALL about thought process. if your exams are so easy that you can grade like this and still have a reasonable grading distribution, you need to severely change your exam design.

>>10276079
>no, that's true
no, it's not. on an exam or on homework, everyone prefers that you just write things in the most clear way possible. i.e. not literally just logical symbols, nor complete english writing. of course one should always \forall and \exists, but sacrificing english like "such that", "thus", or "suppose" makes your work unreadable
>le epic triangle dots therefore
literally the worst garbage

>>10277473
Do you actually just throw in logical symbols in the middle of text?

>>10276075
I'd go even as far as to say write as much English as possible, only using symbols for e.g. set theoretic and functional notations. The actual equations should go in display math mode. The mixing of logic notation or English is common in handwritten stuff (e.g. exams or quick notes) to save your hand some work, but in LaTeX it makes no sense since you're probably spending more time and effort in writing out (and then reading) logic rather than plain paragraphs.

>>10277503
no, i do that during exams and homeworks. that dude is saying he marks people off for dumb shit like that on exams.
this >>10277532 is what i do, in general

>>10277532
This is the convention I learned as well

>>10276073
>only logic or only English
Please do not try to write a proof using only logic and math symbols. As someone else pointed out, you need words like "then", "suppose", "such that", etc to make the proof readable and to contextualize the math you're doing. It's all for the sake of clarity and usually the clearest thing to do is to use English and avoid logic symbols.

>Even though it's implied by your explanation, consider writing out your assumptions explicitly.
I agree with this advice, however I think the "suppose for the sake of contradiction" makes it obvious that OP is assuming the hypotheses and the negation of the conclusion. If he were doing something less direct, such as proving the contrapositive by contradiction or something, then that needs to be made clear. I do mark students all the time for not writing what they're assuming at all because they often mangle the rest of the proof and it's difficult to discern what they are trying to do, but I think OP was okay here.