@KasmirKhaan, I'm interested in taking a stab at your question, but I'm not sure I understand it.
But recognition of "sameness" is something you can't define except by already knowing it. Same with "different" and "similar." You can define how you shall determine for a particular type of mathematical structure whether "sameness" applies or not, but recognition of differences, similarities and identities is totally fundamental to perception of the universe.
And yes, as noted by others, that's not really a math question. :)
@Wildcard I want to understand more about the algabraic structure of a group, so ofc one starts with understanding when things are equal in any new structure, I wanted to know why the criteria of 1-1 , onto function + f(xy) = f(x) f(y) , those three, what defines equiality in groups
@KasmirKhaan first you should realize that "equality" in math is shorthand for "we're not interested in the differences between these things at this time."
@KasmirKhaan So for groups, when we say they're "equal," it's so that we can reason about certain types of similarities between groups. If they're similar in such and so way, then they have certain common properties.
So you're just asking about definitions at that point.
Q. Let X be an infinite set. Let X be endowed with discrete metric. Give an open cover of X which does not admit a finite subcover. now let x belong in X, then if i take a ball of radius greater than or equal to 1, then i get the whole space, so the radius i take must be less than 1, which gives only the centre. so can i write that for each element in X, consider ball of radius 1/2, and take its union, we will get X. is this correct upto now?
So in algebra what we care about is that multiplication table. If we take a given set and operation which generates that table, or another, it doesn't really matter to an algebraist, right?
Ultimately any algebraic properties will still hold
If you have an isomorphism, that means that when we take the table of the first group, it'll have elements $a_1,a_2,...,a_n$ (take a finite group for simplicity).
@orlp However, I had a very odd feeling while I was computing it. I felt like one of the students I mentioned here:
I just sit down the 8th grader and ask him what's $117 \times 277 - 116 \times 277$. A surprising number will compute it the long way. I don't even point it out to them; they often still don't see it. Then I give them progressively bigger numbers, like $13754 \times 347 - 13654 \times 347$ (and they get annoyed at being asked to do this without a calculator) until they suddenly get it. Then we go from there to trickier problems, like $97 \times 103$ without a calculator, then $498 \times 502$, and so on. — WildcardApr 5 at 0:39
@orlp So I think what I really need is the "leading" questions in number theory to compute out longhand that, when I solve them, will make the patterns involved extremely obvious.
I could see some patterns, but not clearly enough to isolate the basic principles involved.
I expect by starting smaller I'll work out the laws involved, though. :)
@TobiasKildetoft then i cannot find a subcover of this because each ball contains only its center, and we have infinite elements and therefore infinite balls, and therefore i cannot have a finite number of balls covering X. is this correct? how can i write this formally?
@orlp I worked out the totient function and Euler's theorem on my own after learning the basic idea of modular arithmetic.
@orlp Are there more laws and trickery involved than just Euler's theorem?
(N.B.: I want more practice with Euler's theorem to prove out to my own satisfaction and full understanding how and why the converse holds (to prove co-primality) and how it can more easily be used in practice. Part is understanding and part is familiarization; both are needful. It's been more than 5 years since I worked it out and played with it and I haven't used it since.)
There's a difference between understanding and following out a proof, and KNOWING to your core that something is true because it would be impossible for it to be otherwise.
Let E be the tangent plane at the graph $f(x, y) = x^2 + 3xy$ at the point $(1, 1, 4)$. At which points does the plane with cartesian equation $5x^2 + 3y^2 + z^2 = 9$ a unit perpendicular vector that is perpendicular to $E$.
We have that $E:z=f(1,1)+f_x(1,1)(x-1)+f_y(1,1)(y-1) \Rightarrow z=5x+3y-4$, so ge get the plane $E:5x+3y-z=4$, right? A normal vector is a multiple of $(5,3,-1)$, or not? How could we continue?
For example, I (and probably everyone here) KNOWS that there are 120 permutations of 5 items. It would be impossible to convince any one of us otherwise, or even to cast the slightest shadow of doubt in my mind that this is exactly correct. No matter how much authority or respectability or anything else someone had, nor how many people were to say, "No it's not; it's 117."
There can be no flaw discovered in the proof.
That's KNOWING something.
Then there's just having proved something symbolically, which can be useful as a route to knowing it, but if you don't actually proceed to then knowing it, you haven't finished with it and it's a much lower-grade action.
