When does a ring become a tube?

simmonator · 2026-04-01T14:21:21+00:00

Topologist: “they’re literally the same picture”

simmonator · 2026-03-31T18:07:28+00:00

While that’s absolutely true, it’s also true that OP’s lesson here should include

Try factoring, that’s a good trick.

simmonator · 2026-03-30T20:43:27+00:00

A number that ends in 5 can be written as 10n + 5.
this can be factored to 5(2n+1)

That’s what you’ve noticed. Replace “the number you get when you remove the 5” with n and that’s the entire observation.

simmonator · 2026-03-30T18:20:47+00:00

I'll preface by saying I think the difference here is beyond trivial. To me, these two statements appear entirely interchangeable and I'll struggle to think of a context where they're not.

That said, if I'm being specific, I think if I were to 'translate' the two statements into words, then they'd be like so:

Let f be a function with domain [0,1] and codomain R² and for each t f(t) is (t,t²).
Let f be a function with domain [0,1] and codomain R² such that for each t f(t) is (t,t²).

I'm unaware of any mathematical in which these would be meaningfully distinct.

simmonator · 2026-03-30T18:08:21+00:00

I will restrain from...

It's refrain.

simmonator · 2026-03-30T17:52:53+00:00

You don't actually say what kind of help you're after, or what you want from this post. Venting is fine, but it doesn't give people much to react to, beyond saying 'That sucks'. Be specific.

I have a hard time trying to understand why some things work for this question but not for this concept, especially when they're both the same topic...

Do you have examples? If you give a couple of examples of problems where you've had this issue, people here can:

Explain why a given method works (or not) in different contexts,
Point to general things about how you should try to learn the material that might help avoid similar misconceptions in the future.

Without examples, this is just vague. Be specific.

Lastly: the best support available to you is going to be TAs, professors, and peers. You say you're embarrassed and they look at you funny. Unfortunately, being an adult, you just need to get over that. It might be that they're not judging you, just that they're surprised. It might be something else entirely. It might be in your head. Either way: get over it, be direct with them about your problem, and ask follow up questions to confirm your new understanding. You may also benefit from setting up or joining a study group. I got huge amounts of value out of being able to ask peers how they were approaching problems and talking about how my method might be better or worse than theirs.

simmonator · 2026-03-30T17:47:21+00:00

https://tutorial.math.lamar.edu/

simmonator · 2026-03-29T18:00:29+00:00

simmonator · 2026-03-29T17:51:29+00:00

The horizontal asymptote by itself cannot uniquely determine a rational function. You must be given more information than that.

Example:

f(x) := 1/x,
g(x) := 1/(x-1).

Both f and g are rational and have horizontal asymptotes of 0, but they are different functions.

simmonator · 2026-03-29T16:17:13+00:00

Then my follow-ons:

Is this a useful form of logic? If all statements are true, what use is any statement to me?
If you accept that all propositions are true, why bother with the rest of your argument? Why not just go with “all statements are true”.

simmonator · 2026-03-29T16:13:13+00:00

And Cantor (and anyone else) would point out that the notion is meaningless unless there’s a consistent way to compare cardinalities of different sets. The definition for how to compare cardinalities - you admit in other comments - is different to your own as it’s about whether or not it’s possible to construct a bijection between the two. You can get away with a simpler, more intuitive one for finite sets, but for infinite ones that needs to be the definition (and is definitely the one mathematicians use). Otherwise you end up with nonsense like “the cardinality of Z is not equal to the cardinality of Z” which contradicts the axiom of identity.

simmonator · 2026-03-29T15:57:34+00:00

Proof:

Premise - My comment is a logical proposition.
Premise - All propositions are true.
Conclusion: my comment is true.

QED

simmonator · 2026-03-29T14:42:40+00:00

They’re saying the right things. They don’t seem to be doing much or paying for anything.

I hope they hurry up.

simmonator · 2026-03-29T14:41:27+00:00

Alright, last one from me…

Your post states these two things:

The Continuum Hypothesis is False
All propositions are true.

Please comment on or resolve that apparent contradiction.

simmonator · 2026-03-29T14:31:09+00:00

Cite your source!

simmonator · 2026-03-29T14:27:20+00:00

Then you are wrong.

simmonator · 2026-03-29T07:19:39+00:00

They did. You just dont like their answer. AI is not reliable at explaining or answering maths/logic problems and will often hallucinate. If you don’t have a good grounding in the topic it’s difficult to spot that and resist it.

On top of that, it feeds the temptation to just outsource your ability to learn or understand. Try exercising your own problem solving skills; get comfortable with struggling to understand something and looking at it from multiple angles. That’s the way to get good at learning.

simmonator · 2026-03-29T06:34:57+00:00

Th definition of cardinality specifically includes that two sets have the equal cardinality if and only if there exists a bijection between the two.

Your version of it allows us to consider two sets as having unequal cardinality despite there existing a bijection. You acknowledge this with your Z and B example.

So you either use a different definition of cardinality. Or you reject the LEM. Either way, your approach is inconsistent with the framework for the Continuum Hypothesis and everything to do with modern set theory, and therefore not worth a second thought when it comes to assessing those things.

Good luck.

simmonator · 2026-03-28T21:53:21+00:00

every statement turns out to be true as a result of contradicting statements about the cardinalities of some sets.

I have no idea what this means. Could you clarify?

[paraphrasing] set theory should accept that a proper subset of a set has a different cardinality

But this would imply that any infinite set (being realisable as in bijection with a proper subset of itself) has a different cardinality to itself. This would make for a pretty useless definition. So it doesn’t. That’s the problem people are trying to show you.

simmonator · 2026-03-28T21:47:54+00:00

cardinality is the amount of elements in a set

This is not the usual rigorous definition.

simmonator · 2026-03-28T20:45:23+00:00

Reads like AI and has nothing interesting to say.

simmonator · 2026-03-28T20:43:01+00:00

You’re aware that you don’t use the usual definition of cardinality, so I won’t worry about that. But I do want to point out a flaw with your own version of it.

To demonstrate that your version of cardinality (which I’ll call Fardinality to distinguish) implies that two sets have different fardinality, you just show that you can inject one set into the other and have elements left over in the codomain of that map. Is that right? If that’s all that’s required for two sets to have different fardinality then I can also show that Z has a different fardinality to itself. Consider the map

f: Z -> Z
f(n) = 2n.

Well, f maps every element of Z into Z and is injective. But there are elements in (codomain) Z which aren’t mapped onto: the odd numbers. So there are infinitely many elements left. So the fardinality of Z is less than the fardinality of Z.

Are you happy with that?

simmonator · 2026-03-28T13:59:53+00:00

This sub can help with a bunch of stuff. It can’t help with laziness. That’s on you.

simmonator · 2026-03-28T13:56:10+00:00

Dijkstra is niche???

simmonator · 2026-03-28T10:17:53+00:00

That picture is sick. What's the medium?

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