Oh yes... It should be: Let E be the tangent plane at the graph $f(x, y) = x^2 + 3xy$ at the point $(1, 1, 4)$. At which points does the plane with cartesian equation $5x^2 + 3y^2 + z^2 = 9$ have a unit perpendicular vector that is perpendicular to $E$.
@TedShifrin So, because I got sick of my username and changed it once permanently that makes me some evil individual? FYI, I have been offline for the past 24 hours and have not flagged anyone for anything. Even before that, I have been pretty much offline for the past week. So don't be rude and accuse me of something without even consulting a moderator to find out who is actually doing it. That's rude and a moderator made clear last night such posts are not tolerated on here.
If I see you (or anyone else) making such a statement directed towards me without justification I can and certainly will use a custom flag to ask for a moderator or admin to kindly explain why that behavior is not tolerated. You've certainly just lost most of my respect....
apologies for the posts
there was an old message from earlier that was pretty mean and rude
@Typhon while it does not seem like you're the one who did it this time, I don't believe it's particularly serious. It's nothing actionable, and there was one point in time where you and Dodsy had an incident regarding flag abuse, so he was working off some sort of history. Whatever the case, I would recommend not resorting so quickly to belligerence, it won't help you or anyone else.
and I was almost banned last night I think because I happened to mention not to push another user's buttons due to past incidents of anger
so...
him making that accusation needs to at least have something behind it. I did flag a couple posts that were incredibly offensive but only using custom flags and because of the fact that they were truly NSFW posts.
for instance, blatant posts about illegal book sites
What I saw was that this morning there was some nonsense about people being flagged for using the term "normie", and after Dodsy
What I saw was that this morning there was some nonsense about people being flagged for using the term "normie", and after Dodsy said you were not responsible Ted disagreed
To be honest the illegal book sites are not in good taste but not serious
Like, no one is getting hurt or personally attacked
@Daminark true but that was only one time and I flagged using a custom reason. Nobody was banned. Just a delete and verbal warning. In fact, I think the flag came back as "please don't flag unless serious, bla bla bla"
@Daminark I took ted's comment as a personal attack.
@Jasper I'm perfectly calm dude. Just leaving a response telling them to please not do that again. I have no problem with someone having an issue with me, but don't talk about it behind my back or make unsubstantiated remarks.
I was honestly not planning to dwell on it. I said my peace and it's done.
You took it as such but not justifiably, he said that he suspected it was you on basis of his past information and the fact that you have gone by another name. I don't want to get into any political stuff but I just think this is honestly something that doesn't warrant such a response, just let it slide
@Daminark well, mods have said in other chats and deleted posts because of it that no form of accusation like that is allowed on the site. Hence such posts are to be taken as a violation of the be nice policy. As I said though, I am letting it slide with a verbal warning that such posts will not be tolerated in the future.
@Typhon Policies are made by imperfect humans and then interpreted by other imperfect humans. Do what your conscience tells you is right, and you will find happiness in life. =D
@Jasper if you say so. It sounded a bit deep to me. I was just giving the guy a friendly reminder so that way nothing happens in the future. We don't need drama or consciences in a math chat.
"Do what your conscience tells you is right." If I did that I'd be leaving here for good. That's not happening so don't try that on me, lol.
@Typhon I think whatever Ted said seemed pretty harmless to me. I guess you can always clarify things in this room, which is what you just did. So I guess everything's OK now.
@Jasper Everything always was ok. The statement "this was between me and ted" applies really good in this situation. I don't need nor want people commenting on how I should reply to a post that is nobodies business but mine. It's honestly flat out annoying when people here do that. That's why I apologized. It wasn't meant for anyone else to read but Ted. Period. So please drop it.
it only became not ok when daminark took offense to my reply.
@Faust I'm not familiar with graph theory in the math sense but I've used graphs quite a few times in programming stuff. Maybe I can help with the terminology issues?
There exists a set $ X \subset V (G) - \{u,v\} $ such that some vertex in X is adjacent to both u and v. i dont really get this are they claiming that the vertex adjacent to u is still adjacent in X that seems impossible if you removed it...
@Jasper nah. not really. Just annoying how people always do that specifically because there is no private chat where I can tell someone something without half the site commenting on it or butting in. XD
